deformed to the smooth deformations $X_{t}$ of $X$ follows from standard facts
of analytic geometry: the Fano surfaces $F_{t}=F_{1}(X_{t})\subset
G_{1}(\mathbf{P}^{4})$ for $t$ sufficiently close to 0, organize themselves
into an analytic family of surfaces $\pi:\mathcal{F}\rightarrow(\mathbf{C},0)$
(cf. [AK] or [CGHL, §3]). The direct image by $\pi$ of the relative dualizing
sheaf $\omega^{2}_{\mathcal{F}/(\mathbf{C},0)}$ has fiber
$\boldsymbol{H}^{0}(F_{t},\omega^{2}_{F_{t}})$ at any $t$. Since all these
fibers are 10-dimensional (according to (33)), it follows that
$\pi_{*}\big{(}\omega^{2}_{\mathcal{F}/(\mathbf{C},0)}\big{)}$ is locally-
free. Thus any $\eta\in\boldsymbol{H}^{0}(F_{0},\omega^{2}_{F_{0}})$ can be
extended into an analytic section
$t\rightarrow\eta_{t}\in\boldsymbol{H}^{0}(F_{t},\omega^{2}_{F_{t}})$. As
explained just above, this is sufficient to ensure that the following map is
well-defined:

(34) ${\rm
Tr}:\boldsymbol{H}^{0}(F,\omega^{2}_{F})\longrightarrow\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{L}\mathcal{W}}_{X,x}\big{)}\,,\quad\eta\longmapsto\big{(}\ell_{i}^{*}(\eta)\big{)}_{i=1}^{6}\,.$

There is a last subtlety to deal with which concerns the injectivity of this
map, which is not completely obvious. Let us briefly discuss this matter. For
$i=1,\ldots,6$, let $F^{\prime}_{i}$ be the irreducible component of $F$ which
contains ${\rm Im}(\ell_{i})$. Some of the $F^{\prime}_{i}$'s may coincide,
hence these components can be univocally labeled $F_{j}$ for $j$ in some set
$J$ of cardinality at most 6. Let $F^{gen}$ be the union $\cup_{j\in J}F_{j}$
and denote by ${\widetilde{F}}^{gen}$ its image in $\mathbf{P}^{9}$ by the
Plücker embedding $G_{1}(\mathbf{P}^{4})\subset\mathbf{P}^{9}$ (the notation
with the upper gen comes from the fact that $F^{gen}$ can equivalently be
defined as the union of the irreducible components of $F$ whose generic
element meets the generic point of $X$).

The following statements are easily seen to be equivalent:

(35) $\bullet$ the map (34) is injective;
---
$\bullet$ the restriction map
$\boldsymbol{H}^{0}(F,\omega_{F}^{2})\rightarrow\,\oplus_{j\in
J}\Omega^{2}_{\mathbf{C}(F_{j})}$,
$\omega\mapsto\big{(}\omega\lvert_{F_{j}}\big{)}_{j\in J}$ is injective;
$\bullet$ the surface $\widetilde{F}^{gen}$ is non degenerate in
$\mathbf{P}^{9}$, i.e.
$\big{\langle}\widetilde{F}^{gen}\big{\rangle}=\mathbf{P}^{9}$.

That these equivalent properties hold true is immediate for instance when $F$
is irreducible, but needs to be justified for an arbitrary cubic with finitely
many singular points. This is the content of the lemma just below. The
proposition is fully proved. ∎

######  Lemma 3.4.

For $X$ as above (that is, $X$ is an irreducible cubic threefold with finitely
many singular points and such that the lines it contains form a genuine 6-web
whose leaves are generically in general position), then $\widetilde{F}^{gen}$
is non degenerate in $\mathbf{P}^{9}$.

###### Proof.

The proof is easy and goes by reduction ad absurdum. Recall that the Plücker
embedding $P:G_{1}(\mathbf{P}^{4})\subset\mathbf{P}^{9}$ has for coordinates
the Plücker coordinates $\Delta_{ij}$ for $i,j$ such that $1\leq i<j\leq 5$:
if $L$ is the line passing through $[p]$ and $[q]$ for two distinct points
$p,q\in\mathbf{C}^{5}$ with coordinates $p_{i}$ and $q_{i}$ respectively, then
by definition $\Delta_{ij}(L)=\Delta_{ij}(p,q)=p_{i}q_{j}-p_{j}q_{i}$ for any
$i,j$ as above and the point $\big{[}\Delta_{ij}(L)\big{]}_{1\leq i<j\leq
5}\in\mathbf{P}^{9}$ is well-defined (that is depends only on $L$) and
corresponds to $P(L)$.

That ${\widetilde{F}}^{gen}=P(F^{gen})\subset\mathbf{P}^{9}$ is degenerate
means that there exist scalars defining a hyperplane for $[p]\in X$ generic
and any $[q]$ such that the line $L_{p,q}=\langle[p],[q]\rangle$ joining $[p]$
to $[q]$ is contained in $X$, then the following relation is satisfied

(36) $\Big{\langle}H,P(L_{p,q})\Big{\rangle}=\sum_{1\leq i<j\leq
5}h_{ij}\,\Delta_{ij}(p,q)=0.$

Let $F$ be a cubic homogeneous equation in five variables such that $F=0$ be
an equation of the affine cone over $X$ in $\mathbf{C}^{5}$ and denote by
$F(\cdot,\cdot,\cdot)$ the associated symmetric trilinearization. For $[p]\in
X$ generic, the $q$'s such that the line $L_{p,q}$ is included in $X$ are
those satisfying the following set of homogeneous equations (of degree 1,2 and
3 in q respectively):

$F(p,p,q)=2dF_{p}(q)=0\,,\qquad F(p,q,q)=0\quad\mbox{ and }\quad
F(q,q,q)=6F(q)=0\,.$

Assuming that (36) is satisfied as soon as these preceding equations hold true
leads to two possibilities, depending whether the two $q$-linear equations
$dF_{p}(q)=0$ and (36) are colinear or not. If it is the case, then there
exists $\Lambda(p)\in\mathbf{C}^{*}$ such that the equality between linear
forms
$\Lambda(p)\cdot\big{\langle}H,P(L_{p,\cdot})\big{\rangle}=dF_{p}(\cdot)$
holds true. Since this identity holds for $[p]$ generic in $X$ which is non-
degenerate in $\mathbf{P}^{4}$ and because $dF_{p}(\cdot)$ is homogeneous of
degree 2 with respect to $p$, it follows that the $\Lambda(p)$'s can be taken
to be such that $p\mapsto\Lambda(p)$ be a linear form. But then this would
imply that $dF=0$ on the hyperplane section $X\cap\\{\Lambda=0\\}$,
contradicting the assumption that $X$ has isolated singularities.

Thus necessarily (36) cuts a genuine line in the projectivized tangent space
$\mathbf{P}(T_{p}X)$ which obviously contains all the tangent directions at p
of the lines $\ell_{i}(p)$'s. But this would imply in particular that these
six tangent directions span a 2-plane in $T_{p}X$, contradicting the general
position assumption made about the leaves of
$\boldsymbol{\mathcal{L}\mathcal{W}}_{X}$ at a generic point. The lemma
follows. ∎

######  Remark 3.5.

The proof above actually shows that the conclusion of Lemma 3.4 is still
satisfied under the weaker assumptions that $\dim\big{(}X_{sing}\big{)}\leq
1$, $\dim\big{(}F_{1}(X)\big{)}=2$ and that among the lines contained in $X$
through a generic point, at least three are in general position. For instance,
the chordal cubic $\mathscr{C}$ satisfies these hypotheses (but in this case
$F^{gen}$ coincides with $F_{1}(\mathscr{C})$ and it was known that its
Plücker image in $\mathbf{P}^{9}$ is non degenerate according to the
discussion in §3.2.1).

##### On Abel's theorem for abelian 1-forms on the Fano surface.

We believe that some meaningful parts of Proposition 3.1 generalize in a
straightforward way to any cubic threefold $X$ with isolated singular points,
at least as soon as the lines included in it define a genuine 6-web on
it.272727Note that for $X$ a smooth cubic with isolated singularities, it may
happen that $\boldsymbol{\mathcal{L}\mathcal{W}}_{X}$ be a genuine 6-web but
not a skew one (this precisely happens for Segre’s cubic $\boldsymbol{S}$ for
instance). In such a situation, the space of 1-ARs of
$\boldsymbol{\mathcal{L}\mathcal{W}}_{X}$ may be of infinite dimension hence
one cannot expect the whole Proposition 3.1 to be generalized as it is to the
case of such cubic threefolds. However, as is already the case for smooth
cubics (cf. [ClG, p. 338]), everything that concerns the links between some
differential $k$-forms on the Fano surface $F=F_{1}(X)$ and the $k$-ARs of
$\boldsymbol{\mathcal{L}\mathcal{W}}_{X}$ is much more subtle when $k=1$ than
when $k=2$. We discuss below several approaches which could give a proof of
the extension for cubics with isolated singularities of the fact that the
first trace map may induce an isomorphism between a space of 1-differential
forms on the associated Fano surface and the space of 1-ARs of the
corresponding web of lines on the cubic. We warn the reader that nothing is
proved below, rigorous investigations on this are left for a hypothetical
future work.

In what follows, $X$ stands for a fixed cubic threefold in $\mathbf{P}^{4}$
with isolated singularities.

$\star$

As a first attempt, one might naively want to extend in a straightforward
manner the approach discussed in the preceding paragraph to the case when $X$
has isolated singularities: considering a smoothing
$\pi:\mathcal{X}\rightarrow B$ of $X=X_{0}=\pi^{-1}(b_{0})$ over a smooth
1-dimensional base $B$, one defines a coherent sheaf of `relative abelian
1-forms' $\omega^{1}_{\mathcal{F}/B}$ on the total space of the associated
family of Fano surfaces $\pi:\mathcal{F}\rightarrow B$ by requiring that the
following sequence of sheaves is exact
$0\rightarrow\pi^{*}(\Omega_{B}^{1})\rightarrow\omega_{\mathcal{F}}^{1}\rightarrow\omega^{1}_{\mathcal{F}/B}\rightarrow
0$.282828Here $\omega_{\mathcal{F}}^{1}$ stands for the sheaf of ‘Abelian (or
Barlet [Bar]) differential 1-forms on $\mathcal{F}$’, which can be
characterized as the meromorphic 1-forms on (analytic open subsets of)
$\mathcal{F}$ inducing $\overline{\partial}$-closed currents on their
definition domains. Then setting $F_{0}=F_{1}(X_{0})=F_{1}(X)$, the same
arguments as in the preceding paragraph would give that the trace induces an
injective linear map
$\boldsymbol{H}^{0}(F_{0},\omega_{F_{0}}^{1})\rightarrow\boldsymbol{AR}^{(1)}\big{(}\boldsymbol{\mathcal{L}\mathcal{W}}_{X}\big{)}$,
provided that the two following facts hold true:

$(i).\,\mbox{ one has
}\,\big{(}\omega^{1}_{\mathcal{F}/B}\big{)}\big{\lvert}_{F_{0}}=\omega^{1}_{F_{0}}\,;\qquad\mbox{
and }\qquad(ii).\hskip
11.38092pth^{0}\big{(}F_{0},\omega^{1}_{F_{0}}\big{)}=5\,.$

If one expects that these two facts are indeed satisfied, verifying it is not
obvious. Abelian differentials are not known to behave particularly well under
pull-back hence $(i).$ needs to be justified. As for $(ii).$, it leads to the
following interesting questions: according to [AK, (1.10).(ii)], denoting by
$F_{0}^{*}$ the smooth locus of $F_{0}$, one has
$\Omega^{1}_{F_{0}^{*}}\simeq\big{(}\mathcal{T}^{\vee}\big{)}\lvert_{F_{0}^{*}}$
(compare with (28) in the smooth case). Does this identification extend
through the singularities of $F_{0}$ into an isomorphism
$\omega^{1}_{F_{0}}\simeq\big{(}\mathcal{T}^{\vee}\big{)}\lvert_{F_{0}}$? And
if so, is the restriction map
$V^{\vee}\simeq\boldsymbol{H}^{0}\big{(}G_{1}(\mathbf{P}(V)),\mathcal{T}^{\vee}\big{)}\rightarrow\boldsymbol{H}^{0}(F_{0},\omega^{1}_{F_{0}})$
injective or even an isomorphism? It does not seem easy to answer these
questions, a reason for that being that $F_{0}$ has rather involved
singularities (in particular, its singular locus $F_{0}\setminus F_{0}^{*}$ is
1-dimensional). It is known that $h^{1}(\mathcal{O}_{F_{0}})=5$ (cf. [AK,
Prop. (1.15)]) but one cannot deduce directly from this that
$h^{0}(\omega^{1}_{F_{0}})=5$ since it is not clear if/how the pure Hodge
structures on the first cohomology spaces
$\boldsymbol{H}^{1}(F_{b},\mathbf{C})=\boldsymbol{H}^{1,0}(F_{b})\oplus\boldsymbol{H}^{0,1}(F_{b})$
for $b\in B$ with $F_{b}$ smooth degenerate when $b$ tends to $b_{0}$.

It seems to us that one might be able to deduce from the (rather technical)
Hodge-theoretic considerations in [Col1] that everything discussed above
indeed holds true when considering the chordal cubic $\mathscr{C}$ and its non
irreducible hence singular Fano surface $F_{1}(\mathscr{C})$ (this
independently of the fact that the lines in $\mathscr{C}$ do not define a
6-web in it, but only a 3-web). This makes us confident that everything go as
sketched above in the simpler case of cubics with only finitely many singular
points.

$\star$

Constructing the 1-ARs of $\boldsymbol{\mathcal{L}\mathcal{W}}_{X}$ by means
of the abelian 1-forms of the Fano surface $F$ is certainly interesting
conceptually, but not really necessary if mainly interested in the 1-rank of
this web. Considering a smoothing $\pi:\mathcal{X}\rightarrow B$ as above, it
would already be satisfying to be able to describe an analytic family of
spaces $\boldsymbol{H}^{1}_{b}\subset\Omega_{\mathbf{C}(F_{b})}^{1}$ of
rational 1-forms on $F_{b}=F_{1}(X_{b})$ with $X_{b}=\pi^{-1}(b)$ for any
$b\in B$, with
$\boldsymbol{H}^{1}_{b}\simeq\boldsymbol{H}^{0}(F_{b},\Omega_{F_{b}}^{1})$ for
any $b$ such that $X_{b}$ is smooth and such that the trace induces an
injective map
$\boldsymbol{H}^{1}_{b}\rightarrow\boldsymbol{AR}^{(1)}\big{(}\boldsymbol{\mathcal{L}\mathcal{W}}_{X_{b}}\big{)}$
for every $b\in B$, in particular for $b=b_{0}$. We see several ways to
construct such an analytic family of vector spaces
$\\{\boldsymbol{H}^{1}_{b}\\}_{b\in B}$:

  * •

If $X_{b}$ is smooth, the map
$V^{\vee}\rightarrow\boldsymbol{H}^{0}(F_{b},\Omega_{F_{b}}^{1})$ of v page 27
above can be made rather explicit (see around (12.8) in [ClG]). If by chance
the construction by means of residues might be extended to $X_{0}$, it would
possibly give a map
$V^{\vee}\rightarrow\boldsymbol{H}^{0}\big{(}F_{0}^{*},\Omega^{1}_{F_{0}^{*}}\big{)}$
whose elements in the image extend as rational 1-forms on $F_{0}$ and deform
into holomorphic 1-forms on the smooth nearby fibers $X_{b}$ for
$b\in(B,b_{0})$. Denoting by
$\boldsymbol{H}^{1}_{b_{0}}\subset\Omega_{\mathbf{C}(F_{0})}^{1}$ their span,
we would thus obtain a linear map
$\boldsymbol{H}^{1}_{b_{0}}\rightarrow\boldsymbol{AR}^{(1)}\big{(}\boldsymbol{\mathcal{L}\mathcal{W}}_{X}\big{)}$
whose image is expected to be 5-dimensional.

  * •

Instead of trying to deal with 1-forms on the singular Fano surface $F_{0}$,
one can rather consider working with the global holomorphic forms on a semi-
abelian (hence smooth) variety naturally associated with it. Indeed, 1. for
$b\in B$ with $X_{b}$ smooth, the Albanese map ${\bf alb}:F_{b}\rightarrow{\rm
Alb}(F_{b})$ induces an isomorphism ${\bf
alb}^{*}:\boldsymbol{H}^{0}\big{(}\Omega^{1}_{{\rm
Alb}(F_{b})}\big{)}\rightarrow\boldsymbol{H}^{0}(F_{b},\Omega_{F_{b}}^{1})$;
2. an Albanese variety with an associated Albanese map has been constructed by
several authors for cubics with singularities and this construction has been
proven to behave well for suitable degenerating families (see e.g. [Gw1] for
cubic with ordinary nodes only, or [CGHL] and the reference therein for more
involved isolated singularities).292929Actually, in most papers dealing with
singular cubics, what is constructed is a ‘generalized intermediate Jacobian’
$J(X_{0})$ associated to a suitable singular cubic threefold $X_{0}$, which is
a semi-abelian variety of dimension 5 (actually, the semi-abelian intermediate
Jacobian $J(X_{0})$ is constructed by relating it to a Prym variety of a non
ramified 2-1 cover of a plane quintic curve associated (in a non canonical
way) to $X_{0}$, see [CGHL, §3.2]). For a smooth cubic $X$, there is a well-
understood isomorphism $J(X)\simeq{\rm Alb}(F_{1}(X))$ and it is expected (and
proved in many cases) that this extends to singular cubics, or at least to
some of them. The fact that most of these constructions for singular cubics
are in terms of ‘intermediate Jacobians’ and not of ‘Albanese varieties’
actually is not important for our purpose.. In the case under scrutiny,
$\\{{\rm Alb}(F_{1}(X_{b}))\\}_{b\in B}$ would be an analytic family of
5-dimensional semi-abelian varieties from which it should be possible to
construct an analytic flat family of spaces $\boldsymbol{H}^{1}_{b}$ with the
required properties. We believe that such an approach should work for cubics
with ordinary nodes [Gw1] or for those with `allowable singularities'
according to the terminology used in [CGHL].

  * •

In the case when $X_{0}$ has isolated singularities of the simplest type,
namely ordinary nodes, an approach might be to find a nice desingularization
$\tilde{\pi}:\widetilde{\mathcal{F}}\rightarrow B$ of the relative Fano
surface $\pi:\mathcal{F}\rightarrow B$ such that the central fiber
$\tilde{F}_{0}=\tilde{\pi}^{-1}(0)$ be a reduced surface with normal crossing.
In such a situation, the $(1,0)$-component of the Hodge structure on
$\boldsymbol{H}^{1}(\tilde{F}_{0},\mathbf{C})$ obtained as the limit of the
standard one of the smooth Fano surfaces $\tilde{F}_{b}$ for $b\in(B,b_{0})$
coincides with the space of global sections of a certain sheaf
$\Lambda_{\tilde{F}_{0}}^{1}$ of meromorphic 1-forms on $\tilde{F}_{0}$ (see
[Fr, §3]). The arguments sketched at the beginning of this paragraph (cf. our
`first attempt' above), if applied not to $\mathcal{F}$ but to the
desingularized family $\tilde{\pi}:\widetilde{\mathcal{F}}\rightarrow B$,
should provide the sought-after injective morphism
$\boldsymbol{H}^{0}(\tilde{F}_{0},\Lambda_{\tilde{F}_{0}}^{1}\big{)}\rightarrow\boldsymbol{AR}^{(1)}\big{(}\boldsymbol{\mathcal{L}\mathcal{W}}_{X}\big{)}$.

In [Col2], Collino describes quite explicitly the desingularization
$\tilde{\pi}:\widetilde{\mathcal{F}}\rightarrow B$ when the initial non smooth
$X_{0}$ is Segre's cubic primal and the limit Hodge structure on
$\tilde{F}_{0}$ (see Appendix B at the end of this text for more details in
the case of Segre's cubic). We believe that most of the results in this paper
may be adapted to follow the strategy sketched just above when $X_{0}$ is any
cubic threefold with ordinary nodes.

