# ON INVARIANT OF THE REGULARITY INDEX OF
FAT POINTS

Phan Van Thien
Department of Mathematics,
University of Education, Hue University, Vietnam
E-mail<EMAIL_ADDRESS>

###### Abstract

We prove invariant of the regularity index of fat points under changes of the
linear subspace containing the support of the fat points (Theorem 3.5). Then
we show that Segre’s bound is attained by any set of $s$ non-degenerate
equimultiple fat points in $\mathbb{P}^{n}$, $s\leq n+3$ (Theorem 4.3). We
also give an example showing that there always exists a set of $n+4$ non-
degenerate equimultiple fat points in $\mathbb{P}^{n}$ such that Segre’s bound
is not attained (Example 4.4).

Key words and phrases. Regularity index; Fat points.

2010 Mathematics Subject Classification. Primary 14C20; Secondary 13D40.

##  1 Introduction

In this paper, we denote by $\mathbb{P}^{n}:=\mathbb{P}^{n}_{K}$ the
$n$-dimensional projective space over an algebraically closed $K$ of arbitrary
characteristic, and denote by $R:=K[X_{0},\ldots,X_{n}]$ the polynomial ring
in variables $X_{0},\ldots,X_{n}$ over $K$. If $P_{1},\ldots,P_{s}$ are
distinct points in $\mathbb{P}^{n}$; we denote by $\wp_{1},\ldots,\wp_{r}$ the
defining homogeneous prime ideals of $P_{1},\ldots,P_{s}$ in $R$,
respectively.

Let $P_{1},\ldots,P_{s}$ be distinct points in $\mathbb{P}^{n}$, and let
$m_{1},\ldots,m_{s}$ be positive integers. Then the set of all homogeneous
polynomilas that vanish at $P_{i}$ to order $m_{i}$; for $i=1,\ldots,s$; is
the homogeneous ideal $I:=\wp^{m_{1}}_{1}\cap\cdots\cap\wp^{m_{s}}_{s}$. We
call the zero-scheme defined by $I$ to be fat points (or a set of fat points)
in $\mathbb{P}^{n}$, and we denote it by

$Z:=m_{1}P_{1}+\cdots+m_{s}P_{s}.$

The integer $m_{i}$ is called the multiplicity of $P_{i}$; $i=1,\ldots,s$. If
$m_{1}=\cdots=m_{s}=2$, then fat points $Z=2P_{1}+\cdots+2P_{s}$ is said to be
double points. If $m_{1}=\cdots=m_{s}=m$, then fat points
$Z=mP_{1}+\cdots+mP_{s}$ is said to be equimultiple. We call the points
$P_{i}$ to be the points of support of $Z$. If the support of $Z$ span
$\mathbb{P}^{n}$; then $P_{1},\ldots,P_{s}$ is called non-degenerate in
$\mathbb{P}^{n}$; and we also call $Z$ a set of non-degenerate fat points in
$\mathbb{P}^{n}$.

The homogeneous coordinate ring $R/I$ of $Z$ is a graded ring,
$R/I=\underset{t\geq 0}{\oplus}(R/I)_{t}$. We call the function

$H_{R/I}(t):=\dim_{K}(R/I)_{t}$

to be the Hilbert function of $Z$ in $\mathbb{P}^{n}$. Since

$\dim_{K}I_{t}=\dim_{K}R_{t}-\dim_{K}(R/I)_{t}=\binom{t+n}{n}-H_{R/I}(t),$

we know the dimension of linear space of all homogeneous polynomials of degree
$t$ vanishing at $P_{i}$ to order $m_{i}$; $i=1,\ldots,s$; if we know the
Hilbert function $H_{R/I}(t)$. But it is a fairly difficult problem to
determine the Hilbert function of fat points in $\mathbb{P}^{n}$.

The ring $R/I$ has the multiplicity

$e(R/I):=\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+n-1}{n}.$

The Hilbert function $H_{R/I}(t)$ strictly increases until it reaches the
multiplicity $e(R/I)$, at which it stabilizes. We call the least integer $t$
such that $H_{R/I}(t)=e(R/I)$ to be the regularity index of $Z$ in
$\mathbb{P}^{n}$, and we denote it by $\operatorname{reg}(Z)$. It is well
known that $\operatorname{reg}(Z)=\operatorname{reg}(R/I)$, the Castelnuovo-
Mumford regularity of the ring $R/I$.

If $t\geq\operatorname{reg}(Z)$, we have

$\dim_{K}I_{t}=\binom{t+n}{n}-\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+n-1}{n}.$

So, it is interesting to determine $\operatorname{reg}(Z)$ or, even less, an
upper bound for it.

From now on if $a$ is a rational number, we denote by $[a]$ it’s integer part.