$\star$

Finally, as a last possible way to construct the 1-ARs of
$\boldsymbol{\mathcal{L}\mathcal{W}}_{X}$, we would like to mention the old
and not well-known but very interesting paper [BW]. In it (in §6 more
precisely), Blaschke and Walberer construct quite directly from any cubic
homogeneous form defining a given cubic threefold $X\subset\mathbf{P}^{4}$,
the 1-ARs not of the web by lines on $X$ but of an associated 3-web, namely
the curvilinear 3-web $\boldsymbol{\mathcal{T}\mathcal{W}}_{X}$ naturally
defined on the variety $\boldsymbol{\mathcal{T}}_{X}$ of `triangles included
in $X$'.303030We will define this variety and discuss the 3-web naturally
defined on it in the case of the chordal cubic $\mathscr{C}$ in Appendix B to
which we refer the reader for more details. It would be interesting to check
whether their approach can be adapted to build by hand an explicit family of
dimension 5 of 1-RAs, but for $\boldsymbol{\mathcal{L}\mathcal{W}}_{X}$ this
time. Note that in [BW], essentially no assumption about the singularities of
$X$ are made; it seems to be only required that
$\boldsymbol{\mathcal{T}\mathcal{W}}_{X}$ is a skew curvilinear 3-web whose
leaves are in general position generically. It would be interesting to revisit
[BW] with a modern and rigorous approach, for instance to determine precisely
under which hypothesis the results contained in it hold true.

###  3.3. Segre's cubic (case $n=3$).

We now focus on the case of Segre's cubic. Since it is a very well-known and
studied threefold, no proof of its properties stated below will be given, the
interested reader will refer to the specific references indicated in the
following subsection.

####  3.3.1. References.

Segre's cubic has been discussed in many papers since its discovery by Segre
in 1887. As classical references, in addition to Segre's original paper [Se]
(§24 to §27 therein), one can mention §19 to §31 in `Capitolo $8^{\rm o}$' of
Bertini's book [B], [Sn], [SR, Chap. VIII, §2.32]. As for other recent
references, the following ones are interesting considering our purpose but
there may exist many others we are not aware of: Segre's cubic primal is
discussed in several papers by Dolgachev, such as [Do1] (in particular §2 and
§4 therein) or [Do2, §5]. Other modern references are: [Hu, §3.2], [Gw1, Gw2],
[Col1], [Col2] and [DFL, §2]. Among the previous references, [Col2] is perhaps
the most relevant one since Collino describes there the Fano surface of lines
of Segre's cubic quite explicitly and studies some of its features. It is
unfortunate that such an interesting text has not be published in proper form.

####  3.3.2. Segre's cubic

We recall well-known classical facts about Segre's cubic.

First, we introduce some notation: Let $[U_{1},U_{2},U_{3},U_{4}]$ be some
fixed homogeneous coordinates on $\mathbf{P}^{3}$ and $x,y,z$ stand for the
affine coordinates on $\mathbf{C}^{3}=\mathbf{P}^{3}\setminus\\{U_{4}=0\\}$
corresponding to the embedding $(x,y,z)\mapsto[x:y:z:1]$. As above, the
$p_{i}$'s (for $i=1,\ldots,5$) stand for the following points in
$\mathbf{P}^{3}$: $p_{i}=[\delta^{i}_{j}]_{j=1}^{4}$ for $i=1,\ldots,4$ and
$p_{5}=[1,\ldots,1]$. We set $P=\\{p_{i}\\}_{i=1}^{5}$.

In what follows, whitout any supplementary precision, $i,j,k,l,m$ will stand
for pairwise distinct elements of $\\{1,\ldots,5\\}$ (hence such that
$\\{i,j,k,l,m\\}=\\{1,\ldots,5\\}$). For such indices, we denote by

  * •

$L_{ij}$ the line passing through $p_{i}$ and $p_{j}$: i.e.
$L_{ij}=\big{\langle}p_{i},p_{j}\big{\rangle}\subset\mathbf{P}^{3}$;

  * •

$P_{ijk}$ or $P^{lm}$ the 2-plane containing $p_{i},p_{j}$ and $p_{k}$: i.e.
$P_{ijk}=P^{lm}=\big{\langle}p_{i},p_{j},p_{k}\big{\rangle}\subset\mathbf{P}^{3}$.

Recall (cf. §2.2.1 above) that $\boldsymbol{\mathcal{L}}_{3}$ is the linear
system of quadric surfaces in $\mathbf{P}^{3}$ passing through all the
$p_{i}$'s: $\boldsymbol{\mathcal{L}}_{3}=\lvert 2H-\sum_{i=1}^{5}p_{i}\lvert$.
Its elements are exactly the surfaces in $\mathbf{P}^{3}$ cut out by
homogeneous equations $\sum_{i<j}a_{ij}U_{i}U_{j}=0$ for some coefficients
$a_{ij}$'s whose total sum is zero.

In the coordinates $x,y,z$, the rational map associated to
$\boldsymbol{\mathcal{L}}_{3}$ can be taken to be

(37)
$\varphi_{3}:(x,y,z)\dashrightarrow\big{[}\,x(z-y)\,:\,x(1-y)\,:\,y(z-x)\,:\,y(1-x)\,:\,z-xy\,\big{]}\,.$

Its base-points in $\mathbf{C}^{3}$ are origin $\boldsymbol{0}=(0,0,0)$ and
the point $\boldsymbol{1}=(1,1,1)$ which correspond to $p_{4}$ and $p_{5}$
respectively. We will again denote by $\varphi_{3}$ the extension to
$\mathbf{P}^{3}$ of the above map (as a rational map). By definition, the
variety $V_{3}$ is the closure of $\varphi_{3}\big{(}\mathbf{P}^{3}\setminus
P\big{)}$ in $\mathbf{P}^{4}$. It can be verified that it is the cubic
hypersurface cut out by the following equation:

$X_{1}X_{2}X_{3}-X_{1}X_{2}X_{4}-X_{1}X_{3}X_{4}+X_{1}X_{4}X_{5}+X_{2}X_{3}X_{4}-X_{2}X_{3}X_{5}=0\,.$

For any distinct $p_{i}$ and $p_{j}$, the line $L_{ij}=\langle
p_{i},p_{j}\rangle$ is contracted onto a singular point of $V_{3}$, which will
be denoted by $p_{ij}$. The $p_{ij}$'s are easy to determine: they are the six
vertices of the canonical simplex in $\mathbf{P}^{4}$ (namely the points
$\big{[}\delta_{i}^{j}\big{]}_{j=1}^{5}$ for $i=1,\ldots,5$ and $[1:1:1:1:1]$)
and the following four other points $[0:0:1:1:1]$, $[0:1:0:1:1]$,
$[1:1:0:0:1]$ and $[1:0:1:0:1]$. Hence the $p_{ij}$'s are 10 pairwise distinct
singular points of $V_{3}$. On another hand, the map $\varphi_{3}$ has rank 3
outside the union of the 10 lines $L_{ij}$'s, hence $V_{3}$ is smooth outside
the $p_{ij}$'s. Because these singular points all are ordinary nodes (as an
easy analysis shows), one deduces that $V_{3}$ coincides with Segre's 10-nodal
cubic hypersurface. For that reason, we will denote this threefold by
$\boldsymbol{S}$ from now on. We will write
$N_{\boldsymbol{S}}=\\{\,p_{ij}\,\lvert\,1\leq i<j\leq 5\,\\}$ for the set of
its nodes.

Let us describe the 2-planes and some lines contained in $\boldsymbol{S}$ and
their incidence with the nodes. For any $i,j,k$ (pairwise distinct), the
quadric of $\boldsymbol{\mathcal{L}}_{3}$ passing through a generic point
$p\in P_{ijk}$ is the union of $P_{ijk}$ itself with the 2-plane passing
through $p_{l}$, $p_{m}$ and $p$. It follows that $\varphi_{3}$ maps $P_{ijk}$
onto a plane in $\boldsymbol{S}$, that we will denote $P_{ijk}$ too. These ten
2-planes $P_{ijk}$'s are pairwise distinct. In addition to them,
$\boldsymbol{S}$ contains five other planes, which can be described as
follows: let $\pi:\widetilde{\boldsymbol{S}}={\rm
Bl}_{P}\mathbf{P}^{3}\rightarrow\mathbf{P}^{3}$ be the blow-up at the five
base-points of $\lvert\boldsymbol{\mathcal{L}}_{3}\lvert$. Then
$\widetilde{\varphi}_{3}=\varphi_{3}\circ\pi:\widetilde{\boldsymbol{S}}\rightarrow\boldsymbol{S}$
is a morphism and if $E_{i}$ stands for the exceptional divisor corresponding
to $p_{i}$ (i.e. $E_{i}=\pi^{-1}(p_{i})$) for any $i$, then
$P_{i}=\widetilde{\varphi}_{3}(E_{i})$ is a 2-plane in Segre's cubic. In this
way, we get five other planes included in $\boldsymbol{S}$ which with the 15
described above form the whole set of 2-planes contained in Segre's cubic,
which therefore has cardinality 16. In order to have an uniform notation, we
set $P^{ij}=P_{klm}$ and $P^{i6}=P_{i}$, here for any $i,j$ such that $1\leq
i\leq j\leq 5$.

######  Remark 3.6.

There is a nice uniform interpretation of the planes $P^{u,v}$ for any $1\leq
u<v\leq 6$ in terms of Knudsen-Mumford modular compactification
$\overline{\mathcal{M}}_{0,6}$. According to a well-known result of Kapranov,
the blow-up of $\widetilde{\boldsymbol{S}}$ along the union of the strict
transforms by $\pi$ of the lines $L_{i,j}$ is isomorphic to
$\overline{\mathcal{M}}_{0,6}$ and any plane $P^{uv}$ in Segre's cubic is the
image by the morphism $\overline{\mathcal{M}}_{0,6}\rightarrow\boldsymbol{S}$
associated to $\varphi_{3}$, of the boundary divisor $\Delta_{uv}$ of
$\overline{\mathcal{M}}_{0,6}$ (naturally isomorphic to
$\overline{\mathcal{M}}_{0,5}$) formed by limits of configurations of 6 points
on the projective line obtained when making the $u$-th and $v$-points of the
configurations collapse. A generic point of $\Delta^{uv}$ is a rational curve
$C_{1}\cup C_{2}$ with two irreducible components $C_{1},C_{2}$ intersecting
at a node, with three marked points on $C_{1}$, namely the $u$-th, the $v$-th
and the node.

One easily determines the incidence relations between the nodes of
$\boldsymbol{S}$ and the 2-planes included in it: for any indices $i,j,k,l,m$
such that their set coincides with $\\{1,\ldots,5\\}$, one has

  * •

$P^{ij}\cap N_{\boldsymbol{S}}=\\{\,p_{ij},p_{kl},p_{km},p_{lm}\,\\}$ and
$P^{i6}\cap N_{\boldsymbol{S}}=\\{\,p_{ij},p_{ik},p_{il},p_{im}\,\\}$;
moreover, the four nodes in a plane included in $\boldsymbol{S}$ are in
general position in it.

  * •

The set of 2-planes in $\boldsymbol{S}$ adjacent to $p_{ij}$ is
$\\{\,P^{ij},P^{kl},P^{km},P^{lm},P^{i6},P^{j6}\,\\}$.

Finally, for any distinct pairs $(i,j)$ and $(k,l)$ (with $1\leq i<j\leq 5$
and similarly for $k$ and $l$), the line $\ell_{ij,kl}=\langle
p_{ij},p_{kl}\rangle\subset\mathbf{P}^{4}$ is included in $\boldsymbol{S}$ as
well. From the preceding point, it follows that these lines are pairwise
distinct hence are 45 in number. These lines serve to describe the
intersections of two distinct planes of Segre's cubic. For $i,j,k,l$ in
$\\{1,\ldots,5\\}$ such that $\\{i,j\\}\neq\\{k,l\\}$, the intersection of
$P^{ij}$ with $P^{kl}$ is either a point or a line:

(38) $P^{ij}\cap P^{i6}=\\{\,p_{ij}\,\\}$, $P^{i6}\cap
P^{j6}=\\{\,p_{ij}\,\\}$ and $P^{ij}\cap P^{ik}=\\{\,p_{lm}\,\\}$
(intersection of dim. 0);
---
$P^{ij}\cap P^{kl}=\ell_{ij,kl}=\langle\,p_{ij},p_{kl}\rangle$, $P^{ij}\cap
P^{k6}=\ell_{kl,km}=\langle\,p_{kl},p_{km}\rangle$ (intersection of dim. 1).

######  Remark 3.7.

It may be helpful to indicate how the lines $\ell_{ij,kl}$'s can be realized
`at the source' (that is, in terms of the $\mathbf{P}^{3}$ serving as source
of the rational parametrization $\varphi_{3}$ of $\boldsymbol{S}$). When
$\\{i,j\\}\cap\\{k,l\\}$ is empty, $\ell_{ij,kl}$ is just the image of the
line $P^{ij}\cap P^{kl}\subset\mathbf{P}^{3}$ by $\varphi_{3}$. Giving an
interpretation of $\ell_{ij,ik}$ for $i,j,k$ pairwise distinct is a bit more
involved: the closure of the set of tangent directions at $p_{i}$ of the lines
joining $p_{i}$ to a generic point of $\langle p_{j},p_{k}\rangle$ is a line
in $\mathbf{P}(T_{p_{i}}\mathbf{P}^{3})$. We denote by $l_{i,jk}$ the
corresponding line in the exceptional divisor
$E_{i}=\pi^{-1}(p_{i})\subset\widetilde{\boldsymbol{S}}$. Then an easy
computation shows that, by restriction, $\widetilde{\varphi}_{3}$ induces a
linear isomorphism from $l_{i,jk}$ onto $\ell_{ij,ik}\subset\boldsymbol{S}$.

Beware that our notations above for the lines through two nodes of
$\boldsymbol{S}$ are lightly different from the ones, namely $L[(ij),(kl)]$,
used in [Col2]. In Collino's text, $i,j,k,l$ stand for indices ranging from 1
to 6 (with $\\{i,j\\}\neq\\{k,l\\}$ of course). When all fourth are less than
or equal to 5, then Collino's $L[(ij),(kl)]$ coincides with the line we denote
by $\ell_{ij,kl}$. But if one among these indices is equal to 6, for example
$l$, then one has instead $L[(ij),(k6)]=\langle
p_{ik},p_{jk}\rangle=\ell_{ik,jk}$.

####  3.3.3. Fano surface of Segre's cubic

Everything here is taken from [Col2, §2.2.8]. For a line $L$ in
$\mathbf{P}^{4}$, we write $[L]$ for the point of the Grassmann manifold
$G_{1}(\mathbf{P}^{4})$ to which it corresponds. From the preceding
description of the nodes and of the planes in $\boldsymbol{S}$, one can deduce
a combinatorial description of its Fano surface denoted by $\Sigma$, that we
will often identify with its image by the Plücker embedding

(39) $\Sigma=F_{1}(\boldsymbol{S})\subset
G_{1}(\mathbf{P}V)\hookrightarrow\mathbf{P}\big{(}\wedge^{2}V\big{)}\simeq\mathbf{P}^{9}\,.$

It is a non irreducible surface with 21 irreducible components, all of which
are rational and can be described easily:

  * •

for $1\leq i<j\leq 6$, the set of lines of $P^{ij}$ forms an irreducible
component of $\Sigma$, denoted by $\Sigma^{ij}$. Any such component is
isomorphic to $\mathbf{P}^{2}$ and there are 15 components of this type;

  * •

for $i=1,\ldots,6$, the images by $\varphi_{3}$ of the lines through $p_{i}$
in $\mathbf{P}^{3}$ if $i\leq 5$, or of the twisted cubics through all the
$p_{k}$'s when $i=6$, give rise to an irreducible component of $\Sigma$,
denoted by $\Sigma(i)$. For $i\leq 5$, it identifies with the blow up at the
$[L_{ij}]$'s (with $j\in\\{1,\ldots,5\\}\setminus\\{i\\}$) of the projective
plane parametrizing the lines of $\mathbf{P}^{3}$ passing through $p_{i}$.
Therefore $\Sigma(i)$ is isomorphic to $\overline{\mathcal{M}}_{0,5}$. It can
be verified that the same holds true for $\Sigma(6)$ too;

  * •

the images of the components of $\Sigma$ by the Plücker embedding are not
difficult to describe:

    * –

for any $i,j$ distinct, $\Sigma^{ij}$ is the set of lines of a 2-plane in
$\mathbf{P}V\simeq\mathbf{P}^{4}$ hence its image in $\mathbf{P}^{9}$ is a
2-plane as well;

    * –

for each $i$, the restriction of (39) to $\Sigma(i)$ coincides with the
anticanonical embedding hence its image is a quintic del Pezzo surface (which
spans a 5-plane in $\mathbf{P}^{9}$).

Our notations are unfortunately not well suited to deal with the birational
$\mathfrak{S}_{6}$-symmetries of $\boldsymbol{S}$ and of its Fano surfaces. In
order to describe the incidence relations between the 21 components of
$\Sigma$, it is much more convenient to use Collino's notations $L[(ij),(kl)]$
(recalled above) for the lines joigning two nodes of $\boldsymbol{S}$. Then
the intersections between the different components of $\Sigma$ can be seen to
be as follows where now $i,j,k,l,m,n$ stand for elements ranging in
$\\{1,\ldots,6\\}$ (with the pairs $\\{i,j\\}$ and $\\{k,l\\}$ distinct, etc)
:

  * $-$

the relations (38) can be stated in equivalent terms for the $\Sigma^{ij}$'s:
the two planes $\Sigma^{ij}$ and $\Sigma^{kl}$ do not meet if
$\\{i,j\\}\cap\\{k,l\\}\neq\emptyset$ whereas one has
$\Sigma^{ij}\cap\Sigma^{kl}=\\{\,L[(ij),(kl)]\,\\}$ otherwise;

  * $-$

as for the intersection of two del Pezzo components, when $i$ and $j$ are
distinct then $\Sigma(i)\cap\Sigma(j)$ is the set of three points of $\Sigma$
corresponding to the lines $L[(kl),(mn)]$ for all $k,l,m,n$ such that
$\\{i,j,k,l,m,n\\}=\\{1,\ldots,6\\}$;

  * $-$

there are two possibilities for the intersection of a plane $\Sigma^{ij}$ with
a del Pezzo component $\Sigma(k)$: first, one obviously has
$\Sigma^{ij}\cap\Sigma(k)=\emptyset$ if $k\in\\{i,j\\}$. At the opposite, when
$i,j$ and $k$ are pairwise distinct, then $\Sigma(k)$ intersects $\Sigma^{ij}$
along a line denoted by $R(k,ij)$, which is isomorphic to
$\overline{\mathcal{M}}_{0,4}$. In our non symmetric presentation, for $k<6$
the line $R(ij,k)$ corresponds to the (images by $\varphi_{3}$ of the) lines
joining $p_{k}$ to a point of $L_{lm}=\langle
p_{l},p_{m}\rangle\subset\mathbf{P}^{3}$ and the three boundary points
correspond to the lines passing through $p_{l}$, $p_{m}$ and the point of
intersection in $\mathbf{P}^{3}$ of the line $L_{ij}=\langle
p_{i},p_{j}\rangle$ with the plane $P^{ij}=\langle p_{k},p_{l},p_{m}\rangle$.

A noteworthy last point concerns the incidence between the lines joining two
nodes of $\boldsymbol{S}$ but considered as points of the Fano surface
$\Sigma$, and the 2-dimensional components of the latter: using Collino's
notation, when $i,j,k,l,m,n$ stand for the elements of $\\{1,\ldots,6\\}$ then
the two del Pezzo's $\Sigma(m)$, $\Sigma(n)$ and the two planes $\Sigma^{ij}$
and $\Sigma^{kl}$ are exactly the components of $\Sigma$ to which
$L[(ij),(kl)]\in\Sigma$ belongs. Essentially quoting Collino here, one
concludes by saying that `this shows that $\Sigma=F(\boldsymbol{S})$ is not a
normal crossing surface: at any point $L[(ij),(kl)]$, four irreducible
components of $\Sigma$ meet, and moreover these components split in two
couples which intersect locally only at this point'.

####  3.3.4. Abelian relations and rank of $\mathcal{W}_{{0,6}}$

We have seen above that $\boldsymbol{\mathcal{W}}_{0,6}$ identifies with the
6-web by lines $\boldsymbol{\mathcal{L}\mathcal{W}}_{\boldsymbol{S}}$ on
Segre's cubic. Our goal here is to establish that Proposition C of the
Introduction, which is stated for a `sufficiently generic' hypercubic of
$\mathbf{P}^{4}$, holds true for $\boldsymbol{S}$ as well:

######  Proposition 3.8.

1\. The trace induces an isomorphism ${\rm
Tr}:\boldsymbol{H}^{0}\big{(}\Sigma,\omega_{\Sigma}^{2}\big{)}\simeq\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{L}\mathcal{W}}_{\boldsymbol{S}}\big{)}$.

2\. Therefore $\boldsymbol{\mathcal{W}}_{0,6}$ is an algebraizable 6-web of
maximal 2-rank ${\bf
rk}\big{(}\boldsymbol{\mathcal{W}}_{0,6}\big{)}=h^{0}\big{(}\Sigma,\omega_{\Sigma}^{2}\big{)}=10$.

##### An elementary and explicit proof of Proposition 3.8

The key fact behind the elementary effective approach discussed below is point
2. in (33), namely that the dualizing sheaf $\omega_{F}^{2}$ of the Fano
surface $F$ of a cubic threefold $X$ with finitely many ordinary singular
points coincides with the pull-back of $\mathcal{O}_{\mathbf{P}^{9}}(1)$ under
the Plücker embedding $F\subset
G_{1}(\mathbf{P}^{4})\hookrightarrow\mathbf{P}^{9}$. Since the dualizing sheaf
$\omega_{F}$ naturally identifies with the sheaf $\omega^{2}_{F}$ of abelian
differential 2-forms on $F$ (cf. [Bar]), the aforementioned fact allows to get
explicit expressions for a basis of $\boldsymbol{H}^{0}(F,\omega^{2}_{F})$ as
soon as the embedding $F\hookrightarrow\mathbf{P}^{9}$ is explicitly known.
One is reduced to verify whether the elements of this basis give rise to
abelian relations for the associated linear web
$\boldsymbol{\mathcal{L}\mathcal{W}}_{{X}}$, which can be done through direct
computations. In the sequel, we apply this approach to Segre's cubic
$\boldsymbol{S}$ in order to describe the ARs then to compute the rank of
$\boldsymbol{\mathcal{W}}_{0,6}\simeq\boldsymbol{\mathcal{L}\mathcal{W}}_{{\boldsymbol{S}}}$.

Recall that to a rank 2 matrix $M\in{\rm Mat}_{2,5}(\mathbf{C})$, one can
associate the line in $\mathbf{P}^{4}$, denoted by $\langle M\rangle$, passing
through the two points whose homogeneous coordinates are the two lines of $M$.
hereafter, we will use the well-known fact that ${\rm
Mat}_{2,5}(\mathbf{C})\dashrightarrow G_{1}(\mathbf{P}^{4})$, $M\mapsto\langle
M\rangle$ is a birational chart.

Let $(x,y,z)$ stand for the coordinates of a generic point $p$ of
$\mathbf{P}^{3}$; here are the rational first integrals for
$\boldsymbol{\mathcal{W}}_{0,6}$ we will work with: $U_{1}=(y,z)$,
$U_{2}=(x,z)$, $U_{3}=(x,y)$, $U_{4}=(x/z,y/z)$,
$U_{5}=((x-1)/(z-1),(y-1)/(z-1))$ and
$U_{6}=(x(z-1)/(z(x-1)),y(z-1)/(z(y-1)))$.

For each $i=1,\dots,6$, solving $U_{i}(x,y,z)=(u_{i},v_{i})$ with
$u_{i},v_{i}\in\mathbf{Q}(x,y,z)$, one gets easily a parametrization
$\xi_{i,p}:\mathbf{P}^{1}\rightarrow\mathbf{P}^{3}$ of the $i$-th leaf of
$\boldsymbol{\mathcal{W}}_{0,6}$ through $p$, whose coefficients are rational
in $u_{i}$ and $v_{i}$. Then evaluating
$\varphi\circ\xi_{i,p}:\mathbf{P}^{1}\rightarrow\mathbf{P}^{4}$ at two generic
points, for instance $0$ and $\infty$, one easily computes a $2\times 5$
matrix $M_{i}\in{\rm Mat}_{2,5}\big{(}\mathbf{Q}(u_{i},v_{i})\big{)}$ such
that the $i$-th leaf of
$\boldsymbol{\mathcal{L}\mathcal{W}}_{{\boldsymbol{S}}}$ through
$\varphi_{3}(p)$ is the line passing through the two points whose homogeneous
coordinates are the two lines of $M_{i}$. Of course, the $M_{i}$'s are not
unique but some elementary computations give that they can be taken to be the
following ones:

$\displaystyle M_{1}=$
$\displaystyle\,\begin{bmatrix}0&0&u_{1}v_{1}&u_{1}&v_{1}\\\
u_{1}-v_{1}&u_{1}-1&u_{1}&u_{1}&u_{1}\end{bmatrix}$ $\displaystyle M_{2}=$
$\displaystyle\,\begin{bmatrix}u_{2}v_{2}&u_{2}&0&0&v_{2}\\\
u_{2}&u_{2}&u_{2}-v_{2}&u_{2}-1&u_{2}\end{bmatrix}$ $\displaystyle M_{3}=$
$\displaystyle\,\begin{bmatrix}u_{3}v_{3}&u_{3}(v_{3}-1)&u_{3}v_{3}&v_{3}(u_{3}-1)&v_{3}\\\
u_{3}&0&v_{3}&0&1\end{bmatrix}$ $\displaystyle M_{4}=$
$\displaystyle\,\begin{bmatrix}0&u_{4}&0&v_{4}&1\\\
u_{4}(1-v_{4})&-u_{4}v_{4}&v_{4}(1-u_{4})&-u_{4}v_{4}&-u_{4}v_{4}\end{bmatrix}$
$\displaystyle M_{5}=$
$\displaystyle\,\begin{bmatrix}u_{5}(v_{5}-1)&u_{5}v_{5}&v_{5}(u_{5}-1)&u_{5}v_{5}&u_{5}v_{5}\\\
(v_{5}-1)(1-u_{5})&v_{5}(1-u_{5})&(v_{5}-1)(1-u_{5})&u_{5}(1-v_{5})&(v_{5}-1)(1-u_{5})\end{bmatrix}$
$\displaystyle\mbox{and }\quad M_{6}=$
$\displaystyle\,\begin{bmatrix}0&u_{6}&0&v_{6}&1\\\
u_{6}(v_{6}-1)&0&v_{6}(u_{6}-1)&0&(v_{6}-1)(u_{6}-1)\end{bmatrix}\,.$

The map $\mathbf{C}^{2}\dashrightarrow\Sigma(i)$, $(u_{i},v_{i})\mapsto\langle
M_{i}\rangle$ is easily seen to be birational, from which it follows that
$(u_{i},v_{i})$ can be taken as global rational coordinates on $\Sigma(i)$ for
any $i=1,\ldots,6$. It is then immediate to compute a parametrization of the
image $\widetilde{\Sigma}(i)$ of the component $\Sigma(i)\subset\Sigma$ by the
Plücker embedding $\rho:\Sigma\subset
G_{1}(\mathbf{P}^{4})\hookrightarrow\mathbf{P}^{9}$. Indeed, denoting by
$\Delta_{a,b}(M_{i})$ the $2\times 2$ minor of $M_{i}$ corresponding to its
$a$-th and $b$-th columns (in this order) for any $(a,b)$ such that $1\leq
a<b\leq 5$, the aforementioned parametrization is given by

$\displaystyle\kappa_{i}:\big{(}u_{i},v_{i}\big{)}\dashrightarrow\Big{[}\Delta_{a,b}(M_{i})\Big{]}_{1\leq
a<b\leq 5}\in\mathbf{P}^{9}\,.$

Since $\kappa_{i}$ corresponds to the expression, in the rational coordinates
$u_{i},v_{i}$, of the restriction on $\Sigma(i)$ of the canonical map
$\kappa=\kappa_{\lvert\omega^{2}_{\Sigma}\lvert}:\Sigma\dashrightarrow\mathbf{P}H^{0}(\Sigma,\omega^{2}_{\Sigma})^{\vee}$,
there exists a rational function $\sigma_{i}=\sigma_{i}(u_{i},v_{i})$ (to be
made explicit further below) such that

$\Big{\\{}\,\Delta_{a,b}(M_{i})\,\big{(}\sigma_{i}\,du_{i}\wedge
dv_{i}\big{)}\,\big{\lvert}\,1\leq a<b\leq b\,\Big{\\}}$

spans of the space of rational 2-forms
$\iota_{i}^{*}\big{(}\boldsymbol{H}^{0}(\Sigma,\omega_{\Sigma}^{2})\big{)}$ on
$\Sigma(i)$.313131More prosaically,
$\iota_{i}^{*}\big{(}\boldsymbol{H}^{0}(\Sigma,\omega_{\Sigma}^{2})\big{)}$ is
the image of the restriction map
$\boldsymbol{H}^{0}\big{(}\Sigma,\omega_{\Sigma}^{2}\big{)}\rightarrow\Omega^{2}_{\mathbf{C}(\Sigma(i))}$,
$\omega\mapsto\omega\lvert_{\Sigma(i)}$. Actually, working a bit further, one
verifies that

  1. (1)

for any $i$, the composition
$\rho\circ\iota_{i}:\Sigma(i)\subset\Sigma\rightarrow\mathbf{P}^{9}$ is given
by the complete anticanonical linear system $\lvert K_{\Sigma(i)}^{-1}\lvert$,
hence $\widetilde{\Sigma}(i)=\rho(\Sigma(i))$ is a quintic del Pezzo surface
which spans a $\mathbf{P}^{5}$ in the ambiant $\mathbf{P}^{9}$;

  2. (2)

the union of the $\widetilde{\Sigma}(i)$'s spans the whole ambiant projective
space, that is
$\big{\langle}\cup_{i=1}^{6}\widetilde{\Sigma}(i)\big{\rangle}=\mathbf{P}^{9}$.
It follows that the linear map
$\boldsymbol{H}^{0}\big{(}\Sigma,\omega_{\Sigma}^{2}\big{)}\longrightarrow\oplus_{i=1}^{6}\Omega^{2}_{\mathbf{C}(\Sigma(i))}$,
$\omega\longmapsto\big{(}\omega\lvert_{\Sigma(i)}\big{)}_{i=1}^{6}$ is
injective and consequently, the set of 6-tuples of rational 2-forms

(40) $\bigg{\\{}\,\Big{(}\Delta_{a,b}(M_{i})\,\big{(}\sigma_{i}\,du_{i}\wedge
dv_{i}\big{)}\Big{)}_{i=1}^{6}\in\oplus_{i=1}^{6}\Omega^{2}_{\mathbf{C}(\Sigma(i))}\,\big{\lvert}\,1\leq
a<b\leq b\,\bigg{\\}}$

can be identified with a basis of the space of global abelian differential
2-forms on $\Sigma$.

At this point, it is straightforward to give an explicit expression in the
rational coordinates $x,y,z$, for the trace of the 2-form
$\omega_{a,b}\in\boldsymbol{H}^{0}\big{(}\Sigma,\omega_{\Sigma}^{2}\big{)}$
corresponding to the 6-tuple in (40) associated to the $(a,b)$-minor: for any
$a,b$ such that $1\leq a<b\leq 5$, one has

(41) ${\rm
Tr}\big{(}\omega_{a,b}\big{)}=\sum_{i=1}^{6}\Delta_{a,b}(M_{i})\,\big{(}\sigma_{i}\,du_{i}\wedge
dv_{i}\big{)}$

where all the $u_{i}$'s and $v_{i}$'s are now viewed as rational functions of
$x,y,z$, hence any term of the right-hand side sum has to be considered as a
rational 2-form in the variables $x,y$ and $z$.

An easy computation gives that the only 6-tuple of functions
$(\sigma_{i})_{i=1}^{6}$ with $\sigma_{i}=\sigma_{i}(u_{i},v_{i})$ such that
the RHS of any expression (41) vanishes identically is a nonzero constant
multiple of

$\left(\frac{1}{m_{5}(u_{i},v_{i})}\right)_{i=1}^{6}$

where $m_{5}$ stands for the `$\mathcal{M}_{0,5}$-polynomial' defined by
$m_{5}(u,v)=uv(u-1)(v-1)(u-v)$.

We thus have obtained the following result which has to be seen as an explicit
version of Proposition 3.8:

######  Proposition 3.9.

In the coordinates $x,y,z$ and relatively to the first integrals
$U_{i}=(u_{i},v_{i})$, $i=1,\ldots,6$, the ARs of
$\boldsymbol{\mathcal{W}}_{0,6}$ are all rational and their space admits as a
basis the set of $6$-tuples of rational 2-forms (for all $a,b$ such that
$1\leq a<b\leq 5$)

(42) $\left(\frac{\Delta_{a,b}(M_{i})}{m_{5}(u_{i},v_{i})}\Big{(}du_{i}\wedge
dv_{i}\Big{)}\right)_{i=1}^{6}$

which all satisfy the relation
$\sum_{i=1}^{6}\big{(}{\Delta_{a,b}(M_{i})}/{m_{5}(u_{i},v_{i})}\big{)}\big{(}du_{i}\wedge
dv_{i}\big{)}=0$ identically.

Since the $U_{i}$'s as well as the $M_{i}$'s are all explicit, is it just an
elementary computational matter to make the ARs (42) explicit. For instance,
the one corresponding to $(a,b)=(1,2)$ is

$\displaystyle\frac{u_{2}\,du_{2}\wedge
dv_{2}}{v_{2}(u_{2}-1)(u_{2}-v_{2})}-\frac{u_{3}\,du_{3}\wedge
dv_{3}}{v_{3}(u_{3}-1)(u_{3}-v_{3})}+\frac{u_{4}\,du_{4}\wedge
dv_{4}}{v_{4}(u_{4}-1)(u_{4}-v_{4})}-\frac{u_{6}\,du_{6}\wedge
dv_{6}}{v_{6}(u_{6}-1)(u_{6}-v_{6})}=0$

and that associated to $(a,b)=(4,5)$ is

(43) $\displaystyle\frac{du_{1}\wedge dv_{1}}{v_{1}(u_{1}-1)(v_{1}-1)}-$
$\displaystyle\frac{du_{2}\wedge
dv_{2}}{u_{2}(v_{2}-1)(u_{2}-v_{2})}+\frac{du_{3}\wedge
dv_{3}}{u_{3}(v_{3}-1)(u_{3}-v_{3})}$ $\displaystyle-$
$\displaystyle\frac{du_{4}\wedge
dv_{4}}{(u_{4}-1)(u_{4}-v_{4})}+\frac{du_{5}\wedge
dv_{5}}{(u_{5}-1)(u_{5}-v_{5})}+\frac{du_{6}\wedge
dv_{6}}{u_{6}(u_{6}-v_{6})}=0\,.$

Realizing that all the ARs of $\boldsymbol{\mathcal{W}}_{0,6}$ were rational
in some natural coordinates led us to doubt about the validity of the
decomposition (8) of $\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,6})$ as a
$\mathfrak{S}_{6}$-module claimed in [Da2]. In order to clear this up, we have
investigated the birational action of $\mathfrak{S}_{6}$ on the ARs (42) to
come to the conclusion that there was indeed an error on this point.

####  3.3.5. The representation of $\mathfrak{S}_{6}$ on
${AR}_{C}(\mathcal{W}_{{0,6}})$

We use below notations almost identical to those used in the preceding
subsection. Here $x,y,z$ stand for the rational coordinates corresponding to
the map $\mathbf{C}^{3}\dashrightarrow\boldsymbol{\mathcal{M}}_{0,6}$,
$(x,y,z)\mapsto[\infty,0,1,x,y,z]$. The first integrals we work with here for
$\boldsymbol{\mathcal{W}}_{{0,6}}$ are the six rational first integrals
$U_{i}=(u_{i},v_{i}):\mathbf{C}^{3}\dashrightarrow\mathbf{C}^{2}$ of the
previous subsection, with the only difference that we have exchanged $U_{1}$
and $U_{3}$ (namely, one takes $U_{1}=(x,y)$ and $U_{3}=(y,z)$ below while the
others $U_{i}$'s are left unchanged).

The associated normals $\Omega_{i}=du_{i}\wedge dv_{i}$ for $i=1,\ldots,6$ are
easy to compute: the sheaf of 2-forms on $\mathbf{C}^{2}$ is globally spanned
by $\Omega_{1}=dx\wedge dy$, $\Omega_{2}=dx\wedge dz$ and $\Omega_{3}=dy\wedge
dz$ and as global rational 2-forms, the others $\Omega_{i}$'s are determined
by the following matricial relation:

(44) $\begin{bmatrix}\Omega_{4}\\\ \Omega_{5}\\\
\Omega_{6}\end{bmatrix}=\begin{bmatrix}\frac{1}{z^{2}}&-\frac{y}{z^{3}}&\frac{x}{z^{3}}\\\
\frac{1}{(z-1)^{2}}&-\frac{y-1}{(z-1)^{3}}&\frac{x-1}{(z-1)^{3}}\\\
\frac{(z-1)^{2}}{(x-1)^{2}(y-1)^{2}z^{2}}&-\frac{y(z-1)}{(x-1)^{2}(y-1)z^{3}}&\frac{x(z-1)}{(x-1)(y-1)^{2}z^{3}}\end{bmatrix}\,\begin{bmatrix}\Omega_{1}\\\
\Omega_{2}\\\ \Omega_{3}\end{bmatrix}$

The transformations

$\mathscr{T}:\,[p_{i}]_{i=1}^{6}\longmapsto[p_{1},p_{2},p_{3},p_{5},p_{4},p_{6}]\qquad\mbox{
and
}\qquad\mathscr{C}:\,[p_{i}]_{i=1}^{6}\longmapsto[p_{6},p_{1},\ldots,p_{5}]$

are two automorphisms of $\boldsymbol{\mathcal{M}}_{0,6}$ which correspond to
the transposition $\tau=(45)$ and to the cycle $c=(1,\ldots,6)$ of
$\mathfrak{S}_{6}$ respectively. The corresponding rational maps in the affine
coordinates $(x,y,z)$ are given respectively by

(45) $\mathcal{T}:(x,y,z)\mapsto(y,x,z)\qquad\mbox{ and
}\qquad\mathcal{C}:(x,y,z)\mapsto\left(\frac{1}{x}\,,\,\frac{x-z}{(x-1)z}\,,\,\frac{y-z}{(y-1)z}\right)\,.$

Clearly, $\mathcal{T}$ and $\mathcal{C}$ generate the image of ${\rm
Aut}(\boldsymbol{\mathcal{M}}_{0,6})\simeq\mathfrak{S}_{6}$ in ${\bf
Bir}_{3}$. Their actions under pull-backs on the $\Omega_{i}$'s can be easily
computed. One verifies that they are given by the following relations:

(46)
$\displaystyle\Big{(}\mathcal{T}^{*}\big{(}\Omega_{i}\big{)}\Big{)}_{i=1}^{6}=$
$\displaystyle\,\Big{(}\,-\Omega_{1}\,,\,\Omega_{3}\,,\,\Omega_{2}\,,\,-\Omega_{4}\,,\,-\Omega_{5}\,,\,-\Omega_{6}\,\Big{)}$
$\displaystyle\mbox{and
}\quad\Big{(}\mathcal{C}^{*}\big{(}\Omega_{i}\big{)}\Big{)}_{i=1}^{6}=$
$\displaystyle\,\Big{(}\,\eta_{2}\,\Omega_{2}\,,\,\eta_{3}\,\Omega_{3}\,,\,\Omega_{6}\,,\,\widetilde{\eta}_{5}\,\Omega_{5}\,,\,\eta_{1}\,\Omega_{1}\,,\,\widetilde{\eta}_{4}\,\Omega_{4}\Big{)}\,,$

with $\eta_{i}=\eta(U_{i})=\eta(u_{i},v_{i})$ for any $i$ (and similarly for
$\widetilde{\eta}_{i}$), where $\eta$ and $\widetilde{\eta}$ are standing for
the rational functions $\eta(u,v)=(v-1)/(v^{3}(u-1)^{2})$ and
$\widetilde{\eta}(u,v)=\eta(1+u,1-v)=v/(u^{2}(v-1)^{3})$.

From (46), it follows that the action on
$\boldsymbol{\mathcal{W}}_{0,n+3}=(\mathcal{F}_{i})_{i=1}^{6}$ induced by
$\mathcal{T}$ and $\mathcal{C}$ are given respectively by the transposition
$\tau=(23)$ and the 6-cycle $c=(123645)$: for $i=1,\ldots,6$, one has

(47)
$\mathcal{T}^{*}\big{(}\mathcal{F}_{i}\big{)}=\mathcal{F}_{\tau(i)}\qquad\mbox{
and }\qquad\mathcal{C}^{*}\big{(}\mathcal{F}_{i}\big{)}=\mathcal{F}_{c(i)}\,.$

##### An explicit combinatorial basis of
$\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,6})$.

Some of the ARs (42) are combinatorial but not all of them (cf. (43)). But
since these abelian relations form a basis of
$\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,6})$, it is just a matter of
computing to construct all the combinatorial ARs of
$\boldsymbol{\mathcal{W}}_{0,6}$. We want to do that explicitly here.

We set $I=\\{1,\ldots,6\\}$ and $I^{2}_{\neq}$ stands for the set of all pairs
$(i,j)\in I^{2}$ such that $i<j$. For any $(i,j)$ in this set, one denotes by
$AR_{ij}$ the non trivial AR for the 4-subweb
$\boldsymbol{\mathcal{W}}_{\widehat{\imath\jmath}}=\boldsymbol{\mathcal{W}}(U_{k}\,\big{\lvert}\,k\in
I\setminus\\{i,j\\}\,\big{)}$ of $\boldsymbol{\mathcal{W}}_{0,6}$. Actually,
$AR_{ij}$ is only defined up to multiplication by a non zero scalar but this
ambiguity will be irrelevant with regard to what we are going to discuss in
the sequel.