For generic fat points $Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$ in $\mathbb{P}^{2}$
with $m_{1}\geq\cdots\geq m_{s}$, Segre [8] showed that

$\operatorname{reg}(Z)\leq\max\left\\{m_{1}+m_{2}-1,\left[\frac{m_{1}+\cdots+m_{s}}{2}\right]\right\\}.$

It was conjectured by Trung (see [9]) and, independently, by Farabbi and
Lorenzini (see [6]) that

$\operatorname{reg}(R/I)\leq\max\\{T_{j}(Z)|\ j=1,\ldots,n\\},$

where

$T_{j}(Z):=\max\left\\{\left[\frac{\sum_{l=1}^{q}m_{i_{l}}+j-2}{j}\right]|\
P_{i_{1}},\ldots,P_{i_{q}}\text{ on a }j\text{-dimensional linear
subspace}\right\\}.$

The number

$T(Z):=\max\\{T_{j}(Z)|\ j=1,\ldots,n\\}$

is called the Segre’s bound because it generalizes Segre’s upper bound. There
were many different results for proving the Segre’s bound. In 2016, Ballico et
al. [2] succesfully proved the Segre’s bound for $n+3$ non-degenerate fat
points in $\mathbb{P}^{n}$. Recently, Nagel and Trok (see [7, Theorem 5.3])
have succefully proved the Segre’s bound for arbitrary fat points $Z$ in
$\mathbb{P}^{n}$.

The problem to exactly determine $\operatorname{reg}(R/I)$ is more fairly
difficult. So far, there are only a few results about computing
$\operatorname{reg}(R/I)$.

For abitrary fat points $Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$ in $\mathbb{P}^{n}$,
Davis and Geramita [5, Corollary 2.3] proved that

$\operatorname{reg}(R/I)=m_{1}+\cdots+m_{s}-1$

if and only if points $P_{1},\ldots,P_{s}$ lie on a line in $\mathbb{P}^{n}$.

The points $P_{1},\ldots,P_{s}$ in $\mathbb{P}^{n}$ is said to be in linearly
general position if no $j+2$ of the points $P_{1},\ldots,P_{s}$ are on any
$j$-dimensional linear subspace for $j<n$, then we also call the fat points
$Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$ to be in general position in
$\mathbb{P}^{n}$. A rational normal curve in $\mathbb{P}^{n}$ to be a curve of
degree $n$ that may be given parametrically as the image of the map

$\displaystyle\mathbb{P}^{1}$ $\displaystyle\to\mathbb{P}^{n}.$
$\displaystyle(s,t)$ $\displaystyle\mapsto(s^{n},s^{n-1}t,\ldots,t^{n})$

For fat points $Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$ in $\mathbb{P}^{n}$ with
$m_{1}\geq\cdots\geq m_{s}$, Catalisano et al. [4] showed formulas to compute
$\operatorname{reg}(Z)$ in the following two cases:

If $s\geq 2$ and $P_{1},\ldots,P_{s}$ are on a rational normal curve in
$\mathbb{P}^{n}$ [4, Proposition 7], then

$\operatorname{reg}(R/I)=\max\left\\{m_{1}+m_{2}-1,\left[(\sum_{i=1}^{s}m_{i}+n-2)/n\right]\right\\}.$

If $n\geq 3$, $2\leq s\leq n+2$, $2\leq m_{1}\geq\cdots\geq m_{s}$ and
$P_{1},\ldots,P_{s}$ are in general position in $\mathbb{P}^{n}$, then

$\operatorname{reg}(R/I)=m_{1}+m_{2}-1.$

For fat points $Z=m_{1}P_{1}+\cdots+m_{s+2}P_{s+2}$ in $\mathbb{P}^{n}$ with
$P_{1},\ldots,P_{s+2}$ not in a linear $(s-1)$-space in $\mathbb{P}^{n}$,
$s\leq n$, Thien [10, Theorem 3.4] showed

$\operatorname{reg}(R/I)=T(Z).$

For equimutiple fat points $Z=mP_{1}+\cdots+mP_{s}$ in $\mathbb{P}^{n}$ with
$m\neq 2$; $P_{1},\ldots,P_{s}$ are not on a linear $(r-1)$-space and $s\leq
r+3$, Thien and Sinh [11, Theorem 4.5] showed

$\operatorname{reg}(R/I)=T(Z).$

Assume that $\alpha$ is a $r$-dimensional linear subspace of $\mathbb{P}^{n}$
containing the points $P_{1},\ldots,P_{s}$. We may consider the
$r$-dimensional linear subspace $\alpha$ as a projective space
$\mathbb{P}^{r}$ containing the points
$P_{1\alpha}:=P_{1},\ldots,P_{s\alpha}:=P_{s}$, and we consider
$Z_{\alpha}:=m_{1}P_{1\alpha}+\cdots+m_{s}P_{s\alpha}$ as a zero-scheme in
$\mathbb{P}^{r}$. What is the relation between $\operatorname{reg}(Z)$ and
$\operatorname{reg}(Z_{\alpha})$?