As it can easily be verified, each row of the matrix page 3.3.5, denoted by
$M_{AR}$ in what follows, encodes an abelian relation $AR_{ij}$ (where
therefore $i$ and $j$ correspond to the zero coefficients): denoting by
$m_{ab}$ (for $a$ and $b$ ranging from 1 to 10 and 6 respectively) its
coefficients, then $m_{ab}$ is a function of $U_{b}$ for any $b$ (hence one
can write $m_{ab}=m_{ab}(U_{b})$) and for any $a=1,\ldots,10$, the following
relation $\sum_{b=1}^{10}m_{ab}(U_{b})\,\Omega_{b}=0$ holds true identically.

One denotes by $AR(a)$ the combinatorial AR corresponding to the previous
relation (hence $AR(1)=AR_{13}$, $\ldots$, $AR(10)=AR_{56}$). By elementary
linear algebra, one verifies that the $AR(a)$'s for $a=1,\ldots,10$ are
linearly independent abelian relations, from which we get the

######  Proposition 3.10.

1\. The space $\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,6})$ has
dimension 10.

2\. Consequently
$\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,6})=\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,6})$
and $\boldsymbol{\mathcal{W}}_{0,6}$ has maximal rank.

We remark that the five combinatorial abelian relations $AR_{ij}$ which do not
correspond to one of the lines of $M_{AR}$ correspond to the pairs (12), (23),
(36), (46) and (45) hence one deduces a more invariant way to describe the
basis of $\boldsymbol{AR}(\mathcal{W}_{0,6})$ formed by the $AR(a)$'s for
$a=1,\ldots,10$:

######  Fact 3.11.

The family of $AR_{ij}$'s for all pairs $(i,j)\in I^{2}_{\neq}$ minus the five
abelian relations $(\mathcal{C}^{\ell})^{*}(AR_{12})$ for $\ell=0,\ldots,4$,
form a basis of $\boldsymbol{AR}(\mathcal{W}_{0,6})$.

The interest of formulating things that way is that it naturally asks whether
the following straightforward generalization holds true or not:

######  Question 3.12.

For any odd integer $n\geq 2$, does the family of $AR_{ij}$'s for all pairs
$(i,j)\in I^{2}_{\neq}$ minus the $n+2$ abelian relations
$(\mathcal{C}^{\ell})^{*}(AR_{12})$ for $\ell=0,\ldots,n+1$, form a basis of
$\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,n+3})$?

##### The representation of $\mathfrak{S}_{6}$ on
$\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,6})$.

Looking at (47) and because the space of ARs of
$\boldsymbol{\mathcal{W}}_{\widehat{\imath\jmath}}$ is 1-dimensional for any
$(i,j)\in I^{2}_{\neq}$, it is clear that

$\mathcal{T}^{*}(AR_{ij})=\lambda_{ij}\,AR_{\tau(i)\tau(j)}\qquad\mbox{ and
}\qquad\mathcal{C}^{*}(AR_{ij})=\gamma_{ij}\,AR_{c(i)c(j)}$

for some non zero scalars $\lambda_{ij}$ and $\gamma_{ij}$ which are easy to
make explicit using formulas (46). One gets that relatively to the basis
$(AR(a))_{a=1}^{10}$, the actions on
$\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,6})$ induced by the pull-backs
under $\mathcal{T}$ and $\mathcal{C}$ (defined in (45)) are given by the two
following matrices:

$T={\begin{bmatrix}\hskip 2.84544pt1&1&-1&-1&0&0&0&0&0&0\hskip 2.84544pt\\\
\hskip 2.84544pt0&-1&0&0&0&0&0&0&0&0\hskip 2.84544pt\\\ \hskip
2.84544pt0&0&-1&0&0&0&0&0&0&0\hskip 2.84544pt\\\ \hskip
2.84544pt0&0&0&-1&0&0&0&0&0&0\hskip 2.84544pt\\\ \hskip
2.84544pt0&0&0&0&0&0&0&1&0&0\hskip 2.84544pt\\\ \hskip
2.84544pt0&0&0&0&0&0&0&0&1&0\hskip 2.84544pt\\\ \hskip
2.84544pt0&1&-1&-1&-1&1&1&1&-1&0\hskip 2.84544pt\\\ \hskip
2.84544pt0&0&0&0&1&0&0&0&0&0\hskip 2.84544pt\\\ \hskip
2.84544pt0&0&0&0&0&1&0&0&0&0\hskip 2.84544pt\\\ \hskip
2.84544pt0&0&0&0&0&0&0&0&0&-1\hskip 2.84544pt\end{bmatrix}}\hskip
17.07182pt\mbox{ and }\hskip 11.38092ptC={\begin{bmatrix}\hskip
2.84544pt0&0&1&0&0&1&0&0&0&0\hskip 2.84544pt\\\ \hskip
2.84544pt0&0&1&0&0&0&0&0&0&-1\hskip 2.84544pt\\\ \hskip
2.84544pt0&0&-1&0&0&0&0&0&0&0\hskip 2.84544pt\\\ \hskip
2.84544pt0&0&-1&0&0&0&0&0&1&0\hskip 2.84544pt\\\ \hskip
2.84544pt0&0&0&-1&0&0&0&0&0&0\hskip 2.84544pt\\\ \hskip
2.84544pt0&-1&0&0&0&0&0&0&0&0\hskip 2.84544pt\\\ \hskip
2.84544pt-1&0&0&0&0&0&0&0&0&0\hskip 2.84544pt\\\ \hskip
2.84544pt0&0&0&0&0&0&-1&0&0&0\hskip 2.84544pt\\\ \hskip
2.84544pt0&0&0&0&-1&0&0&0&0&0\hskip 2.84544pt\\\ \hskip
2.84544pt0&0&0&0&0&0&0&1&0&0\hskip 2.84544pt\end{bmatrix}}\,.\vskip 12.0pt
plus 4.0pt minus 4.0pt$

Their traces are given by

${\rm Tr}(T)=-2\qquad\mbox{ and }\qquad{\rm Tr}(C)=-1\,.$

The representation of
$\langle\mathcal{T},\mathcal{C}\rangle\simeq\mathfrak{S}_{6}$ on the vector
space
$\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,6})=\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,6})$
of dimension 10 is isomorphic to the one defined by the group morphism
$\rho:\mathfrak{S}_{6}\rightarrow{\rm GL}_{10}$ charaterized by

$\rho(\tau)=T\qquad\mbox{ and }\qquad\rho(c)=C\,.$

$\begin{bmatrix}\quad
0&{\frac{v_{2}-u_{2}}{u_{2}v_{2}\left(u_{2}-1\right)\left(v_{2}-1\right)}}&0&{\frac{{u_{4}}-1}{{u_{4}}\,\left({u_{4}}-{v_{4}}\right)\left({\it
v_{4}}-1\right)}}&{\frac{1-{u_{5}}}{{u_{5}}\,\left({u_{5}}-{v_{5}}\right)\left({v_{5}}-1\right)}}&{\frac{1-{u_{6}}}{{u_{6}}\,\left({u_{6}}-{v_{6}}\right)\left({v_{6}}-1\right)}}\quad\vskip
12.0pt plus 4.0pt minus 4.0pt\\\ \quad
0&{\frac{v_{2}-1}{\left(u_{2}-1\right)\left(u_{2}-v_{2}\right)v_{2}}}&{\frac{1-v_{3}}{\left(u_{3}-1\right)\left(u_{3}-v_{3}\right)v_{3}}}&0&{\frac{-1}{\left({\it
u_{5}}-1\right)\left({\it v_{5}}-1\right){\it u_{5}}\,{\it
v_{5}}}}&{\frac{1}{\left({\it v_{6}}-1\right)\left({\it
u_{6}}-1\right)}}\quad\vskip 12.0pt plus 4.0pt minus 4.0pt\\\ \quad
0&{\frac{v_{2}}{u_{2}\left(u_{2}-v_{2}\right)\left(v_{2}-1\right)}}&{\frac{-v_{3}}{u_{3}\left(u_{3}-v_{3}\right)\left(v_{3}-1\right)}}&{\frac{-1}{u_{{4}}v_{{4}}\left(u_{{4}}-1\right)\left(v_{{4}}-1\right)}}&0&{\frac{1}{u_{{6}}v_{{6}}\left(u_{{6}}-1\right)\left(v_{{6}}-1\right)}}\quad\vskip
12.0pt plus 4.0pt minus 4.0pt\\\ \quad
0&{\frac{-1}{v_{2}\left(u_{2}-v_{2}\right)\left(v_{2}-1\right)}}&{\frac{1}{v_{3}\left(u_{3}-v_{3}\right)\left(v_{3}-1\right)}}&{\frac{1}{\left(u_{{4}}-1\right)\left(v_{{4}}-1\right)}}&{\frac{-1}{\left(u_{{5}}-1\right)\left(v_{{5}}-1\right)}}&0\quad\vskip
12.0pt plus 4.0pt minus 4.0pt\\\
\quad{\frac{v_{1}-1}{v_{1}\left(u_{1}-1\right)\left(u_{1}-v_{1}\right)}}&0&{\frac{1-u_{3}}{u_{3}\left(u_{3}-v_{3}\right)\left(v_{3}-1\right)}}&0&{\frac{-v_{{5}}}{u_{{5}}\left(u_{{5}}-v_{{5}}\right)\left(v_{{5}}-1\right)}}&{\frac{1}{v_{{6}}\left(u_{{6}}-v_{{6}}\right)\left(v_{{6}}-1\right)}}\quad\vskip
12.0pt plus 4.0pt minus 4.0pt\\\
\quad{\frac{v_{1}}{u_{1}\left(v_{1}-1\right)\left(u_{1}-v_{1}\right)}}&0&{\frac{-u_{3}}{v_{3}\left(u_{3}-1\right)\left(u_{3}-v_{3}\right)}}&{\frac{-v_{{4}}}{u_{{4}}\left(u_{{4}}-v_{{4}}\right)\left(v_{{4}}-1\right)}}&0&{\frac{v_{{6}}}{u_{{6}}\left(u_{{6}}-v_{{6}}\right)\left(v_{{6}}-1\right)}}\quad\vskip
12.0pt plus 4.0pt minus 4.0pt\\\
\quad{\frac{-1}{v_{1}\left(u_{1}-v_{1}\right)\left(v_{1}-1\right)}}&0&{\frac{1}{u_{3}\left(u_{3}-1\right)\left(u_{3}-v_{3}\right)}}&{\frac{1}{v_{{4}}\left(u_{{4}}-v_{{4}}\right)\left(v_{{4}}-1\right)}}&{\frac{-1}{v_{{5}}\left(u_{{5}}-v_{{5}}\right)\left(v_{{5}}-1\right)}}&0\quad\vskip
12.0pt plus 4.0pt minus 4.0pt\\\
\quad{\frac{u_{1}-1}{u_{1}\left(u_{1}-v_{1}\right)\left(v_{1}-1\right)}}&{\frac{1-u_{2}}{u_{2}\left(u_{2}-v_{2}\right)\left(v_{2}-1\right)}}&0&0&{\frac{-u_{{5}}}{\left(u_{{5}}-1\right)\left(u_{{5}}-v_{{5}}\right)v_{{5}}}}&{\frac{1}{u_{{6}}\left(u_{{6}}-v_{{6}}\right)\left(u_{{6}}-1\right)}}\quad\vskip
12.0pt plus 4.0pt minus 4.0pt\\\
\quad{\frac{u_{1}}{v_{1}\left(u_{1}-1\right)\left(u_{1}-v_{1}\right)}}&{\frac{-u_{2}}{v_{2}\left(u_{2}-1\right)\left(u_{2}-v_{2}\right)}}&0&{\frac{-u_{{4}}}{v_{{4}}\left(u_{{4}}-1\right)\left(u_{{4}}-v_{{4}}\right)}}&0&{\frac{u_{{6}}}{v_{{6}}\left(u_{{6}}-1\right)\left(u_{{6}}-v_{{6}}\right)}}\quad\vskip
12.0pt plus 4.0pt minus 4.0pt\\\
\quad{\frac{-1}{u_{1}v_{1}}}&{\frac{1}{u_{2}v_{2}}}&{\frac{-1}{u_{3}v_{3}}}&{\frac{1}{u_{{4}}v_{{4}}}}&0&0\quad\end{bmatrix}$

From the explicit expression for $T$ and $C$ above and considering basic facts
of the linear repre- -sentation theory of $\mathfrak{S}_{6}$, it is not
difficult to recognize to which representation $\rho$ corresponds. First,
elementary computations show that $T$ and $C$ do not have any eigenvector in
common. Hence if the representation under scrutiny is not irreducible, then it
is necessarily the sum of two irreducible $\mathfrak{S}_{6}$-modules both of
dimension 5. Under this assumption, looking at the character table of the
$\mathfrak{S}_{6}$-representations, one deduces from ${\rm Tr}(C)=-1$ that the
representation is the direct sum of the standard representation with another
representation with Young diagram either $[3^{2}]$ or $[2^{3}]$.

In these cases, the trace of the matrix corresponding to a transposition would
be $3-1=2$ and $3+1=4$ respectively, which would contradict ${\rm Tr}(T)=-2$.
This implies that the representation $\rho$ is irreducible. From ${\rm
Tr}(T)=-2$, one deduces that it is the third exterior power of the standard
representation of $\mathfrak{S}_{6}$. We have proved the

######  Proposition 3.13.

As a $\mathfrak{S}_{6}$-module, the 10-dimensional space
$\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,6})=\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,6})$
is irreducible with associated Young diagram $[31^{3}]$.

Proposition 3.10 and the one just above contradict some of the claims in
[Da2]. The first thing one may think of for explaining this would be that
there is a problem with the construction given by Damiano of Euler's abelian
relation $\boldsymbol{\mathcal{E}}_{3}$ and that this AR actually does not
exist. It turns out that this is not the case, Damiano's construction of
Euler's abelian relation is valid for any $n\geq 2$: we will discuss this in
depth in Section §5.1. According to us, he problem lies in the fact that,
unaware of the dichotomy according to the parity of $n$ in what concerns the
properties of web $\boldsymbol{\mathcal{W}}_{0,n+3}$, Damiano missed an
elementary but crucial fact when he studied the $\mathfrak{S}_{n+3}$-module
structure of the space of combinatorial abelian relations
$\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$. It is what we
are going to discuss in the next section.

##  4\. The space of combinatorial abelian relations as a
$\mathfrak{S}_{n+3}$-representation

We would like to get a better understanding of the space of combinatorial ARs
of $\boldsymbol{\mathcal{W}}_{0,n+3}$. Damiano used a powerful combinatorial
and representation theoretic approach to do so. Using it, he obtained very
interesting and almost complete results about the structure of
$\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,n+3})$ as a
$\mathfrak{S}_{n+3}$-module. However, since he was not aware of the dichotomy
between the cases when $n$ is even or odd in what concerns the
$\mathfrak{S}_{n}$-module structure of
$\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$, he did not go
all the way and missed a complete description of it. On another hand, the
proofs in Damiano's article [Da2] are very concise and/or a bit elliptical in
some places. More details are given in his thesis [Da1] but this one is less
easily available. For these reasons and especially for the sake of
completeness, we thought it relevant to take up Damiano's approach with as
much details as possible and it is the purpose of the current section.

Most of the material below, the statements as the proofs, are due to Damiano
and are taken with essentially no change either from the fourth chapter of
Damiano's thesis [Da1] or from Section §3 of the corresponding paper [Da2]. We
only have added a few details in some places. Proofs have been included below
mainly for the sake of completeness but also because these results are
particularly important regarding the main purpose of this text and are quite
nice, as well as their proofs. The unique genuine novelty in this section is
Lemma 4.10 which together with Damiano's result and although it is elementary,
allows us to show that actually all ARs of $\boldsymbol{\mathcal{W}}_{0,n+3}$
are combinatorial when $n$ is odd.

The plan of the sequel is as follows: after having introduced some notations
in §4.1, we state Theorem 4.2 which is the main result of this section. Proofs
are given in the following subsections §4.3 and §4.4. Finally in §4.5, we
discuss explicit formulas for the components of the combinatorial ARs of
$\boldsymbol{\mathcal{W}}_{0,n+3}$, which will be used later on to make
explicit the components of Euler's abelian relation in Section §5 (see
Proposition 5.22 more specifically).

###  4.1. Some Notations.

Here, $n$ is a fixed integer bigger than or equal to 2 and
$\boldsymbol{I}_{n}$ stands for the set of pairs $(i,j)$ with $1\leq i<j\leq
n+3$. We will take $\mathbf{C}$ as base field. Since we are only dealing with
rational quantities, this choice actually is not relevant and we could have
chosen to work over $\mathbf{R}$ instead (at the cost of some cosmetic changes
that we will not mention).

We use the coordinates system $(x_{1},\ldots,x_{n})$ associated to the
birational map
$\psi:\mathbf{C}^{n}\dashrightarrow\mathcal{M}_{0,n+3},\,x=(x_{i})_{i=1}^{n}\mapsto[0,1,\infty,x_{1},\ldots,x_{n}]$
and we will work with the following cross-ratio

${\rm
cr}(z_{1},z_{2},z_{3},z_{4})=-\frac{(z_{1}-z_{4})(z_{2}-z_{3})}{(z_{1}-z_{2})(z_{3}-z_{4})}$

(chosen because of the normalization ${\rm cr}(0,1,\infty,z)=z$ for any
$z\in\mathbf{C}\setminus\\{0,1\\}$).

We identify $\boldsymbol{\mathcal{W}}_{0,n+3}$ with its pull-back under
$\psi$. As first integrals in the coordinates $x_{i}$'s for the foliations of
this web, we take the following rational maps (for $i$ ranging from 1 to $n$):

(48) $\displaystyle U_{i}(x)=$ $\displaystyle\,\,\Big{(}x_{j}\Big{)}_{j\leq
n,j\neq i}$ $\displaystyle
U_{n+1}(x)=\left(\frac{x_{j}-1}{x_{n}-1}\right)_{j\leq n-1}\vskip 12.0pt plus
4.0pt minus 4.0pt$ $\displaystyle U_{n+2}(x)=$
$\displaystyle\,\left(\frac{x_{j}}{x_{n}}\right)_{j\leq n-1}$ $\displaystyle
U_{n+3}(x)=\left(\frac{x_{j}(x_{n}-1)}{x_{n}(x_{j}-1)}\right)_{j\leq n-1}\,.$

We will denote by $\mathcal{F}_{i}$ the foliation admitting $U_{i}$ as a first
integral. For $i=1,\ldots,n+3$, one sets

(49) $\Omega_{i}=dU_{i,1}\wedge\ldots\wedge dU_{i,n-1}$

where $U_{i,s}$ stands for the $s$-th component of $U_{i}$ for any $s\leq
n-1$. Then one has $\Omega_{k}=\wedge_{l=1,\ldots,n,\,l\neq k}dx_{k}$ for
$k=1,\ldots,n$ hence these forms form a basis of the module of
$(n-1)$-differential forms on any open domain of $\mathbf{C}^{n}$.

We will use the following automorphisms of $\mathcal{M}_{0,n+3}$:

  * •

the cyclic shift
$\mathcal{C}:\,[0,1,\infty,x_{1},\ldots,x_{n}]\mapsto[1,\infty,x_{1},\ldots,x_{n},0]$;

  * •

the transposition
${{\mathcal{T}}}:\,[0,1,\infty,x_{1},\ldots,x_{n}]\mapsto[0,\infty,1,x_{1},\ldots,x_{n}]$.

We denote by $C$ and $T$ the two birational maps corresponding to
$\mathcal{C}$ and $\mathcal{T}$ but expressed in the coordinates $x_{i}$'s
(formally $C=\psi^{-1}\circ\mathcal{C}\circ\psi$ and similarly for $T$ with
respect to $\mathcal{T}$). One verifies easily that $C$ and $T$ are given by
the following formulas:

(50)
$C(x)=\bigg{(}\,\frac{1-x_{2}}{x_{1}-x_{2}},\frac{1-x_{3}}{x_{1}-x_{3}},\ldots,\frac{1-x_{n}}{x_{1}-x_{n}},\frac{1}{x_{1}}\,\bigg{)}\qquad\mbox{and}\qquad
T(x)=\bigg{(}\,\frac{x_{1}}{x_{1}-1}\,,\,\ldots\,,\,\frac{x_{n}}{x_{n}-1}\,\bigg{)}\,\,.$

######  Lemma 4.1.

The maps $C$ and $T$ are automorphisms of $\boldsymbol{\mathcal{W}}_{0,n+3}$
and their actions on the foliations $\mathcal{F}_{i}$ are as follow: for any
$i=1,\ldots,n+3$, one has

$C^{*}\big{(}\mathcal{F}_{i}\big{)}=\mathcal{F}_{c(i)}\qquad\mbox{ and }\qquad
T^{*}\big{(}\mathcal{F}_{i}\big{)}=\mathcal{F}_{\tau(i)}\,$

where $c$ stands for the $(n+3)$-cycle $(12\cdots(n+3))$ and $\tau$ for the
transposition $((n+2)(n+3))$.