In this paper we prove that
$\operatorname{reg}(Z)=\operatorname{reg}(Z_{\alpha})$ (Theorem 3.5). Then we
calculate regularity index of a set of $n+3$ non-degenerate double points in
$\mathbb{P}^{n}$ (Proposition 4.2) and show that Segre’s bound is attained by
any set of $s$ non-degenerate equimultiple fat points in $\mathbb{P}^{n}$,
$s\leq n+3$ (Theorem 4.3). We also show that there always exists a set of
$n+4$ non-degenerate equimultiple fat points in $\mathbb{P}^{n}$ such that
Segre’s bound is not attained (Corollary 4.5).

##  2 Preliminaries

We use the following lemmas which have been proved yet. The first lemma allows
us to compute the regularity index by induction.

######  Lemma 2.1.

[4, Lemma 1] Let $P_{1},\ldots,P_{r},P$ be distinct points in $\mathbb{P}^{n}$
and $\wp$ be the defining ideals in $R$ corresponding to the point $P$. If
$m_{1},\ldots,m_{r},a$ are positive integers,
$J:=\wp^{m_{1}}_{1}\cap\cdots\cap\wp^{m_{r}}_{r}$, and $I=J\cap\wp^{a}$, then

$\operatorname{reg}(R/I)=\max\left\\{a-1,\operatorname{reg}(R/J),\operatorname{reg}(R/(J+\wp^{a}))\right\\}.$

Note that $R/(J+\wp^{a})$ is a zero-dimensional graded ring algebra, the
regularity index $\operatorname{reg}(R/(J+\wp^{a}))$ is the least integer such
that $[R/(J+\wp^{a}))]_{t}=0$. To estimate $\operatorname{reg}(R/(J+\wp^{a}))$
we shall use the following lemma.

######  Lemma 2.2.

[4, Lemma 3] Let $P_{1},\ldots,P_{r},P$ be distinct points in $\mathbb{P}^{n}$
and $\wp$ be the defining ideal in $R$ corresponding to the point $P$. If
$m_{1},\ldots,m_{r},a$ are positive integers,
$J=\wp^{m_{1}}_{1}\cap\cdots\cap\wp^{m_{r}}_{r}$ and
$\wp=(X_{1},\ldots,X_{n})$, then

$\operatorname{reg}(R/(J+\wp^{a}))\leq b$

if and only if $X^{b-i}_{0}M\in J+\wp^{i+1}$ for every monomial $M$ of degree
$i$ in $X_{1},\ldots,X_{n}$; $i=0,\ldots,a-1$.

######  Lemma 2.3.

[4, Corollary 2] Let $s\geq 2$; $P_{1},\ldots,P_{s}$ be distinct points in
$\mathbb{P}^{n}$ and $m_{1}\geq\cdots\geq m_{s}$ be positive integers. If
$I=\wp_{1}^{m_{1}}\cap\cdots\cap\wp_{s}^{m_{s}}$, then

$\operatorname{reg}(R/I)\geq m_{1}+m_{2}-1.$

######  Lemma 2.4.

[4, Corollary 8] Let $n\geq 3$, $2\leq s\leq n+2$, and let
$P_{1},\ldots,P_{s}$ be distinct points in general position in
$\mathbb{P}^{n}$. If $2\leq m_{1}\geq\cdots\geq m_{s}>0$ are given integers
and $I=\wp_{1}^{m_{1}}\cap\cdots\cap\wp_{s}^{m_{s}}$, then

$\operatorname{reg}(R/I)=m_{1}+m_{2}-1.$

In [2], Ballico et al. succesfully proved the Segre’s for non-degenerate $n+3$
fat points in $\mathbb{P}^{n}$.

######  Lemma 2.5.

[2, Theorem 2.1] Let $Z=m_{1}P_{1}+\cdots+m_{n+3}P_{n+3}$ be a scheme of fat
points supported on a non-degenerate set of distinct points in
$\mathbb{P}^{n}$. Then $Z$ sastifies the Segre’s bound, namely

$\operatorname{reg}(Z)\leq\max\\{T(Z,L):L\subseteq\mathbb{P}^{n}\\},$

where $L$ is a $j$-dimensional linear subspace of $\mathbb{P}^{n}$;
$j=1,\ldots,n$; and

$T(Z,L):=\max\left\\{\left[\frac{\sum_{l=1}^{q}m_{i_{l}}+j-2}{j}\right]|\
P_{i_{1}},\ldots,P_{i_{q}}\in L\right\\}.$

We see that the number $\max\\{T(Z,L):L\subseteq\mathbb{P}^{n}\\}$ is the
Segre’s bound $T(Z)$ in the Trung’s conjecture.

The following lemma was proved in [10], it also has been shown in [7, Lemma
3.1].

######  Lemma 2.6.

[10, Lemma 3.3] Let $X=\\{P_{1},\ldots,P_{s}\\}$ be a set of distinct points
in $\mathbb{P}^{n}$ and $m_{1},\ldots,m_{s}$ be positive integers. Put
$I=\wp^{m_{1}}_{1}\cap\cdots\cap\wp^{m_{s}}_{s}$. If $Y$ is a subset of $X$
and $J=\underset{P_{i}\in Y}{\cap}\wp_{i}^{m_{i}}$, then

$\operatorname{reg}(R/J)\leq\operatorname{reg}(R/I).$

This implies that if $Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$ is the set of fat points
defining by $I$, and $U=\underset{P_{i}\in Y}{\sum}m_{i}P_{i}$ is the set of
fat points defining by $J$, then

$\operatorname{reg}(U)\leq\operatorname{reg}(Z).$

######  Lemma 2.7.

[10, Theorem 3.4] Let $P_{1},\ldots,P_{s+2}$ be distinct points not on a
$(s-1)$-dimensional linear subspace in $\mathbb{P}^{n}$; $s\leq n$; and
$m_{1},\ldots,m_{s+2}$ be positive integers. Put
$I=\wp_{1}^{m_{1}}\cap\cdots\cap\wp_{s+2}^{m_{s+2}}$. Then,

$\operatorname{reg}(R/I)=\max\\{T_{j}|\ j=1,\ldots,n\\},$

where

$T_{j}:=\max\left\\{\left[\frac{\sum_{l=1}^{q}m_{i_{l}}+j-2}{j}\right]|\
P_{i_{1}},\ldots,P_{i_{q}}\text{ on a }j\text{-dimensional linear
subspace}\right\\}.$

If $P_{1},\ldots,P_{s}$ are on a $r$-dimensional linear subspace
$\alpha\subset\mathbb{P}^{n}$ and there are no $j+2$ of the points
$P_{1},\ldots,P_{s}$ on any $j$-dimensional linear subspace for $j<r$; then we
call $P_{1},\ldots,P_{s}$ to be in linearly general position on $\alpha$. The
following lemmas were proved in [11].

######  Lemma 2.8.

[11, Theorem 3.1] Let $P_{1},\ldots,P_{s}$ be distinct points in general
position on an $r$-dimensional linearly linear subspace $\alpha$ in
$\mathbb{P}^{n}$, $s\leq r+3$. Let $m_{1},\ldots,m_{s}$ be positive integers
and $Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$. Then,

$\operatorname{reg}(Z)=\max\\{T_{1},T_{r}\\},$

where

$\displaystyle T_{1}$ $\displaystyle=\max\\{m_{i}+m_{j}-1\ |\ i\neq
j;i,j=1,\ldots,s\\},$ $\displaystyle T_{r}$
$\displaystyle=\left[\frac{m_{1}+\cdots+m_{s}+r-2}{r}\right].$

We see that the Segre’s bound is attained for this case. If $r=n$ and $s=n+3$,
the $P_{1},\ldots,P_{n+3}$ are in linearly general position in
$\mathbb{P}^{n}$. So, there exists a rational normal curve passing through
$P_{1},\ldots,P_{n+3}$. Then we get Proposition 7 in [4].

######  Lemma 2.9.

[11, Corollary 3.2] Let $P_{1},\ldots,P_{s}$ be distinct points in
$\mathbb{P}^{n}$, $s\leq 5$. Let $m$ be a positive integer and
$Z=mP_{1}+\cdots+mP_{s}$. Then

$\operatorname{reg}(Z)=\max\\{T_{j}|\ j=1,\ldots,n\\},$

where

$T_{j}:=\max\left\\{\left[\frac{qm+j-2}{j}\right]|\
P_{i_{1}},\ldots,P_{i_{q}}\text{ on a }j\text{-dimensional linear
subspace}\right\\}.$

######  Lemma 2.10.

[11, Theorem 4.5] Let $X=\\{P_{1},\ldots,P_{s+3}\\}$ be a set of distinct
points not on a $(s-1)$-dimensional linear subspace in $\mathbb{P}^{n}$,
$s\leq n$ and $m$ be a positive integer, $m\neq 2$. Let
$Z=mP_{1}+\cdots+mP_{s+3}$ be a set equimultiple fat points. Then,

$\operatorname{reg}(Z)=\max\\{T_{j}|\ j=1,\ldots,n\\},$

where

$T_{j}=\max\left\\{\left[\frac{mq+j-2}{j}\right]|\
P_{i_{1}},\ldots,P_{i_{q}}\text{ lie on a linear }j\text{-space}\right\\},$

We see that in case $s=n$ and $m\neq 2$, the regularity index of $n+3$ non-
degenerate equimultiple fat points $Z=m1P_{1}+\cdots+mP_{n+3}$ in
$\mathbb{P}^{n}$ reaches the Segre’s bound.