The group generated by $C$ and $T$, denoted by $S_{n+3}$, is isomorphic to
$\mathfrak{S}_{n+3}$. More precisely, for any $G$ in this group, let
$\delta_{G}$ be the permutation of $\\{1,\ldots,n+3\\}$ such that
$G^{*}(\mathcal{F}_{i})=\mathcal{F}_{\delta_{G}(i)}$ for $i=1,\ldots,n+3$.
Then $\delta:S_{n+3}\rightarrow\mathfrak{S}_{n+3}$, $G\mapsto\delta_{G}$ is an
isomorphism.323232For $G,G^{\prime}\in S_{n+3}$, one has
$\delta_{G^{\prime}\circ G}=\delta_{G^{\prime}}\cdot\delta_{G}$ where $\cdot$
denotes the product on $\mathfrak{S}_{n+3}$ given by applying the permutations
according to their order of appearance from the right to the left, i.e. one
has
$(\delta_{G^{\prime}}\cdot\delta_{G})(k)=\delta_{G}(\delta_{G^{\prime}}(k))$
for any $k$. We will denote by $G_{\sigma}$ the birational map of
${\mathcal{M}}_{0,n+3}$ corresponding to $\sigma\in\mathfrak{S}_{n+3}$ up to
this isomorphism, that is

$G_{\sigma}=\delta^{-1}(\sigma)\in{\bf
Bir}\big{(}{\mathcal{M}}_{0,n+3}\big{)}\,.$

Note that $S_{n+3}$ identifies with the group of birational automorphisms of
$\boldsymbol{\mathcal{W}}_{0,n+3}$ (which coincides with the group of
automorphisms of ${\mathcal{M}}_{0,n+3}$).

The $\mathfrak{S}_{n+3}$-module structure of
$\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$ is as follows:
for any permutation $\sigma\in\mathfrak{S}_{n+3}$ and any abelian relation
$A=(A^{i}\cdot\Omega_{i})_{i=1}^{n+3}\in\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$,
one has

(51) $\sigma\cdot
A=G_{\sigma}^{*}\big{(}A\big{)}=\Big{(}G_{\sigma}^{*}\big{(}A^{i}\Omega_{i}\big{)}\Big{)}_{i=1}^{n+3}\,.$

For any $k=1,\ldots,n+3$ and any $(i,j)\in\boldsymbol{I}_{n}$, we denote by
$\boldsymbol{\mathcal{W}}_{\widehat{k}}$ and
$\boldsymbol{\mathcal{W}}_{\widehat{\imath\jmath}}$ the subwebs of
$\boldsymbol{\mathcal{W}}_{0,n+3}$ defined by the first integrals $U_{l}$ for
$l\in\\{1,\ldots,n+3\\}\setminus\\{k\\}$ and
$l\in\\{1,\ldots,n+3\\}\setminus\\{i,j\\}$ respectively. The corresponding
spaces of abelian relations will be denoted by $\boldsymbol{A}_{\widehat{k}}$
and $\boldsymbol{A}_{\widehat{\imath\jmath}}$ respectively. These vector
spaces can naturally be seen as subspaces of
$\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,n+3})$.

###  4.2. The main theorem.

Here we aim to give an almost complete proof of the theorem below, which
describes $\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,n+3})$ as a
$\mathfrak{S}_{n+3}$-module. This result contradicts, when $n$ is odd, one of
the main results claimed by Damiano. However the proof, whether $n$ is odd or
not, is essentially Damiano's one! The single novelties here are the few
details we have added and also a final argument, elementary but missed by
Damiano, showing that all ARs of $\boldsymbol{\mathcal{W}}_{0,n+3}$ are
combinatorial when $n$ is odd.

The approach we will follow is Damiano's and can be summarised as such:

  * $-$

we start by considering the $(n+1)$-subwebs
$\boldsymbol{\mathcal{W}}_{\widehat{\imath\jmath}}$ and their abelian
relations;

  * $-$

then we turn to the $(n+1)$-subwebs $\boldsymbol{\mathcal{W}}_{\widehat{k}}$
whose structure of the space of its abelian relations will be determined as an
$\mathfrak{S}_{n+2}$-module;

  * $-$

eventually we consider
$\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$ and establish
Theorem 4.2 below.

For any pair $(i,j)$ in $\boldsymbol{I}_{n}$, the space
$\boldsymbol{A}_{\widehat{\imath\jmath}}=\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{\widehat{\imath\jmath}}\big{)}$
is a subspace of dimension at most 1 of
$\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,n+3})$. Our goal here is first to
give an effective construction of a non trivial element in
$\boldsymbol{A}_{\widehat{\imath\jmath}}$. We will thus get a explicit family
of ARs generating $\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,n+3})$
which will make it possible to study this $\mathfrak{S}_{n+3}$-module more in
depth.

We first consider some explicit scalar quantities $A_{n+2,n+3}^{i}$'s for
$i=1,\ldots,n+3$: one has $A_{n+2,n+3}^{k}=0$ for $k=n+2,n+3$ and the other
$A_{n+2,n+3}^{i}$'s are given by the following formulas:

$\displaystyle
A_{n+2,n+3}^{i}=\frac{(-1)^{i-1}}{\prod_{k=1}^{n-1}\big{(}U_{i,k}-1\big{)}}\quad\mbox{for
}\;i=1,\ldots,n\quad\qquad\mbox{ and }\qquad\quad
A_{n+2,n+3}^{n+1}=\frac{(-1)^{n}}{\prod_{k=1}^{n-1}U_{n+1,k}}\,.$

Then it can be verified that the following $(n+3)$-tuple of differential forms

$AR_{n+2,n+3}=\left(A_{n+2,n+3}^{i}\cdot\Omega_{i}\right)_{i=1}^{n+3}$

is an abelian relation for $\boldsymbol{\mathcal{W}}_{\widehat{n+2,n+3}}$
(i.e. one has $\sum_{i=1}^{n+1}A_{n+2,n+3}^{i}\cdot\Omega_{i}=0$ as a rational
$(n-1)$-form on $\mathbf{C}^{n}$). Our approach to build the ARs of the other
subwebs $\boldsymbol{\mathcal{W}}_{\widehat{\imath\jmath}}$ consists in taking
the pull-backs of $AR_{n+2,n+3}$ under some birational isomorphisms the
constructions of which are not difficult to show.

One sets

  * •

$T_{k-1,k}=\big{(}C\big{)}^{-k}\circ T\circ\big{(}C\big{)}^{k}$ for
$k=2,\ldots,n+4$; and $T_{1,n+3}=T_{n+3,n+4}$;

  * •

$C_{1}={\bf I}{\rm d}$ and $C_{l}=T_{l-1,l}\circ T_{l-2,l-1}\cdots\circ
T_{1,2}$ for $l=2,\ldots,n+3$;

  * •

$T_{1,\ell}=\big{(}C_{\ell-1}\big{)}^{-1}\circ T_{\ell-1,\ell}\circ
C_{\ell-1}$ and $T_{2,\ell}=\big{(}T_{1,2}\big{)}^{-1}\circ T_{1,\ell}\circ
T_{1,2}$ for $\ell=3,\ldots,n+3$;

  * •

$T_{i,j}=T_{2,j}\circ T_{1,i}$ for any pair $(i,j)$ such that $3\leq i<j\leq
n+3$.

It follows from elementary standard computations in $\mathfrak{S}_{n+3}$ that
for any $(i,j)\in\boldsymbol{I}_{n}$, $T_{i,j}$ is an involution such that
$\delta(T_{i,j})=(1i)(2j)$. Consequently, for any such pair, the pull-back

$AR_{i,j}=\Big{(}T_{i,j}\circ T_{n+2,n+3}\Big{)}^{*}\left(AR_{n+2,n+3}\right)$

is a non trivial abelian relation of
$\boldsymbol{\mathcal{W}}_{\widehat{\imath\jmath}}$. Thus any space
$\boldsymbol{A}_{\widehat{\imath\jmath}}$ is spanned by $AR_{i,j}$ hence has
dimension 1 and $\\{AR_{{i,j}}\,\lvert\,(i,j)\in\boldsymbol{I}_{n}\,\\}$ is a
generating family for the space of combinatorial ARs. Because the
$AR_{{i,j}}$'s have been constructed effectively, this gives a way to
investigate
$\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,n+3})=\big{\langle}\,{AR}_{i,j}\,\lvert\,(i,j)\in\boldsymbol{I}_{n}\big{\rangle}$
as a $\mathfrak{S}_{n+3}$-module.

It will be useful to set ${AR}_{k,l}={AR}_{l,k}$ for any $k,l=1,\ldots,n+3$
with $n+3\geq k>l\geq 1$.

$\star$

For $k\leq n+1$, one denotes by $\boldsymbol{I}_{n}^{k}$ the set of $(i,j)$'s
in $\boldsymbol{I}_{n}$ such that $i=1,\ldots,k$ and $j=i+1,\ldots,n+2$, and
one defines

$\boldsymbol{J}_{n}=\begin{cases}\,\boldsymbol{I}_{n}^{n}\hskip
19.91684pt\mbox{when }n\mbox{ is even };\\\ \,\boldsymbol{I}_{n}^{n+1}\hskip
11.38092pt\mbox{when }n\mbox{ is odd }.\end{cases}$

Remark that one has
$\boldsymbol{I}_{n}^{n+1}=\boldsymbol{I}_{n}^{n}\cup\\{(n+1,n+2)\\}$ and ${\rm
Card}\big{(}\boldsymbol{I}_{n}^{n+1}\big{)}={\rm
Card}\big{(}\boldsymbol{I}_{n}^{n}\big{)}+1=(n+1)(n+2)/2$.

Our goal below is to prove the following

######  Theorem 4.2.

The following assertions hold true for any integer $n\geq 2$:

  1. 1.

If $n$ is odd, then:

    1. –

the abelian relations $AR_{ij}$'s for $(i,j)\in\boldsymbol{J}_{n}$ form a
basis of $\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,n+3})$ which has
dimension $(n+1)(n+2)/2$. Consequently
$\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,n+3})=\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,n+3})$
and the web $\boldsymbol{\mathcal{W}}_{0,n+3}$ has maximal rank;

    2. –

as a $\mathfrak{S}_{n+3}$-representation, the space
$\boldsymbol{AR}(\boldsymbol{\mathcal{W}}_{0,n+3})=\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,n+3})$
is irreducible and isomorphic to the $n$-th wedge product of the standard
$\mathfrak{S}_{n+3}$-representation (with associated Young symbol
$[3,1^{n}]$).333333Remember that for $m\geq 3$ and any $p\in\\{1,\ldots,m\\}$,
the $p$-th wedge product of the standard $\mathfrak{S}_{m}$-representation
with Young symbol $[m-1,1]$ is irreducible with Young symbol $[m-p,1^{p}]$
(see Proposition 3.12 and Exercise 4.6 in [FH]).

  2. 2.

When $n$ is even:

    1. –

the abelian relations $AR_{ij}$'s for $(i,j)\in\boldsymbol{J}_{n}$ form a
basis of $\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,n+3})$. Thus this
space has dimension $n(n+3)/2=(n+1)(n+2)/2-1$.

    2. –

as a $\mathfrak{S}_{n+3}$-representation, the space
$\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,n+3})$ of combinatorial ARs
is irreducible with associated Young symbol $\big{[}221^{n-1}\big{]}$.

###  4.3. The subwebs ${\mathcal{W}}_{\widehat{\imath}}$'s and their abelian
relations

Since $\mathfrak{S}_{n+3}$ acts transitively on the foliations of
$\boldsymbol{\mathcal{W}}_{0,n+3}$, any $(n+2)$-web
$\boldsymbol{\mathcal{W}}_{\widehat{\imath}}$ is isomorphic to
$\boldsymbol{\mathcal{W}}_{\widehat{n+3}}$. The latter is the linear web
formed by the $n+2$ linear families of lines passing through $n+2$ points in
general position in $\mathbf{P}^{n}$ (which can be taken to be the points
$p_{k}$ of §2.1.3, cf. (15)).

For any $i$, the subgroup $\mathfrak{S}_{n+2}\simeq{\rm
Fix}(i)<\mathfrak{S}_{n+3}$ acts on the space of combinatorial ARs

$\boldsymbol{A}_{C}(i)=\Big{\langle}\,AR_{i,j}\,\big{\lvert}\,j=1,\ldots,n+3,\,j\neq
i\,\Big{\rangle}\subset\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{\widehat{\imath}}\big{)}=\boldsymbol{A}(i)\,.$

From the action (51) of $\mathfrak{S}_{n+3}$ on
$\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$, one gets
a $\mathfrak{S}_{n+2}$-module structure on $\boldsymbol{A}_{C}(i)$, whose
structure is made clear thanks to the following

######  Proposition 4.3 (Damiano).

  1. 1.

The space $\boldsymbol{A}_{C}(i)$ has dimension $n+1$ and the abelian
relations $AR_{i,j}$'s for $j=1,\ldots,n+3$ with $j\neq i$ are in general
position in it: any set of $n+1$ of them form a basis of this space.

  2. 2.

Consequently, one has $\boldsymbol{A}_{C}(i)=\boldsymbol{A}(i)$ and
$\boldsymbol{\mathcal{W}}_{\widehat{\imath}}$ has maximal rank $n+1$.

  3. 3.

As a $\mathfrak{S}_{n+2}$-representation, $\boldsymbol{A}(i)$ is irreducible
with Young symbol $[2,1^{n}]$.

To prove this result, we will need the following lemma which will be used
again at the end:

######  Lemma 4.4.

For any elements $(i,j)$ and $(k,j)$ of $\boldsymbol{I}_{n}$ with
$\\{i,j\\}\cap\\{k,l\\}=\emptyset$, one has

(52) $(i,j)\cdot AR_{i,j}=(-1)^{n-1}\,AR_{i,j}\qquad\quad\mbox{ and
}\qquad\quad(k,l)\cdot AR_{i,j}=-AR_{i,j}\,.$

###### Proof.

Considering the action of $\mathfrak{S}_{n+3}$ on the foliations of
$\boldsymbol{\mathcal{W}}_{0,n+3}$, one can restrict oneself to the case when
$(i,j)=(n+2,n+3)$ and $(k,l)=(1,2)$.

In the rational coordinates $x_{1},\ldots,x_{n}$ we are working with, the
birational maps associated to the two corresponding transpositions are given
by

$G_{n+2,n+3}(x)=T(x)=\Big{(}{x_{i}}/({x_{i}-1})\Big{)}_{i=1}^{n}\qquad\mbox{
and }\qquad G_{1,2}(x)=\big{(}x_{2},x_{1},x_{3},\ldots,x_{n}\big{)}\,.$

Let $\tau$ stand for $(n+2,n+3)$ or $(1,2)$. Since $G_{\tau}$ leaves
$\boldsymbol{\mathcal{W}}_{\widehat{\imath\jmath}}$ invariant and because
$\tau$ is an involution, one has $\tau\cdot
AR_{n+2,n+3}=G_{\tau}^{*}\big{(}AR_{n+2,n+3}\big{)}=\epsilon_{\tau}\,AR_{n+2,n+3}$
for $\epsilon_{\tau}\in\\{\pm 1\\}$ to be determined.

To do so, it suffices to compare $G_{\tau}^{*}(A_{n+2,n+3}^{3}\,\Omega_{3})$
with $A_{n+2,n+3}^{3}\Omega_{3}$, which is easy to do. Indeed, one verifies
easily that $A_{n+2,n+3}^{3}\Omega_{3}$ is equal to the wedge product of the
logarithmic 1-forms $d\hskip 0.85355pt{\rm Log}(x_{\ell}-1)$ for
$\ell=1,2,\ldots,n$ distinct from 3:

$A_{n+2,n+3}^{3}\hskip 1.70709pt\Omega_{3}=d\hskip 0.85355pt{\rm
Log}\big{(}x_{1}-1\big{)}\wedge d\hskip 0.85355pt{\rm
Log}\big{(}x_{2}-1\big{)}\wedge\left(\,\bigwedge_{\ell=4}^{n}d\hskip
0.85355pt{\rm Log}\big{(}x_{\ell}-1\big{)}\,\right)\,.$

Since $G_{1,2}$ is just exchanging $x_{1}$ and $x_{2}$, one gets $(1,2)\cdot
A_{n+2,n+3}^{3}\,\Omega_{3}=-A_{n+2,n+3}^{3}\,\Omega_{3}$ hence
$\epsilon_{1,2}=-1$. Because taking the pull-bak under $T$ consists in
replacing $x_{i}$ by $x_{i}/(x_{i}-1)$, one has

${\rm Log}\left(\frac{x_{i}}{x_{i}-1}-1\right)={\rm
Log}\left(\frac{1}{x_{i}-1}\right)=-{\rm Log}\big{(}x_{i}-1\big{)}$

for any $i=1,\ldots,n$, which immediately gives us that
$T^{*}\big{(}A_{n+2,n+3}^{3}\,\Omega_{3}\big{)}=(-1)^{n-1}A_{n+2,n+3}^{3}\,\Omega_{3}$.
Hence one has $\epsilon_{n+2,n+3}=(-1)^{n-1}$, which finishes the proof of the
lemma. ∎

We now turn to the proof of the preceding proposition. The one below is
entirely taken from [Da1] (cf. §4.1.4 in it) and is reproduced here for the
sake of completeness. Next, in §4.3.1, we will describe (without proof) a more
geometric approach of Proposition 4.3 which is much more in the spirit of
classical web geometry.

###### Proof of Proposition 4.3.

It suffices to prove the proposition for $i=n+3$. To simplify the notation, we
set
$\boldsymbol{A}=\boldsymbol{A}(n+3)=\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{\widehat{n+3}}\big{)}$
and
$\boldsymbol{A}_{C}=\boldsymbol{A}_{C}(n+3)=\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{\widehat{n+3}}\big{)}$.
For $l=1,\ldots,n+2$, we denote by $q_{l}$ the point of
$\mathbf{P}(\boldsymbol{A})$ corresponding to $AR_{l,n+3}$. By definition,
$\mathbf{P}(\boldsymbol{A}_{C})$ is the subspace of
$\mathbf{P}(\boldsymbol{A})$ spanned by all the $q_{l}$'s and from Proposition
1.1, we know that $\dim\big{(}\mathbf{P}(\boldsymbol{A})\big{)}\leq n$.

Given $\sigma\in\mathfrak{S}_{n+2}$ and $q=[Q]\in\mathbf{P}(\boldsymbol{A})$
with $Q\in\boldsymbol{AR}(\boldsymbol{\mathcal{W}})\setminus\\{0\\}$, we set
$\sigma\cdot q=[\sigma\cdot Q]$. Then for any $i,j$ such that $1\leq i<j\leq
n+2$, the transposition $\sigma=(i,j)$ exchanges $q_{i}$ and $q_{j}$ and
admits all the other $q_{l}$'s (for $l\in\\{1,\ldots,n+2\\}$ distinct from $i$
and $j$) as fixed points.

Let $\kappa\in\\{1,\ldots,n\\}$ be the largest integer satisfying the property
that there exists a subset
$J=\\{{j_{1}},\ldots,j_{\kappa+2}\\}\subset\\{1,\ldots,n+2\\}$ of cardinality
$\kappa+2$ such that the points $q_{j_{1}},\ldots,q_{j_{\kappa+2}}$ are in
general position in their span $\mathbf{P}(J)=\langle q_{j}\,,\,j\in J\rangle$
which is of dimension $\kappa$. If $\kappa=n$ then 1. is proved hence assume
that it is not the case. Then for any $i\in\\{1,\ldots,n+2\\}\setminus J$, the
transposition $\sigma_{i}=(i,j_{1})$ lets $q_{j_{s}}$ fixed for any
$s=2,\ldots,\kappa+2$. Since these points form a projective frame of
$\mathbf{P}(J)$ according to the definition of $J$, we have that
$\sigma_{i}\cdot\mathbf{P}(J)=\mathbf{P}(J)$. On the other hand, $\sigma_{i}$
exchanges $q_{i}$ with $q_{j_{1}}$ hence we get that $q_{i}$ belongs to
$\mathbf{P}(J)$. Because this applies to any $i$ not in $J$, we obtain that
$\mathbf{P}(J)=\mathbf{P}(\boldsymbol{A}_{C})$. Assume that $\kappa<n-1$. Then
there exist $i,i^{\prime}\in\\{1,\ldots,n+2\\}\setminus J$ distinct. On the
one hand, the permutation $(i,i^{\prime})$ lets $q_{j}$ invariant for all
$j\in J$. Since these points are in general position in $\mathbf{P}(J)$, it
follows that $(i,i^{\prime})$ acts as the identity on this space. On the other
hand, $(i,i^{\prime})$ exchanges $q_{i}$ and $q_{i^{\prime}}$, which is a
contradiction. Thus necessarily, one has $\kappa=n-1$ or $\kappa=n$.