##  3 Invariant of the regularity index under changes of linear subspace
containing the support of fat points

Let $Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$ be a set of fat points in
$\mathbb{P}^{n}$, and $I$ be the ideal consists all homogeneous polynomials in
$R$ vanishing at $P_{i}\in\mathbb{P}^{n}$ to order $m_{i}$; $i=1,\ldots,s$.
The regularity index $\operatorname{reg}(Z)$ is the least integer such that
the Hilbert function

$H_{R/I}(t)=\dim_{K}(R/I)_{t}=\binom{t+n}{n}-\dim_{K}I_{t}$

reaches the multiplicity
$e(R/I)=\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+n-1}{n}.$ Note that the
Hilbert function of an ideal is invariant under projective automorphisms. So,
the regularity index of a set of fat points is also invariant under projective
automorphisms.

Assume that $\alpha$ is a $r$-dimensional linear subspace of $\mathbb{P}^{n}$
containing the points $P_{1},\ldots,P_{s}$. We consider the $r$-dimensional
linear subspace $\alpha$ as a projective space $\mathbb{P}^{r}$ containing the
points $P_{1\alpha}:=P_{1},\ldots,P_{s\alpha}:=P_{s}$, and consider
$Z_{\alpha}:=m_{1}P_{1\alpha}+\cdots+m_{s}P_{s\alpha}$ as a set of fat points
in $\mathbb{P}^{r}$. We denote by $I_{\alpha}$ the ideal consists all
homogeneous polynomials in $R_{\alpha}:=K[X_{0},\ldots,X_{r}]$ that vanish at
$P_{i\alpha}$ to order $m_{i}$; $i=1,\ldots,s$. The Hilbert function of
$Z_{\alpha}$ in $\mathbb{P}^{r}$ is

$H_{Z_{\alpha}}(t):=H_{R_{\alpha}/I_{\alpha}}(t)=\dim_{K}(K[X_{0},\ldots,X_{r}]/I_{\alpha})_{t}=\binom{t+r}{r}-\dim_{K}(I_{\alpha})_{t}.$

The multiplicity number of $Z_{\alpha}$ in $\mathbb{P}^{r}$ is

$e(R_{\alpha}/I_{\alpha}):=\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+r-1}{r}.$

The regularity index of $Z_{\alpha}$ in $\mathbb{P}^{r}$ is the least integer
$t$ such that $H_{R/I_{\alpha}}(t)=e(R_{\alpha}/I_{\alpha})$, and we denote it
by $\operatorname{reg}(Z_{\alpha})$.

We have $e(R_{\alpha}/I_{\alpha})\leq e(R/I)$,
${\displaystyle\binom{t+r}{r}\leq\binom{t+n}{n}}$ and
$\dim_{K}(I_{\alpha})_{t}\leq\dim_{K}I_{t}$. Here there are two questions:

1) Question 1: $H_{Z_{\alpha}}(t)=H_{Z}(t)$?

2) Question 2: $\operatorname{reg}(Z_{\alpha})=\operatorname{reg}(Z)$?

For Question 1, we now do not know about the relation between $H_{Z}(t)$ and
$H_{Z_{\alpha}}(t)$ if $t<\operatorname{reg}(Z)$. But if
$t\geq\operatorname{reg}(Z)$, then $H_{Z_{\alpha}}(t)\leq H_{Z}(t)$.
Furthermore, if there exists $m_{i}\geq 2$ and $r<n$, then
$H_{Z_{\alpha}}(t)<H_{Z}(t)$ for $t\geq\operatorname{reg}(Z)$. (see Corollary
3.6).

For Question 2, due to a result of Benedetti et al. (see [3, Lemma 4.4]), we
know that if $Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$ is a set of fat points in
$\mathbb{P}^{n}$ whose support is contained in a $(n-1)$-dimensional linear
subspace $\alpha\cong\mathbb{P}^{n-1}$ and if there is a non-negative integer
$t$ such that $\operatorname{reg}(Z_{\alpha})\leq t,$ then

$\operatorname{reg}(Z)\leq t.$

By inductive argument on the dimension of linear subspace, we get

######  Lemma 3.1.

Let $Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$ be a set of fat points in
$\mathbb{P}^{n}$ whose support is contained in a linear $r$-space
$\alpha\cong\mathbb{P}^{r}$. We may consider the linear $r$-space $\alpha$ as
a $r$-dimensional projective space $\mathbb{P}^{r}$ containing the points
$P_{1\alpha}:=P_{1},\ldots,P_{s\alpha}:=P_{s}$ and consider
$Z_{\alpha}=m_{1}P_{1\alpha}+\cdots+m_{s}P_{s\alpha}$ as a set of fat points
in $\mathbb{P}^{r}$. If there is a non-negative integer $t$ such that

$\operatorname{reg}(Z_{\alpha})\leq t,$

then

$\operatorname{reg}(Z)\leq t.$

Now we prove that the converse is also true. The fisrt, we prove the following
lemma.

######  Lemma 3.2.

Let $\alpha$ be a $(n-1)$-dimensional linear subspace in $\mathbb{P}^{n}$, and
let $P_{1},\ldots,P_{s}$ be distinct points on $\alpha$. Let
$m_{1},\ldots,m_{s}$ be positive integers. We may consider the
$(n-1)$-dimensional linear subspace $\alpha$ as a $(n-1)$-dimensional
projective space $\mathbb{P}^{r}$ containing the points
$P_{1\alpha}:=P_{1},\ldots,P_{s\alpha}:=P_{s}$, and consider
$Z_{\alpha}=m_{1}P_{1\alpha}+\cdots+m_{s}P_{s\alpha}$ as a set of fat points
in $\mathbb{P}^{n-1}$. Put $R_{\alpha}=K[X_{0},\ldots,X_{n-1}]$. Let
$\wp_{j}\subset R$ be the defining prime ideal of $P_{j}\in\mathbb{P}^{n}$,
and $\wp_{j\alpha}\subset R_{\alpha}$ be the defining prime ideal of
$P_{j\alpha}\in\mathbb{P}^{r}$; $j=1,\ldots,s$. Put
$J=\wp_{1}^{m_{1}}\cap\cdots\cap\wp_{s-1}^{m_{s-1}}$ and
$J_{\alpha}=\wp_{1\alpha}^{m_{1}}\cap\cdots\cap\wp_{(s-1)\alpha}^{m_{s-1}}$.
If there is a non-negative integer $b$ such that

$\operatorname{reg}(R_{\alpha}/(J_{\alpha}+\wp_{s\alpha}^{m_{s}}))>b,$

then

$\operatorname{reg}(R/(J+\wp_{s}^{m_{s}}))>b.$

###### Proof.

Consider distinct points $P_{1},\ldots,P_{s}$ on $\alpha$, an
$(n-1)$-dimensional linear subspace in $\mathbb{P}^{n}$. We may assume that
$n$ points
$(\underset{1}{\underbrace{1}},0,\ldots,0,0,0),\ldots,(0,0,\ldots,0,\underset{n}{\underbrace{1}},0)$
span $\alpha$. Put $P_{s}=(1,0,\ldots,0)$, then
$\wp_{s}=(X_{1},\ldots,X_{n})$,
$P_{s\alpha}=(1,0,\ldots,0)\in\mathbb{P}^{n-1}$ and
$\wp_{s\alpha}=(X_{1},\ldots,X_{n-1})$. If

$\operatorname{reg}(R_{\alpha}/(J_{\alpha}+\wp_{s\alpha}^{m_{s}}))>b,$

then by Lemma 2.2 there exists a monomial $M=X_{1}^{c_{1}}\cdots
X_{n-1}^{c_{n-1}}\in\wp_{s\alpha}$, $c_{1}+\cdots+c_{r}=i$,
$i\in\\{0,\ldots,m_{s}-1\\}$, such that

$X_{0}^{b-i}M\notin J_{\alpha}+\wp_{s\alpha}^{i+1}.$

Now we consider $M=X_{1}^{c_{1}}\cdots X_{n-1}^{c_{n-1}}$ in
$K[X_{0},\ldots,X_{n}]$. We claim that

$X_{0}^{b-i}M\notin J+\wp_{s}^{i+1}.$

In fact, if $X_{0}^{b-i}M\in J+\wp_{s}^{i+1}$, then
$X_{0}^{b-i}M\in\left[J+\wp_{s}^{i+1}\right]_{b}$. Consider the finite
dimensional vector space $[J+\wp_{s}^{i+1}]_{b}$ over the field $K$ with the
monomial $X_{0}^{b-i}M$ being an element of it’s basis, we get