Since we are done in the latter case, let us assume than the former holds
true: one has $\kappa=n-1$. We denote by $i$ the element of
$\\{1,\ldots,n+2\\}$ not in $J$. Then $\mathfrak{S}_{n+1}\simeq{\rm
Fix}_{\mathfrak{S}_{n+2}}(i)$ acts on
$\mathbf{P}(J)=\mathbf{P}(\boldsymbol{A}_{C})\simeq\mathbf{P}^{n-1}$ inducing
the standard permutation action on the $n+1$ points
$q_{1},\ldots,\widehat{q_{i}},\ldots,q_{n+2}$. It is easily verified that
necessarily, the action on $\mathbf{P}^{n-1}$ comes from a linear action on a
vector space of dimension $n$ which is irreducible. But this action also lets
$q_{i}$ invariant, a contradiction.

We thus have proved 1. The second point follows immediately.

Let us turn to the structure of the space $\boldsymbol{A}$ as a
$\mathfrak{S}_{n+2}$-module. Picking a projective basis for
$\mathbf{P}(\boldsymbol{A})$, say the $(n+1)$-tuple $(q_{1},\ldots,q_{n+1})$,
we identify it with $\mathbf{P}^{n}$. From the arguments above, it follows
that the image of $\mathfrak{S}_{n+2}$ in ${\rm PGL}_{n+1}(\mathbf{C})$
corresponding to this identification can be described as the set of projective
transformations leaving $\\{q_{i}\\}_{i=1}^{n+1}$ invariant and inducing a
permutation of it. From this, one gets easily that
$\boldsymbol{A}_{\widehat{n+3}}$ necessarily is an irreducible
$\mathfrak{S}_{n+2}$-representation. Since moreover it has dimension $n+1$,
there are only two possibilities: either it is the standard representation or
its dual (with respective Young symbols $[n+1,1]$ and $[2,1^{n}]$).

From above, we know that the abelian relations $AR_{1,n+3}$ for
$i=1,\ldots,n+1$ form a basis of $\boldsymbol{A}=\boldsymbol{A}_{C}$ which has
dimension $n+1$. In order to recognize it as a
$\mathfrak{S}_{n+2}$-representation, we consider the matrix in this basis of
the linear involution associated to the transposition $(1,2)$, denoted by
$M_{(1,2)}$. From Lemma 4.4, we have $(1,2)\cdot AR_{j,n+3}=-AR_{j,n+3}$ for
any $j=3,\ldots,n+1$ on the one hand, whereas one has $(1,2)\cdot
AR_{k,n+3}=\epsilon_{k}\,AR_{l,n+3}$ for $k,l$ such that
$\\{k,l\\}=\\{1,2\\}$, with $\epsilon_{k},\epsilon_{l}\in\\{\pm 1\\}$ on the
other hand. It follows that ${\rm Tr}\big{(}M_{(1,2)}\big{)}=-(n-1)$. Since
the character map $\chi_{[n+1,1]}$ of the standard representation is such that
$\chi_{[n+1,1]}(\sigma)=\lvert{\rm Fix}(\sigma)\lvert-1$ for any
$\sigma\in\mathfrak{S}_{n+2}$ where ${\rm
Fix}(\sigma)\subset\\{1,\ldots,n+2\\}$ stands for the set of fixed points of
$\sigma$, we deduce that the space $\boldsymbol{A}(n+3)$ is not the standard
representation but its dual, which concludes the proof of the proposition. ∎

####  4.3.1. An alternative view on Proposition 4.3

Here we would like to explain another way to establish Proposition 4.3 much
more in the spirit of classical web geometry and which suggests an interesting
possible development (see §6.2 further). This approach is a rather
straightforward generalization for curvilinear $(n+2)$-web in dimension $n$ of
the classical linearization/algebraization of maximal rank planar 4-webs which
goes back to works by Lie, Poincaré and Blaschke (for which we refer to [PP,
§4.3]).

We denote by $V$ the complex vector space $\mathbf{C}^{n+1}$ the
projectivization of which is the ambiant projective space $\mathbf{P}^{n}$ we
are working in: $\mathbf{P}^{n}=\mathbf{P}(V)$. If $v_{1},\ldots,v_{n+1}\in V$
stand for the vectors of the canonical basis of $V$ and
$v_{n+2}=(1,\ldots,1)\in V$, the. one has $p_{i}=[v_{i}]$ for $i=1,\ldots,n+2$
where $[\cdot]:V\setminus\\{0\\}\rightarrow\mathbf{P}^{n}$ stands for the
standard projectivization map. We denote by $P$ the set of the $p_{i}$'s and
by $\boldsymbol{\mathcal{L}\mathcal{W}}_{P}$ the $(n+2)$-linear web on
$\mathbf{P}^{n}$ formed by the families of lines through the $p_{i}$'s. It is
nothing else but the subweb $\boldsymbol{\mathcal{W}}_{\widehat{n+3}}$ of
$\boldsymbol{\mathcal{W}}_{0,n+3}$.

Here we continue to work with the notations of §4.1. In particular, the
rational coordinates $x_{i}$'s are those associated to the birational map
$\mathbf{P}^{n}\dashrightarrow\overline{\mathcal{M}}_{0,n+3},\,(x_{1},\ldots,x_{n})\rightarrow[0,1,\infty,x_{1},\ldots,x_{n}]$.
The subgroup ${\rm Fix}(3)\subset\mathfrak{S}_{n+3}$ is isomorphic to
$\mathfrak{S}_{n+2}$ hence the latter group acts on $\mathbf{P}^{n}$ by
birational maps which (1) preserve the foliation by RNC of degree $n$ of
$\boldsymbol{\mathcal{W}}_{0,n+3}$ (2) are birational automorphisms of
$\boldsymbol{\mathcal{L}\mathcal{W}}_{P}$. We thus have a representation

(53) $\mathfrak{S}_{n+2}\simeq{\rm Fix}(3)\rightarrow{\rm
Bir}(\mathbf{P}^{n})\,.$

The transpositions $(i,j)$ with $3<i<j\leq n+3$, $(2,4)$ and $(1,2)$ generate
${\rm Fix}(3)$ and the corresponding Cremona transformations $G_{\sigma}$ for
such any transpositions $\sigma$ are easily seen to be given by

$G_{\sigma}(x)=\begin{cases}\hskip
2.84544pt\big{(}\,x_{\sigma(k+3)-3}\,\big{)}_{k=1}^{n}\,\quad\mbox{ for
}\sigma=(i,j)\\\ \hskip 2.84544pt\big{(}1-x_{k}\big{)}_{k=1}^{n}\hskip
28.45274pt\mbox{ for }\sigma=(1,2)\\\ \hskip
2.84544pt\Big{(}\frac{1}{x_{1}},\frac{x_{2}}{x_{1}},\ldots,\frac{x_{n}}{x_{1}}\Big{)}\quad\mbox{
for }\sigma=(2,4)\,.\end{cases}$

These birational maps are projectivizations of linear automorphisms
$\widehat{G}_{\sigma}$ of $V$ which it is straightforward to make explicit.
Using their explicit form, it is not difficult to verify that (53) is induced
by a linear representation which is irreducible. Since moreover one has ${\rm
Tr}(\widehat{G}_{\sigma})=n-1$, we obtain that this representation is
equivalent to the standard one. We thus have established the

Fact: The representation (53) is linear. More precisely, it is the
projectivization of a $\mathfrak{S}_{n+2}$-representation on $V$ which is
irreducible and isomorphic to the standard representation.

As in the preceding section, we denote by $\boldsymbol{A}$ the space of
abelian relations
$\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{L}\mathcal{W}}_{P}\big{)}=\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{\widehat{n+3}}\big{)}$.
For any $i=1,\ldots,n+2$, given $x\in\mathbf{P}^{n}$ generic, the map ${\rm
ev}_{i}:\boldsymbol{A}\rightarrow\mathbf{C},(A_{k}\,\Omega_{k})_{k=1}^{n+2}\mapsto
A_{i}(x)$ is a non trivial linear form hence ${\rm ker}({\rm ev}_{i})$ is a
hyperplane in $\boldsymbol{A}$. After projectivization, we get the $i$-th
canonical map of the web under scrutiny, which by definition is the map

$\textstyle{\kappa_{i}\,:}$$\textstyle{\mathbf{P}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\check{\mathbf{P}}^{n}=\mathbf{P}\big{(}V^{\vee}\big{)}}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\kappa_{i}(x)=\big{[}{\rm
ker}({\rm ev}_{i})\big{]}\,.}$

Clearly, each $\kappa_{i}$ is constant along the leaves of the $i$-th
foliation (which is the bundle $\boldsymbol{\mathcal{L}}_{p_{i}}$ of lines
through $p_{i}$) and, as a rational map, it factorizes through the map
$U_{i}:\mathbf{P}^{n}\rightarrow\mathbf{P}^{n-1}$. The map $\kappa_{i}$ has
maximal rank $n-1$ (generically), it is a canonical first integral for
$\boldsymbol{\mathcal{L}}_{p_{i}}$ and (the closure of) its image is the
hyperplane $H_{i}\subset\check{\mathbf{P}}^{n}$ which is dual to $p_{i}$:
considering the $v_{i}$'s as the canonical coordinates on $V^{\vee}$, $H_{i}$
is cut out by $v_{i}=0$ for $i=1,\ldots,n+1$ and by
$v_{n+2}=\sum_{i=1}^{n+1}v_{i}=0$. The union of the $H_{i}$'s, denoted by $H$,
is a hypersurface of $\check{\mathbf{P}}^{n}$, of degree $n+2$, cut out by
$v=v_{1}\cdots v_{n+2}=0$.

Any AR $(A_{i}\,\Omega_{i})_{i=1}^{n+2}$ is such that
$\sum_{i=1}^{n+2}A_{i}\,\Omega_{i}=0$ as a rational $(n-1)$-form. This
relation is equivalent to $n$ linearly scalar identities, from which it can be
deduced that generically, the $n+2$ points $\kappa_{i}(x)$'s actually span a
line, that we denote by $\boldsymbol{\mathfrak{L}}(x)$: one has

$\boldsymbol{\mathfrak{L}}(x)=\Big{\langle}\,\kappa_{i}(x)\,\big{\lvert}\,i=1,\ldots,n+2\Big{\rangle}\in
G_{1}\big{(}\check{\mathbf{P}}^{n}\big{)}\,.$

We then obtain a map $\boldsymbol{\mathfrak{L}}:\mathbf{P}^{n}\dashrightarrow
G_{1}\big{(}\check{\mathbf{P}}^{n}\big{)}$ which allows to get an interesting
geometric model of $\boldsymbol{\mathcal{L}\mathcal{W}}_{P}$. Indeed, the
closure $Z_{n}=\overline{\boldsymbol{\mathfrak{L}}(\mathbf{P}^{n})}$ is a
$n$-dimensional submanifold of the grassmannian variety of lines in
$\check{\mathbf{P}}^{n}$ and the push-forward onto it of
$\boldsymbol{\mathcal{L}\mathcal{W}}_{P}$ by $\boldsymbol{\mathfrak{L}}$ is
defined by the incidence in $\check{\mathbf{P}}^{n}$ between the lines
belonging to $Z_{n}$ and the points of the hyperplanes $H_{i}$'s. In other
terms, the following holds true:
Fact. The push-forward of $\boldsymbol{\mathcal{L}\mathcal{W}}_{P}$ by
$\boldsymbol{\mathfrak{L}}$ is the trace along $Z_{n}$ of the algebraic web
$\boldsymbol{\mathcal{W}}_{H}$ : one has

$\boldsymbol{\mathfrak{L}}_{*}\Big{(}\boldsymbol{\mathcal{L}\mathcal{W}}_{P}\Big{)}=\big{(}\boldsymbol{\mathcal{W}}_{H}\big{)}\big{\lvert}_{Z_{n}}\,.$

Now, as it classically follows from Abel's theorem and its converse, the ARs
of $\boldsymbol{\mathcal{W}}_{H}$ can be described in terms of the abelian
$(n-1)$-differential forms on $H$ (cf. [Bar, HP]). The maps
$\rho_{i}:G_{1}\big{(}\check{\mathbf{P}}^{n}\big{)}\dashrightarrow
H_{i},\,\ell\,\mapsto\rho_{i}(\ell)$ where $\rho_{i}(\ell)$ stands for the
intersection point of a generic line $\ell$ in $\check{\mathbf{P}}^{n}$ with
the hyperplane $H_{i}$ for each $i$, are natural rational first integrals of
$\boldsymbol{\mathcal{W}}_{H}$. Moreover, the trace gives rise to a linear
isomorphism
$\boldsymbol{H}^{0}\big{(}H,\omega_{H}^{n-1}\big{)}\longrightarrow\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{H}\big{)}$,
$\omega\longmapsto\big{(}\rho_{i}^{*}(\omega)\big{)}_{i=1}^{n+2}$.
Furthermore, it can be verified that the natural map
$\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{H}\big{)}\rightarrow\boldsymbol{AR}\big{(}(\boldsymbol{\mathcal{W}}_{H})\lvert_{Z_{n}}\big{)}$
is injective, hence up to pull-back under $\boldsymbol{\mathfrak{L}}$, we
obtain a linear isomorphism of complex vector spaces

(54)
$\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{L}\mathcal{W}}_{P}\big{)}\simeq\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{H}\big{)}\simeq\boldsymbol{H}^{0}\big{(}H,\omega_{H}^{n-1}\big{)}\,.$

On the other hand, we have the following short exact sequence of sheaves

(55)
$0\rightarrow\Omega^{n}_{\check{\mathbf{P}}^{n}}\longrightarrow\Omega^{n}_{\check{\mathbf{P}}^{n}}\otimes\mathcal{O}_{\check{\mathbf{P}}^{n}}(H)\stackrel{{\scriptstyle
R}}{{\longrightarrow}}\omega^{n-1}_{H}\rightarrow 0$

(where $\omega^{n-1}_{H}$ denotes the sheaf of abelian differentials of top
degree on $H$ and $R$ is the morphism of sheaves induced by Poincaré residue,
see [HP]). Since $\deg(H)=n+2$ and because
$\Omega^{n}_{\check{\mathbf{P}}^{n}}\simeq\mathcal{O}_{\check{\mathbf{P}}^{n}}(-n-1)$,
we have
$\Omega^{n}_{\check{\mathbf{P}}^{n}}\otimes\mathcal{O}_{\check{\mathbf{P}}^{n}}(H)\simeq\mathcal{O}_{\check{\mathbf{P}}^{n}}(1)$.
Since $\check{\mathbf{P}}^{n}=\mathbf{P}\big{(}V^{\vee})$,
$\boldsymbol{H}^{0}\big{(}\check{\mathbf{P}}^{n},\mathcal{O}_{\check{\mathbf{P}}^{n}}(1)\big{)}$
naturally identifies with the dual of $V^{\vee}$. Injecting this into the
first non trivial part of the long cohomology sequence associated to (55)
gives rise to a sequence of natural linear isomorphisms

(56)
$V=\big{(}V^{\vee}\big{)}^{\vee}=\boldsymbol{H}^{0}\Big{(}\check{\mathbf{P}}^{n},\mathcal{O}_{\check{\mathbf{P}}^{n}}(1)\Big{)}\simeq\boldsymbol{H}^{0}\Big{(}H,\omega^{n-1}_{H}\Big{)}\,.$

One verifies that the preceding isomorphisms actually are isomorphisms of
$\mathfrak{S}_{n+2}$-modules, when the action on
$\boldsymbol{H}^{0}\big{(}H,\omega^{n-1}_{H}\big{)}$ is induced by push-
forward by the birational maps $G_{\sigma}$ for $\sigma\in\mathfrak{S}_{n+2}$.
Considering (54), and because the action (51) is induced by pull-backs under
the $G_{\sigma}$'s, we get the

######  Proposition 4.5.

The action of $\mathfrak{S}_{n+2}$ by push-forward on
$\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{L}\mathcal{W}}_{P}\big{)}$ is
isomorphic to the action of $\mathfrak{S}_{n+2}$ on $V$ (hence is isomorphic
to the standard representation). Consequently, the $\mathfrak{S}_{n+2}$-action
(51) on this space of ARs is the dual action (irreducible module with Young
symbol $[2,1^{n}]$).

In addition to giving a more conceptual explanation for Proposition 4.3.3, the
material discussed above is interesting since it asks possibly fruitful
questions, in two distinct directions at least:

  1. $\bullet$

One may think that the construction of canonical maps and of a canonical
geometric model considered above for $\boldsymbol{\mathcal{L}\mathcal{W}}_{P}$
could be generalized to any $(n+2)$-web in dimension $n$ with maximal
$(n-1)$-rank. This suggests a possible approach for algebraizing such webs. We
will say a few words about this problem further in §6.2.

  2. $\bullet$

Another interesting question that emerges from the above considerations
concerns the varieties $Z_{n}$. Given a generic point $p\in\mathbf{P}^{n}$,
let $C_{p}$ be the rational normal curve in $\mathbf{P}^{n}$ passing through
all the $p_{i}$'s and $p$. Denoting by $\mathscr{L}_{p}$ the (embedded)
tangent line to $C_{p}$ at $p$, one gets a rational map
$\mathscr{L}:\mathbf{P}^{n}\dashrightarrow
G_{1}(\mathbf{P}^{n}),\,p\longmapsto\mathscr{L}_{p}$ whose image
$\mathscr{Z}_{n}=\overline{\mathscr{L}(\mathbf{P}^{n})}$ is a $n$-dimensional
submanifold of the grassmannian variety of lines in $\mathbf{P}^{n}$. For
$n=3$ and up to an identification of $\mathbf{P}^{n}$ with its dual
$\check{\mathbf{P}}^{n}$, we have verified that $\mathscr{Z}_{3}$ coincides
with $Z_{3}$ and that it is a degree six complex of lines in $\mathbf{P}^{3}$.
Contrarily to what we expected, we have not found any mention of it in the
classical literature on line complexes. Is it really new? And in higher
dimension, do $Z_{n}$ and $\mathscr{Z}_{n}$ coincide? In any case, we think
that they are interesting subvarieties of grassmannian varieties of lines that
deserve further study.

###  4.4. Proof of Theorem 4.2

This subsection is devoted to proving Theorem 4.2. We have reproduced
Damiano's arguments for completeness, with only minor modifications added in
order to make them clearer. Our only significant (although elementary)
contribution is Lemma 4.10 which definitively settles the case when $n$ is
odd.

We start with two technical lemmata taken from [Da1, §4]. To state the first,
one considers $n+2$ points $q_{1},\ldots,q_{n+2}$ in general position in
$\mathbf{P}^{n}$ together with an irreducible linear
$\mathfrak{S}_{n+2}$-action on this projective space which induces the natural
action by permutations on the $q_{i}$'s (i.e. one has $\sigma\cdot
q_{i}=q_{\sigma(i)}$ for every $i$ and every $\sigma\in\mathfrak{S}_{n+2}$).
Let $m\leq n+1$ be fixed. The subgroup $\cap_{s=m+2}^{n+2}{\rm Fix}(s)$ is
naturally isomorphic to $\mathfrak{S}_{m}$ and will be denoted in the same
way.

######  Lemma 4.6.

For any $q\in\mathbf{P}^{n}\setminus\langle q_{m+1},\ldots,q_{n+2}\rangle$,
one has $\big{\langle}\,\mathfrak{S}_{m}\cdot
q\,,\,q_{m+1}\,,\,\ldots\,,\,q_{n+2}\,\big{\rangle}=\mathbf{P}^{n}$.

###### Proof.

By induction on $m$, one can restrict oneself to only deal with the case when
$m=n+1$.

Let $V$ be the $\mathfrak{S}_{n+2}$-module of dimension $n+1$ whose
projectivization is the considered $\mathfrak{S}_{n+2}$-action on
$\mathbf{P}^{n}$. Since $V$ is irreducible and has dimension $n+1$, its Young
symbol $\lambda$ is either $[n+1,1]$ or $[2,1^{n}]$. It then follows
immediately from the branching rule (see [Sag, Theorem 2.8.3]) that the
restriction $V\downarrow_{\mathfrak{S}_{n+1}}$ is the sum of two irreducible
$\mathfrak{S}_{n+1}$-representations, one with Young symbol $\lambda_{-}$ of
dimension $n$, denoted by $V_{-}$, the other of dimension 1 denoted by $W$.
Since $q_{n+2}$ is fixed by $\mathfrak{S}_{n+1}={\rm Fix}(n+2)$, this point is
the projectivization $\mathbf{P}(W)$ of $W$. Any
$q\in\mathbf{P}^{n}\setminus\\{q_{n+2}\\}$ has a lift in $V$ which can be
written $\hat{q}_{-}+w$ with $\hat{q}_{-}\in V_{-}\setminus\\{0\\}$ and $w\in
W$. Denote by ${q}_{-}$the image of $\hat{q}_{-}$ in $\mathbf{P}^{n}$.

Clearly, one has $\big{\langle}\,\mathfrak{S}_{n+1}\cdot
q\,,\,q_{m+2}\,\big{\rangle}=\big{\langle}\,\mathfrak{S}_{n+1}\cdot
q_{-}\,,\,q_{m+2}\,\big{\rangle}$. On the other hand, since $V_{-}$ is an
irreducible $\mathfrak{S}_{n+1}$-module, the orbit $\mathfrak{S}_{n+1}\cdot
q_{-}$ spans the projectivization $\mathbf{P}(V_{-})$ of $V_{-}$. Thus the
linear span of $\mathfrak{S}_{n+1}\cdot q$ and $q_{m+2}$ contains both
$\mathbf{P}(V_{-})$ and $\mathbf{P}(W)$ hence the lemma follows. ∎

The previous lemma will be used in the proof of the second one, which is the
following:

######  Lemma 4.7.

Let $\\{\,q_{i,j}=q_{j,i}\,\lvert\,(i,j)\in\boldsymbol{I}_{n}\,\\}$ be a
family of $(n+3)(n+2)/2$ points in a projective space $\mathbf{P}^{N}$. Assume
that these points satisfy the following hypotheses:

  1. $(H_{1})$.

For any $i=1,\ldots,n+3$, the $n+2$ points $q_{i,j}$'s for $j\neq i$ are in
general position in their span, which is a linear subspace of dimension $n$ of
$\mathbf{P}^{N}$, denoted by $\mathbf{P}^{n}_{i}$.

  2. $(H_{2})$.

For any ${(i,j)}\in\boldsymbol{I}_{n}$, the intersection of
$\mathbf{P}^{n}_{i}$ with $\mathbf{P}^{n}_{j}$ consists of the point
$q_{i,j}$;

  3. $(H_{3})$.

There is a linear $\mathfrak{S}_{n+3}$-representation on $\mathbf{P}^{N}$ such
that :
(a). for every $\sigma\in\mathfrak{S}_{n+3}$ and any
$(i,j)\in\boldsymbol{I}_{n}$, one has $\sigma\,q_{i,j}=q_{\sigma i,\sigma j}$
;
(b). the restricted action of $\mathfrak{S}_{n+2}\simeq{\rm
Fix}_{\mathfrak{S}_{n+3}}(i)$ on $\mathbf{P}^{n}_{i}$ is irreducible for any
$i\leq n+3$.

Then one necessarily has $N\geq(n+2)(n+1)/2-2$.

###### Proof.

There is no loss of generality by assuming that the ambiant projective space
$\mathbf{P}^{N}$ is spanned by the $q_{i,j}$'s, which we do in what follows.

For $\ell\in\\{1,\ldots,n+3\\}$, we set
$P_{\ell}=\big{\langle}\,\mathbf{P}^{n}_{1},\ldots,\mathbf{P}^{n}_{\ell}\,\big{\rangle}\subset\mathbf{P}^{N}$
and one denotes (a bit abusively) by $\mathfrak{S}_{n+3-\ell}$ the subgroup
$\cap_{i=1}^{\ell}{\rm
Fix}_{\mathfrak{S}_{n+3}}(i)=\mathfrak{S}_{\\{\ell+1,\ldots,n+3\\}}$ of
$\mathfrak{S}_{n+3}$. The $P_{\ell}$'s are linear subspaces of
$\mathbf{P}^{N}=P_{n+3}$ which form an increasing sequence (for the
inclusion). First remark that for any $\ell=1,\ldots,n+2$, since
$q_{i,\ell+1}\in\mathbf{P}^{n}_{i}\subset P_{\ell}$ for $i=1,\ldots,\ell$ and
because these points are in general position in $\mathbf{P}^{n}_{\ell+1}$ (by
$(H_{1})$), one has
$\dim\,\big{(}P_{\ell}\cap\mathbf{P}^{n}_{\ell+1}\big{)}\geq\max(\ell-1,n)$.

One has $\dim\,\big{(}P_{1}\cap\mathbf{P}^{n}_{2}\big{)}=0$ thanks to
$(H_{2})$ hence the set of indices $\ell$ such that
$\dim\,\big{(}P_{\ell}\cap\mathbf{P}^{n}_{\ell+1}\big{)}=\ell-1$ is not empty.
Its maximal element, denoted by $L$, is a well defined element of
$\\{1,\ldots,n+1\\}$. We are going to prove that $P_{L+1}=\mathbf{P}^{N}$ with
$L\geq n-1$, from which it will be easy to get the looked sought-after
minoration.

For $\ell$ such that $1\leq\ell\leq L$, the points
$q_{1,\ell+1},\ldots,q_{\ell,\ell+1}$ all belong to $P_{\ell+1}$. Since
$\dim\big{(}P_{\ell}\cap\mathbf{P}^{n}_{\ell+1}\big{)}$ has dimension $\ell-1$
and because of $(H_{1})$, we get that the points $q_{i,\ell+1}$ for
$i=\ell+1,\ldots,n+2$ are projectively linearly independent modulo $P_{\ell}$.
By induction, we deduce that

  1. 1.

the points $q_{i,j}$'s for $i=1,\ldots,\ell$ and $j=i+1,\ldots,n+2$ form a
projective frame for $P_{\ell}$;

  2. 2.

hence one has $\dim(P_{\ell+1})=N(\ell)$ with
$N(\ell)=\Big{[}\,\big{(}n+1\big{)}+\big{(}n+1-1\big{)}+\cdots+\big{(}n+1-\ell\big{)}\,\Big{]}-1$.

It follows that in order to prove the proposition, one has to establish that
$L\geq n-1$. Assuming the contrary and setting $k=L+1$, one has $k\leq n-1$
and $\dim\,\big{(}P_{k}\cap\mathbf{P}^{n}_{k+1}\big{)}>k-1$ hence there exists
a point $q\in
P_{k}\cap\mathbf{P}^{n}_{k+1}\setminus\big{\langle}q_{1,k+1},\ldots,q_{k,k+1}\big{\rangle}$.
The group $\cap_{i=1}^{k+1}{\rm Fix}(i)\subset\mathfrak{S}_{n+2}$, denoted a
bit abusively by $\mathfrak{S}_{n+3-(k+1)}$ here, acts on
$\mathbf{P}^{n}_{k+1}$. Thanks to hypothesis $(H_{3}.b)$, Lemma 4.6 applies
and gives us that $\mathbf{P}^{n}_{k+1}$ is spanned by the union of
$\\{q_{1,k+1},\ldots,q_{k,k+1}\\}$ with the $\mathfrak{S}_{n+3-(k+1)}$-orbit
of $q$: one has

(57) $\Big{\langle}\,\mathfrak{S}_{n+3-(k+1)}\cdot
q\,\,,\,q_{1,k+1}\,,\,\ldots\,,\,q_{k,k+1}\,\Big{\rangle}=\mathbf{P}^{n}_{k+1}\,.$

Since $q$ belongs to $P_{k}$ which is invariant under the action of
$\mathfrak{S}_{n+3-(k+1)}=\cap_{i=1}^{k}{\rm Fix}(i)$, one has that
$\mathfrak{S}_{n+3-k}\cdot q\subset P_{k}$. Combined with the fact that
$q_{i,k+1}\in\mathbf{P}^{n}_{i}\subset P_{k}$ for $i=1,\ldots,k$, it then
follows from (57) that $\mathbf{P}^{n}_{k+1}\subset P_{k}$. On the other hand,
for any $l>k+1$, the permutation $(k+1,l)\in\mathfrak{S}_{n+3}$ interchanges
$\mathbf{P}^{n}_{k+1}$ and $\mathbf{P}^{n}_{l}$ while leaving $P_{k}$ fixed.
One deduces that $\mathbf{P}^{n}_{l}\subset P_{k}$ for any $l=k+1,\ldots,n+3$
hence $P_{k}=P_{n+3}=\mathbf{P}^{N}$.

Let $F=\big{\\{}\,p\in\mathbf{P}^{N}\,\big{\lvert}\,\tau\cdot
p=p\,\big{\\}}\subset\mathbf{P}^{N}$ be the fixed point set of
$\tau=(n+2,n+3)\in\mathfrak{S}_{n+3}$. From $(H_{1})$, it comes that the set
of $q_{i,j}$'s with $i=1,\ldots,k$ and $j=i+1,\ldots,n+2$ spans $P_{k}$. But
since any such point $q_{i,j}$ belongs to $F$ if $j\leq n+1$, it comes that

(58) $\dim\big{(}P_{k}\big{)}\leq\dim\big{(}F\big{)}+{\rm
Card}\Big{(}\,\big{\\{}\,q_{1,n+2},\ldots,q_{k,n+2}\,\big{\\}}\,\Big{)}=\dim\big{(}F\big{)}+k\,.$

On the other hand, since $\tau\big{(}F\big{)}=F$ and
$\tau\big{(}\mathbf{P}^{n}_{n+3}\big{)}=\mathbf{P}^{n}_{n+2}$, it follows that
$F\cap\mathbf{P}^{n}_{n+3}=\\{\,q_{n+2,n+3}\,\\}$. Hence
$\dim\big{(}F\cap\mathbf{P}^{n}_{n+3}\big{)}=0$ so ${\rm
codim}_{\mathbf{P}^{N}}(F)\geq n$, which can also be written

(59) $\dim\big{(}F\big{)}+n\leq\dim\big{(}P_{k}\big{)}\,.$

From (58) and (59) together, one gets $n\leq k$ contradicting the assumption
$L<n-1$. Thus necessarily $L\geq n$ which gives us the proposition. ∎

This lemma has an interesting consequences when applied to
$\mathbf{P}^{N}=\mathbf{P}\big{(}\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,n+3})\big{)}$
with $q_{i,j}=\mathbf{P}\big{(}\langle AR_{i,j}\rangle\big{)}$ for any
$i,j=1,\ldots,n+3$ with $i\neq j$. For any subset
$J\subset\\{1,\ldots,n+3\\}$, one sets $\boldsymbol{A}_{J}=\langle\hskip
4.26773pt\boldsymbol{A}_{\widehat{\jmath}}\hskip 4.26773pt\lvert\,j\in
J\rangle\subset\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$
and

(60) $\eta=\min_{{J\subset\\{1,\ldots,n+3\\}}}\Big{\\{}\hskip
4.26773pt\lvert\,J\,\lvert\hskip 4.26773pt\big{\lvert}\hskip
4.26773pt\boldsymbol{A}_{J}=\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}\hskip
4.26773pt\Big{\\}}\,.$

Then, from the previous lemma as well as from its proof, one deduces the

######  Corollary 4.8.

  1. 1. Either $\eta=n$ or $\eta=n+1$.

  2. 2.

In both cases, one has
$\boldsymbol{A}_{J}=\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$
for any subset $J$ of cardinality $\eta$ and the $AR_{i,j}$'s for
$i=1,\ldots,\eta$ and $j=i+1,\ldots,n+2$ form a basis of
$\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$.

  3. 3.

Consequently,
$\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$ has
dimension $(n+1)(n+2)-1$ if $\eta=n$ and dimension one more when $\eta=n+1$.
In the latter case, $\boldsymbol{\mathcal{W}}_{0,n+3}$ has maximal rank hence
necessarily
$\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}=\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$.

Another nice feature of Lemma 4.7 is that it is rather easy to deduce from it
a description of
$\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$ as a
$\mathfrak{S}_{n+3}$-module:

######  Corollary 4.9.

  1. 1. As a $\mathfrak{S}_{n+3}$-representation, $\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$ is irreducible.

  2. 2.

Its Young symbol is $[221^{n-1}]$ if $\eta=n$ and $[31^{n}]$ when $\eta=n+1$.

###### Proof.

Assume that
$\boldsymbol{A}_{C}=\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$
is not irreducible. Pick $F\subset\boldsymbol{A}_{C}$, a non trivial and
proper stable sub-$\mathfrak{S}_{n+3}$-module of maximal dimension. Since the
$\mathfrak{S}_{n+3}$-orbit of any $\boldsymbol{A}_{\widehat{\imath}}$
coincides with the whole space
$\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$, we have
that $F\cap\boldsymbol{A}_{\widehat{\imath}}=\\{0\\}$ for any
$i=1,\ldots,n+3$, hence ${\rm codim}(F)\geq n+1$.

Let
$\pi:\mathbf{P}\boldsymbol{A}_{C}\setminus\mathbf{P}F\rightarrow\mathbf{P}(\boldsymbol{A}_{C}/F\big{)}=\mathbf{P}^{M}$
with $M=\dim\big{(}\boldsymbol{A}_{C}\big{)}-\dim(F)-1$. For any
$i=1,\ldots,n+3$, since
$\mathbf{P}F\cap\mathbf{P}\boldsymbol{A}_{\widehat{\imath}}=\emptyset$, it
follows that $\pi\big{(}\mathbf{P}\boldsymbol{A}_{\widehat{\imath}}\big{)}$
has dimension $n$, which justifies denoting this subspace by
$\mathbf{P}_{i}^{n}$. In particular, one has $M\geq n$.

For any $i,j$ distinct, let us prove that
$\mathbf{P}^{n}_{i}\cap\mathbf{P}^{n}_{j}=q_{i,j}=\pi\big{(}p_{i,j}\big{)}$
with $p_{i,j}=[AR_{i,j}]\in\mathbf{P}\boldsymbol{A}_{C}$. Otherwise, there
would exist another point $q$ in this intersection, distinct from $q_{i,j}$.
We denote here by $\mathfrak{S}_{n+2}$ (resp. $\mathfrak{S}_{n+1}$) the
subgroup of $\mathfrak{S}_{n+3}$ formed by permutations fixing $i$ (resp.
fixing both $i$ and $j$): one has $\mathfrak{S}_{n+1}={\rm Fix}(i)\cap{\rm
Fix}(j)<\mathfrak{S}_{n+2}={\rm Fix}(i)<\mathfrak{S}_{n+3}$. Note that the
action of $\mathfrak{S}_{n+2}$ on $\mathbf{P}^{n}_{i}$ is irreducible with
Young symbol $[2,1^{n}]$ according to Proposition 4.3.3. On the other hand

$\big{[}2,1^{n}\big{]}\,\big{\lvert}_{{\mathfrak{S}}_{n+1}}=\big{[}1^{n+1}\big{]}\oplus\big{[}2,1^{n-1}\big{]}$

as it follows immediately from the classical branching rules for the
representations of symmetric groups (see [Sag, §2.8]). This means that the
representation of $\mathfrak{S}_{n+1}$ on $\mathbf{P}^{n}_{i}$ induced by
restriction admits a unique fixed point, which is $q_{i,j}$ of course, and
acts irreducibly on a supplementary hyperplane. It follows that the orbit
$\mathfrak{S}_{n+1}\cdot q$ spans a hyperplane in $\mathbf{P}^{n}_{i}$ which
does not contain $q_{i,j}$ hence one has
$\big{\langle}q_{i,j}\,,\,\mathfrak{S}_{n+1}\cdot
q\big{\rangle}=\mathbf{P}^{n}_{i}$ (cf. also Lemma 4.6). Since all these
arguments also apply when exchanging $i$ and $j$, one has
$\big{\langle}q_{i,j}\,,\,\mathfrak{S}_{n+1}\cdot
q\big{\rangle}=\mathbf{P}^{n}_{j}$ as well, from which it comes that
$\mathbf{P}^{n}_{i}=\mathbf{P}^{n}_{j}$. Since all $\mathbf{P}^{n}_{i}$'s
coincide and because their union spans $\mathbf{P}^{M}$, we get that $M=n$,
hence that ${\rm codim}(F)=n+1$. But this is impossible: indeed, from our
assumption of the maximality of $F$ (as a proper submodule of
$\boldsymbol{A}_{C}$), it comes that $\boldsymbol{A}_{C}/F$ is an irreducible
$\mathfrak{S}_{n+3}$-representation of dimension $n+1$ and such a thing does
not exist when $n\geq 2$.

From the above, it comes that $\mathbf{P}^{M}$ and the points $q_{i,j}$ with
$(i,j)\in\boldsymbol{I}_{n}$ satisfy all the assumptions of Lemma 4.7 which
therefore applies and gives us that $M\geq(n+1)(n+2)-2$. Since $F$ has been
assumed to be non trivial and proper, the single possibility is that
$\boldsymbol{A}_{C}$ and $F$ have dimension $(n+1)(n+2)/2$ and 1 respectively.
But since any $\mathfrak{S}_{n+3}$-representation is completely reducible, $F$
must admit a complementary invariant subspace of dimension
$(n+1)(n+2)-1>1=\dim(F)$, contradicting the maximality of $\dim(F)$ among the
non trivial proper sub-representations of $\boldsymbol{A}_{C}$. Thus there is
no such a sub-representation, which proves the first point.

Let $[\lambda]$ be the Young diagram of
$\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$ as a
$\mathfrak{S}_{n+3}$-module, for a partition $\lambda$ of $n+3$. Since
$\mathfrak{S}_{n+2}={\rm Fix}(n+3)$ acts irreducibly on
$\boldsymbol{A}_{\widehat{n+3}}$ with Young symbol $[2,1^{n}]$ according to
the third point of Proposition 4.3, it comes that
$[\lambda]\,\big{\lvert}_{\mathfrak{S}_{n+2}}$ admits $[2,1^{n}]$ as one of
its irreducible factors. Since $[\lambda]$ is obtained by adding one box to
$[2,1^{n}]$, we deduce that there are only two possibilities for the former
Young symbol: it is either $[3,1^{n}]$ which has dimension $(n+1)(n+2)/2$, or
$[2,2,1^{n}]$, which is of dimension one less. This finishes the proof of the
proposition. ∎

######  Lemma 4.10.

When $n$ is odd, $AR_{n+2,n+3}$ does not belong to
$\Big{\langle}\,AR_{i,j}\hskip
2.84544pt\big{\lvert}\,{\begin{tabular}[]{l}$i=1,\ldots,n$\vspace{-0.09cm}\\\
$$ $j=i+1,\ldots,n+2$\end{tabular}}\Big{\rangle}$.

###### Proof.

Assume that there exist scalars $c_{i,j}$ for $(i,j)\in\boldsymbol{I}_{n}^{n}$
such that

(61) $A_{n+2,n+3}=\sum_{(i,j)\in\boldsymbol{J}_{n}}c_{i,j}\,AR_{i,j}\,.$

From the second point of Corollary 4.8, it follows that this relation is
necessarily unique.

On the other hand, for any $(k,l)$ with $k=1,\ldots,n$ and $l=k+1,\ldots,n+1$,
it follows from Lemma 4.4 that $(k,l)\cdot A_{n+2,n+3}=-A_{n+2,n+3}$ and
$(k,l)\cdot A_{k,l}=(-1)^{n-1}A_{k,l}=A_{k,l}$ (since $n-1$ is even). Applying
$(k,l)$ to (61) would give another such identity if $c_{k,l}$ were not zero,
and this is not possible by the argument given just above. We thus get that
all the constants $c_{i,j}$ are zero for $i=1,\ldots,n$ and $j=1,\ldots,n+1$.

Hence (61) actually is written

$A_{n+2,n+3}=\sum_{i=1}^{n}c_{i,n+2}\,AR_{i,n+2}\,.$

and such a relation does not exist since, as it follows from the first point
of Proposition 4.3, the combinatorial ARs appearing in it form a free family.
Thus there is no identity of the form (61), which gives us the lemma. ∎

With the previous results at hand, it is then easy to establish Theorem 4.2:

###### Proof of Theorem 4.2..

When $n$ is odd, it follows from Lemma 4.10 that the integer $\eta$ defined in
(60) is equal to $n+1$. Hence
$\boldsymbol{AR}_{C}\big{(}\boldsymbol{\mathcal{W}}_{0,n+3}\big{)}$ has
dimension $(n+1)(n+2)/2$ from which it follows that
$\boldsymbol{\mathcal{W}}_{0,n+3}$ has maximal rank with all its ARs
combinatorial. Then Corollary 4.9.2 allows us to conclude.

To prove the theorem in case when $n$ is even, it suffices to prove that
$\eta=n$, which would follow from exhibiting a non trivial abelian relation
which is not combinatorial. We will show that the so-called Euler's abelian
relation does the job, hence the theorem follows. ∎

### 4.5.

Leaving aside the question of the structure of
$\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,n+3})$ as a
$\mathfrak{S}_{n+3}$-module, it is not difficult to describe another rather
efficient computational approach for building the combinatorial abelian
relations. To this end, we first need to set some notation which will be also
be used later on.