$X_{0}^{b-i}M\in J\text{ or }X_{0}^{b-i}M\in\wp_{s}^{i+1}.$

In case of $X_{0}^{b-i}M\in J$, then
$X_{0}^{b-i}M=X_{0}^{b-i}X_{1}^{c_{1}}\cdots X_{n-1}^{c_{n-1}}\in
K[X_{0},\ldots,X_{n-1}]$ and $X_{0}^{b-i}M$ vanishing at the points
$P_{j}\in\alpha$ to order $m_{j}$; $j=1,\ldots,s$. Thus $X_{0}^{b-i}M\in
J_{\alpha}$, this is a contradiction with $X_{0}^{b-i}M\notin
J_{\alpha}+\wp_{s\alpha}^{i+1}$. In case of $X_{0}^{b-i}M\in\wp_{s}^{i+1}$,
then $X_{0}^{b-i}M=X_{0}^{b-i}X_{1}^{c_{1}}\cdots X_{n-1}^{c_{n-1}}\in
K[X_{0},\ldots,X_{n-1}]$ and $X_{0}^{b-i}M$ vanishing at the points
$P_{s}\in\alpha$ to order $i+1$. Thus $X_{0}^{b-i}M\in\wp_{s\alpha}^{i+1}$,
this is a contradiction with $X_{0}^{b-i}M\notin
J_{\alpha}+\wp_{s\alpha}^{i+1}$. Thus, we have the claim

$X_{0}^{b-i}M\notin J+\wp_{s}^{i+1}.$

Then, by Lemma 2.2 we get

$\operatorname{reg}(R/(J+\wp_{s\alpha}^{m_{s}}))>b.$

∎

From the above lemma, by inductive argument on the dimension of linear
subspace, we get the following lemma.

######  Lemma 3.3.

Let $\alpha$ be an $r$-dimensional linear subspace in $\mathbb{P}^{n}$, $1\leq
r\leq n-1$, and let $P_{1},\ldots,P_{s}$ be distinct points on $\alpha$. Let
$m_{1},\ldots,m_{s}$ be positive integers. We may consider the $r$-dimensional
linear subspace $\alpha$ as a $r$-dimensional projective space
$\mathbb{P}^{r}$ containing the points
$P_{1\alpha}:=P_{1},\ldots,P_{s\alpha}:=P_{s}$, and
$Z_{\alpha}=m_{1}P_{1\alpha}+\cdots+m_{s}P_{s\alpha}$ as a zero-scheme in
$\mathbb{P}^{r}$. Put $R_{\alpha}=K[X_{0},\ldots,X_{r}]$. Let $\wp_{j}\subset
R$ be the defining prime ideal of $P_{j}$, and $\wp_{j\alpha}\subset
R_{\alpha}$ be the defining prime ideal of $P_{j\alpha}$; $j=1,\ldots,s$. Put
$J=\wp_{1}^{m_{1}}\cap\cdots\cap\wp_{s-1}^{m_{s-1}}$ and
$J_{\alpha}=\wp_{1\alpha}^{m_{1}}\cap\cdots\cap\wp_{(s-1)\alpha}^{m_{s-1}}$.
If there is a non-negative integer $b$ such that

$\operatorname{reg}(R_{\alpha}/(J_{\alpha}+\wp_{s}^{m_{s}}))>b,$

then

$\operatorname{reg}(R/(J+\wp_{s\alpha}^{m_{s}}))>b.$

By using the above lemmas we prove the following result.

######  Proposition 3.4.

Let $Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$ be fat points in $\mathbb{P}^{n}$ whose
support is contained in an $r$-dimensional linear space
$\alpha\cong\mathbb{P}^{r}$. We may consider the $r$-dimensional linear space
$\alpha$ as a projective space $\mathbb{P}^{r}$ containing the points
$P_{1\alpha}:=P_{1},\ldots,P_{s\alpha}:=P_{s}$, and
$Z_{\alpha}=m_{1}P_{1\alpha}+\cdots+m_{s}P_{s\alpha}$ as a zero-scheme in
$\mathbb{P}^{r}$. If there is a non-negative integer $b$ such that

$\operatorname{reg}(Z_{\alpha})>b,$

then

$\operatorname{reg}(Z)>b.$

###### Proof.

Without loss of generality we may assume that $m_{1}\geq\ldots\geq m_{s}$. Put
$R_{\alpha}=K[X_{0},\ldots,X_{r}]$ and denote by $\wp_{j\alpha}$ the
homogeneous prime ideal in $R_{\alpha}$ defining by $P_{j\alpha}$;
$j=1,\ldots,s$. We argue by induction on $s$. If $s=1$, then $Z=m_{1}P_{1}$ in
$\mathbb{P}^{n}$ and $Z_{\alpha}=m_{1}P_{1\alpha}$ in $\mathbb{P}^{r}$. It is
well known that $\operatorname{reg}(m_{1}P_{1})=m_{1}-1$ and
$\operatorname{reg}(m_{1}P_{1\alpha})=m_{1}-1$. Suppose that the lemma is true
for $s-1$. Put $J=\wp_{1}^{m_{1}}\cap\cdots\cap\wp_{s-1}^{m_{s-1}}$,
$I=J\cap\wp_{s}^{m_{s}}$,
$J_{\alpha}=\wp_{1\alpha}^{m_{1}}\cap\cdots\cap\wp_{(s-1)\alpha}^{m_{s-1}}$,
$I_{\alpha}=J_{\alpha}\cap\wp_{s\alpha}^{m_{s}}$. By Lemma 2.1 we get