Given $x=(x_{1},\ldots,x_{n})$, we use the shorthand
$x^{\prime}=(x_{2},\ldots,x_{n})=U_{1}(x)$; we introduce two new but fixed
variables $x_{n+1}=0$ and $x_{n+2}=1$; we set
$\tilde{x}=(x_{1},\ldots,x_{n},x_{n+1},x_{n+2})$; and for $i,j$ such that
$i=1,\ldots,n+1$ and $j=i+1,\ldots,n+2$, we set
$\tilde{x}_{\widehat{\imath\jmath}}=(x_{s})_{s=1,s\neq i,j}^{n+2}$. We allow
ourselves to combine all these notation: for instance, for $i,j$ as above
$\tilde{x}^{\prime}_{\widehat{\imath\jmath}}$ stands for the $(n-1)$-tuple
obtained by removing the first, the $i$th and the $j$-th entries to
$\tilde{x}$. Since we will always deal with pairs of indices $(i,j)$ with
$2\leq i\leq n$, there will be no ambiguity with this notation in what
follows. We denote by $B_{n}$ the set of pairs we will work with:

$B_{n}=\Big{\\{}\,\big{(}i,j\big{)}\hskip 2.84544pt\big{\lvert}\hskip
2.84544pti=2,\ldots,n,\,j=i+1,\ldots,n+2\,\Big{\\}}\,.$

For any $N>0$, we consider the following homogeneous polynomial of
$z\in\mathbf{C}^{N}$:

$\displaystyle M_{0,N}(z)=$ $\displaystyle\,\prod_{1\leq p<q\leq
N}\big{(}z_{p}-z_{q}\big{)}\,.$

######  Remark 4.11.

For $N=n+3$, one has $M_{0,N}(\tilde{x})=-\prod_{1\leq i\leq
n}x_{i}(x_{i}-1)\prod_{1\leq i<j\leq n}(x_{i}-x_{j})$ for any
$x\in\mathbf{C}^{n}$ hence the equation $M_{0,N}(\tilde{x})=0$ cuts out the
braid arrangement $A_{n+3}$ in $\mathbf{C}^{n}$, whose complement
$\mathbf{C}^{n}\setminus A_{n+3}$ is isomorphic to the moduli space
$\mathcal{M}_{0,n+3}$ (see (143)). This explains the notation
$M_{0,N}=M_{0,n+3}$ for the polynomial above.

We define rational functions $F_{0}$ and $F_{ij}$ for $(i,j)\in B_{n}$, of
$x^{\prime}$, by setting

(62) $\displaystyle F_{0}(x^{\prime})=\frac{1}{x_{2}\ldots x_{n}}\qquad\mbox{
and }\qquad
F_{ij}(x^{\prime})=\Big{(}\tilde{x}_{i}-\tilde{x}_{j}\Big{)}^{n-1}\frac{M_{0,n-1}\Big{(}\tilde{x}^{\prime}_{\widehat{\imath\jmath}}\Big{)}}{M_{0,n+1}\big{(}\tilde{x}^{\prime}\big{)}}\quad\mbox{for
any }\,(i,j)\in B_{n}\,.$

Then we set ${B}_{n}(x^{\prime})=\\{\,F_{ij}\,\\}_{(i,j)\in B_{n}}$ and

(63)
$\displaystyle\boldsymbol{\mathfrak{B}}_{n}(x^{\prime})=\begin{cases}\hskip
2.84544pt{B}_{n}(x^{\prime})\hskip 39.83368pt\mbox{ if }n\mbox{ is
even}\,;\vspace{0.1cm}\\\ \hskip
2.84544pt{B}_{n}(x^{\prime})\cup\big{\\{}\,F_{0}\,\big{\\}}\hskip
5.69046pt\mbox{ if }n\mbox{ is odd}\,.\end{cases}$

The cardinal of $\boldsymbol{\mathfrak{B}}_{n}(x^{\prime})$ is $(n-1)(n+2)/2$
for $n$ even and $n(n+1)/2=(n-1)(n+2)/2+1$ otherwise. In view of describing
the components of the ARs of $\boldsymbol{\mathcal{W}}_{0,n+3}$, we set

(64) $\bullet$ $V_{1}(x)=U_{1}(x)=x^{\prime}=(x_{2},\ldots,x_{n})$ and
$\Pi_{1}=\Omega_{1}=dx_{2}\wedge\cdots\wedge dx_{n}$;
---
$\bullet$ $V_{i}(x)=\big{(}C^{\circ(i-1)}\big{)}^{*}\big{(}V_{1}\big{)}$ and
$\Pi_{i}=\big{(}C^{\circ(i-1)}\big{)}^{*}\big{(}\Pi_{1}\big{)}$ for
$i=2,\ldots,n+3$;
$\bullet$
$\boldsymbol{\mathfrak{B}}_{n}(V_{i})=\big{(}C^{\circ(i-1)}\big{)}^{*}\Big{(}\boldsymbol{\mathfrak{B}}_{n}(x^{\prime})\Big{)}=\big{\\{}\,F\circ
V_{i}\,\big{\\}}_{F\in\boldsymbol{\mathfrak{B}}_{n}(x^{\prime})}$ for
$i=1,\ldots,n+3$.

By considering their respective poles, one gets easily that the elements of
$\boldsymbol{\mathfrak{B}}_{n}(x^{\prime})$ are linearly independent. Then, by
direct elementary but computational checks in what concerns the second point,
we prove the

######  Proposition 4.12.

1. For any $n\geq 2$, the elements of $\boldsymbol{\mathfrak{B}}_{n}(x^{\prime})$ are linearly independent over $\mathbf{C}$;

2. The following holds true for any $n\leq 12$: for any $i=1,\ldots,n+3$, the space $\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,n+3})[i]$ of $V_{i}$-th components of combinatorial abelian relations admits $\big{\\{}\,F_{i}\,\Pi_{i}\,\lvert\,F_{i}\in\boldsymbol{\mathfrak{B}}_{n}(V_{i})\,\big{\\}}$ as a basis.

The interest of this result is threefold: first and of course, we conjecture
that it holds true in full generality. Second, it indicates that
$\boldsymbol{AR}_{C}(\boldsymbol{\mathcal{W}}_{0,n+3})$ can be determined by
pure linear algebra in the finite dimensional direct sum
$\oplus_{i=1}^{n+3}\boldsymbol{\mathfrak{B}}_{n}(V_{i})$. This elementary fact
opens the door for the possibility to arrive at an explicit closed formula for
(a multiple of) the non trivial abelian relation $AR_{i,j}$ of
$\boldsymbol{AR}_{\widehat{\imath\jmath}}$ discussed just above, this for any
$(i,j)\in\boldsymbol{I}_{n}$. Finally, as we will see in the next section, the
proposition above and more generally the notation and formulas of the current
subsection are useful to explicitly describe Euler's abelian relation.

######  Remark 4.13.

One can verify that, if taking instead of the map $C$ in (50) the cyclic
birational automorphism of $\boldsymbol{\mathcal{W}}_{0,n+3}$ given in the
coordinates $x_{i}$'s by

$\tilde{C}(x)=\left(\,\frac{x_{n}}{x_{n}-x_{i-1}}\,\right)_{i=1}^{n}$

(with the convention that $x_{0}=1$), then all the preceding definitions make
sense and all the subsequent results (especially the preceding proposition)
hold true mutatis mutandis. This will be used later on in §5.1.7.

##  5\. On Euler's abelian relation.

In this section, we discuss Euler's AR. Since every AR of
$\boldsymbol{\mathcal{W}}_{0,n+3}$ is combinatorial hence rational when $n$ is
odd, it is certain that there is something wrong in Damiano's claims (8) about
this particular abelian relation for this parity of $n$. In order to make
everything as clear as possible, in this section we present a thorough study
of this AR and before that, of its construction.

Following Damiano but giving more details, we first describe the construction
of a new abelian relation $\boldsymbol{\mathcal{E}}_{n}$ for
$\boldsymbol{\mathcal{W}}_{0,n+3}$ using Gelfand-MacPherson's theory applied
to a characteristic class on the oriented Grassmannian variety
$G_{2}^{or}(\mathbf{R}^{n+3})$ of oriented 2-planes in $\mathbf{R}^{n+3}$. In
§5.1.4, we give a rather concise integral formula for the components of
$\boldsymbol{\mathcal{E}}_{n}$ which we use to determine the fourth-order jet
of $\boldsymbol{\mathcal{E}}_{3}$ at a specific point. Then in §5.1.5, we
discuss some invariance properties satisfied by $\boldsymbol{\mathcal{E}}_{n}$
for a certain dihedral action preserving a particular connected component of
$G_{2}^{or}(\mathbf{R}^{n+3})$ (its positive part
$G_{2}^{or}(\mathbf{R}^{n+3})^{>0}$) and explain how two non trivial
functional identities satisfied by the components of
$\boldsymbol{\mathcal{E}}_{n}$ can be deduced from them.

After having completely explicited $\boldsymbol{\mathcal{E}}_{3}$ in §5.1.6,
we turn to the invariant properties of $\boldsymbol{\mathcal{E}}_{n}$ with
respect to the birational action of the whole symmetric group
$\mathfrak{S}_{n+3}$ on $\mathcal{M}_{0,n+3}$ in §5.1.7. We show that things
actually are more subtile than as described by Damiano, the main and crucial
fact being that, contrarily to the case when $n$ is even, when $n$ is odd
Euler's abelian relation is actually not canonically defined on each connected
component of ${\mathcal{M}}_{0,n+3}$, but only up to sign. We explain that
this fact, that does not seem really significant at first glance, actually has
the consequence that Damiano's construction of a 1-dimensional
$\mathfrak{S}_{n+3}$-representation associated to Euler's abelian relation is
irremediably flawed when $n$ is odd.

Finally, in §5.1.8 and §5.1.9, which concern the cases when $n$ is odd and
even respectively, we give two closed formulas for the components of Euler's
abelian relation. Both formulas are conjectural in full generality but are
proven to hold true indeed for $n$ less than or equal to 12 (by means of
formal computations).

###  5.1. On Gelfand-MacPherson's theory of generalized dilogarithm forms.

Euler's abelian relation is discussed in [Da2, §5] (see also [Da1, Chap. 6]).
Its construction relies on the general theory of 'generalized dilogarithm
forms' exposed in [GM]. Gelfand and MacPherson worked with usual grassmannian
varieties $G_{N}(\mathbf{R}^{N+M})$ whereas Damiano's chose to do so with
oriented grassmannians $G_{N}^{or}(\mathbf{R}^{N+M})$, moreover in the
specific case when $N=2$ and $M=n+1$. In order to show more clearly where the
problem in Damiano's construction of a polylogarithmic AR for
$\boldsymbol{\mathcal{W}}_{0,n+3}$ lies, we give below an overview of the
corresponding constructions. For proofs, we refer the reader to the two
previously cited articles.

#### 5.1.1.

In ordre to consider both cases together, namely the standard and the oriented
grassmannians and associated configuration spaces, we agree that $*$ either
stands for nothing or for ${\rm`or^{\prime}}$, the former notation referring
of course to the standard case and the latter to the oriented one. Considering
this, we set ${\rm O}^{or}_{m}(\mathbf{R})={\rm SO}_{m}(\mathbf{R})$ for any
$m\geq 1$.

Below, `plane' means a subvector space of dimension 2, and when $*={\rm or}$,
a `$*$-plane' is a 2-plane provided with an orientation . We will denote a
$*$-plane by $\xi^{*}$ whereas $\xi$ will stand for the associated (non
oriented) plane.

  * $-$

$G^{*}_{2}(\mathbf{R}^{n+3})={\rm O}^{*}_{n+3}(\mathbf{R})/\big{(}{\rm
O}^{*}_{2}(\mathbf{R})\times{\rm O}^{*}_{n+1}(\mathbf{R})\big{)}$ is the
grassmannian of $*$-planes in $\mathbf{R}^{n+3}$;

  * $-$

${\rm GL}_{n+3}(\mathbf{R})$ acts on $\mathbf{R}^{n+3}$ hence on
$G^{*}_{2}(\mathbf{R}^{n+3})$ as well. This action is not effective: since the
center $\mathbf{R}^{*}{\rm Id}$ acts trivially, this action factors through
$\mathbf{P}{\rm GL}_{n+3}(\mathbf{R})={\rm
GL}_{n+3}(\mathbf{R})/(\mathbf{R}^{*}{\rm Id})$;

  * $-$

$\widehat{G}^{*}_{2}(\mathbf{R}^{n+3})$ stands for the dense open subset of
generic $*$-planes in $\mathbf{R}^{n+3}$, which are those which do not contain
any coordinate line and which are not contained in any coordinate hyperplane;

  * $-$

$\widetilde{H}\subset{\rm GL}_{n+3}(\mathbf{R})$ denotes the abelian subgroup
of diagonal matrices and $H=H/(\mathbf{R}^{*}{\rm Id})$ stands for its image
in $\mathbf{P}{\rm GL}_{n+3}(\mathbf{R})$. The latter group is a maximal torus
isomorphic to $(\mathbf{R}^{*})^{n+2}$ (in natural but non unique ways);

  * $-$

$\widetilde{H}_{0}$ and $H_{0}$ are the connected components of the identity
of $\widetilde{H}$ and $H$ respectively (thus
$H_{0}\simeq(\mathbf{R}_{>0})^{n+2}$ but again not in a canonical way);

  * $-$

$\widetilde{K}$ is the subgroup of $\widetilde{H}$ formed by the matrices with
diagonal coefficients in $\\{\pm 1\\}\simeq\mathbf{Z}_{2}$ and $K$ stands for
its image in $H$. One has $K\simeq(\mathbf{Z}_{2})^{n+2}$ and since $H$ is
commutative, we have that $H\simeq H_{0}\times
K\simeq(\mathbf{R}_{>0})^{n+2}\times(\mathbf{Z}_{2})^{n+2}$;

  * $-$

we define $C_{2}^{*}(n+3)=\widehat{G}^{*}_{2}(\mathbf{R}^{n+3})/H$ (resp.
$EC_{2}^{*}(n+3)=\widehat{G}^{*}_{2}(\mathbf{R}^{n+3})/H_{0}$) as the space of
$*$-configurations (resp. enhanced $*$-configurations);

  * $-$

given an oriented 2-plane $\xi^{or}$, we denote by $\check{\xi}^{or}$ the same
plane but endowed with the opposite orientation, and by $\xi$ the standard
plane (disregarding the orientations);

  * $-$

The map $D:\xi^{or}\rightarrow D(\xi^{or})=\check{\xi}^{or}$ is an involutive
isomorphism which is the deck transformation of the 2-1 (universal) covering
$\nu:G^{or}_{2}(\mathbf{R}^{n+3})\rightarrow G_{2}(\mathbf{R}^{n+3})$,
$\xi^{or}\mapsto\xi$.

The spaces, the quotients maps by $H$ and $H_{0}$ and the coverings induced by
$\nu$ discussed above all fit into the following commutative diagram :

(65)
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\widehat{G}_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{H}}$$\scriptstyle{\pi_{0}}$$\scriptstyle{\nu}$$\textstyle{\widehat{G}_{2}\big{(}\mathbf{R}^{n+2}\big{)}\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{0}}$$\scriptstyle{\pi_{H}}$$\textstyle{EC_{2}^{or}(n+3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\scriptstyle{\hskip
2.84544pt\nu}$$\textstyle{EC_{2}(n+3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\textstyle{C^{or}_{2}(n+3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
2.84544pt\nu}$$\textstyle{C_{2}(n+3)\,.}$
---

In this diagram, he horizontal maps all are non ramified 2-1 coverings induced
by $\nu$ (hence are denoted by the same notation), the maps $\pi_{0}$ and
$\kappa$ correspond to quotienting by $H_{0}$ and $K$ respectively, hence the
compositions $\pi_{H}=\kappa\circ\pi_{0}$ are the maps induced by quotienting
by $H$.

As is well-known, $C^{*}_{2}(n+3)$ naturally identifies with the space
$(\mathbf{P}^{1}_{\mathbf{R}}\big{)}^{n+3}-\Delta$ of $(n+3)$-tuple of
pairwise distinct points on the real projective line353535Here $\Delta$ stands
for the union of all the small diagonals in
$\big{(}\mathbf{P}^{1}_{\mathbf{R}}\big{)}^{n+3}$., quotiented by ${\rm
GL}_{2}^{*}(\mathbf{R})/(\mathbf{R}_{>0}{\bf I}_{2})$ where we agree that
${\rm GL}_{2}^{or}(\mathbf{R})={\rm GL}_{2}^{+}(\mathbf{R})=\\{g\in{\rm
GL}_{2}(\mathbf{R})\,\lvert\,{\rm det}(g)>0\,\\}$. In the standard case when
$*$ is empty, one recovers $\mathcal{M}_{0,n+3}(\mathbf{R})$. When $*=or$, the
quotient $C^{or}_{2}(n+3)=\widehat{G}^{or}_{2}(\mathbf{R}^{n+3})/H$ can be
seen as the set of $(n+3)$-tuples of pairwise distinct points on
$\mathbf{P}^{1}_{\mathbf{R}}\simeq\mathbf{S}^{1}=\\{z\in\mathbf{C}\,,\,\lvert
z\lvert=1\\}\subset\mathbf{C}$ modulo the action of the subgroup ${\rm
Mob}^{+}(\mathbf{S}^{1})$ of Moëbius transformations preserving the standard
orientation of $\mathbf{S}^{1}$. Hence it appears natural to denote this space
of oriented configurations by $\mathcal{M}_{0,n+3}^{\,or}(\mathbf{R})$. The
map

$\nu:\mathcal{M}_{0,n+3}^{\,or}(\mathbf{R})\rightarrow\mathcal{M}_{0,n+3}(\mathbf{R})$

is simply given by identifying configurations when reversing the orientation
of $\mathbf{P}^{1}_{\mathbf{R}}\simeq\mathbf{S}^{1}$.

To explain what an enhanced configuration is, we denote by
${\mathbb{S}}^{1}\subset\mathbf{R}^{2}$ the circle from which
$\mathbf{P}^{1}_{\mathbf{R}}=\mathbf{S}^{1}$ is obtained by identifying
diametrically opposite points. Then an `enhanced configuration' (that is, an
element of $EC_{2}(n+3)$) is a projective equivalence class of a $(n+3)$-tuple
$(c_{i})_{i=1}^{n+3}\in\mathbf{P}^{1}_{\mathbf{R}}-\Delta$ enhanced by a
choice of a lift $\tilde{c}_{i}\in{\mathbb{S}}^{1}$ of $c_{i}$ for each
$i=1,\ldots,n+3$. Of course, there is a similar description of what an
`enhanced oriented configuration' is, we leave it to the interested reader to
elaborate on this.

Recall (see §2.1.5 above) that $\mathcal{M}_{0,n+3}(\mathbf{R})$ is not
connected and that its connected components $\mathcal{M}(\boldsymbol{\sigma})$
are first, biunivocally labeled by classes
$\boldsymbol{\sigma}\in\mathfrak{K}_{n+3}=\mathfrak{S}_{n+3}/D_{0,n+3}$ and
second, all isomorphic to
$\mathcal{M}(\boldsymbol{1})=\mathcal{M}_{0,n+3}^{>0}(\mathbf{R})$. Our
purpose below is to explain Damiano's construction of an AR on any component
$\mathcal{M}(\boldsymbol{\sigma})$ and to describe some of its properties. We
will denote by $\mathcal{E}_{n}(\boldsymbol{\sigma})$ this AR which is an
element of $\boldsymbol{AR}(\boldsymbol{\sigma})$. The question of the nature
of the $\mathcal{E}_{n}(\boldsymbol{\sigma})$'s and how these ARs are related
is crucial and will be discussed. But in a first step, we will only consider
the case of the positive part $\mathcal{M}_{0,n+3}^{>0}(\mathbf{R})$ and of
the AR $\mathcal{E}_{n}(\boldsymbol{1})$ it carries, that we will denote by
$\mathcal{E}_{n}^{>0}$ to simplify.

####  5.1.2. Euler's differential form

When $n$ is even, one could work with usual grassmannians of 2-planes, which
are precisely the manifolds used by Gelfand and MacPherson to develop their
theory of higher polylogarithmic forms and of the differential relations they
satisfy. However this is not possible when $n$ is odd (see Remark 5.19 below
for an explanation of this) and it is then necessary to work at the level of
the oriented grassmannian. In order to give a presentation independent of the
parity of $n$, we will place ourselves within this `oriented framework'.

We assume below that $\mathbf{R}^{n+3}$ is endowed with its standard euclidean