$\operatorname{reg}(R/I)=\max\left\\{m_{s}-1,\operatorname{reg}(R/J),\operatorname{reg}(R/(J+\wp_{s}^{m_{s}}))\right\\}$

and

$\operatorname{reg}(R_{\alpha}/I_{\alpha})=\max\left\\{m_{s}-1,\operatorname{reg}(R_{\alpha}/J_{\alpha}),\operatorname{reg}(R_{\alpha}/(J_{\alpha}+\wp_{s\alpha}^{m_{s}}))\right\\}.$

By Lemma 2.3 we have $\operatorname{reg}(R/J)\geq m_{1}+m_{2}-1$ and
$\operatorname{reg}(R_{\alpha}/J_{\alpha})\geq m_{1}+m_{2}-1$. So,
$\operatorname{reg}(R/J)\geq m_{s}-1$ and
$\operatorname{reg}(R_{\alpha}/J_{\alpha})\geq m_{s}-1$. Therefore,

$\operatorname{reg}(R/I)=\max\left\\{\operatorname{reg}(R/J),\operatorname{reg}(R/(J+\wp_{s}^{m_{s}}))\right\\}$

and

$\operatorname{reg}(R_{\alpha}/I_{\alpha})=\max\left\\{\operatorname{reg}(R_{\alpha}/J_{\alpha}),\operatorname{reg}(R_{\alpha}/(J_{\alpha}+\wp_{s\alpha}^{m_{s}}))\right\\}.$

If $\operatorname{reg}(R_{\alpha}/I_{\alpha})>b$ in $\mathbb{P}^{r}$, then we
consider two following cases:

Case $\operatorname{reg}(R_{\alpha}/J_{\alpha})>b$: Then by inductive
assumption we have $\operatorname{reg}(R/J)>b.$ Therefore,

$\operatorname{reg}(R/I)>b.$

Case $\operatorname{reg}(R_{\alpha}/(J_{\alpha}+\wp_{s\alpha}^{m_{s}}))>b$:
Then by using Lemma 3.3 we get $\operatorname{reg}(R/(J+\wp_{s}^{m_{s}}))>b$.
Therefore,

$\operatorname{reg}(R/I)>b.$

∎

From Lemma 3.1 and Proposition 3.4 we get the following theorem which answers
Question 2.

######  Theorem 3.5.

Let $Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$ be fat points in $\mathbb{P}^{n}$ whose
support is contained in a linear $r$-space $\alpha\cong\mathbb{P}^{r}$. We may
consider the linear $r$-space $\alpha$ as a $r$-dimensional projective space
$\mathbb{P}^{r}$ containing the points
$P_{1\alpha}:=P_{1},\ldots,P_{s\alpha}:=P_{s}$, and consider
$Z_{\alpha}=m_{1}P_{1\alpha}+\cdots+m_{s}P_{s\alpha}$ as a zero-scheme in
$\mathbb{P}^{r}$. Denote $\operatorname{reg}(Z)$ the regularity index of $Z$
in $\mathbb{P}^{n}$, and denote $\operatorname{reg}(Z_{\alpha})$ the
regularity of $Z_{\alpha}$ in $\mathbb{P}^{n}$. Then

$\operatorname{reg}(Z_{\alpha})=\operatorname{reg}(Z).$

Hence, the regularity index of a set of fat points is invariant under changes
of linear subspace containing the support of fat points.

######  Corollary 3.6.

Let $Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$ be fat points in $\mathbb{P}^{n}$ whose
support is contained in a linear $r$-space $\alpha\cong\mathbb{P}^{r}$. We may
consider the linear $r$-space $\alpha$ as a $r$-dimensional projective space
$\mathbb{P}^{r}$ containing the points
$P_{1\alpha}:=P_{1},\ldots,P_{s\alpha}:=P_{s}$, and consider
$Z_{\alpha}=m_{1}P_{1\alpha}+\cdots+m_{s}P_{s\alpha}$ as a zero-scheme in
$\mathbb{P}^{r}$. If $t\geq\operatorname{reg}(Z)$, then

$H_{Z_{\alpha}}(t)\leq H_{Z}(t).$

Furthermore, if there exists $m_{i}\geq 2$ and $r<n$, then

$H_{Z_{\alpha}}(t)<H_{Z}(t)$

for $t\geq\operatorname{reg}(Z)$.

###### Proof.

We know that the Hilbert function $H_{Z}(t)$ strictly increases until it
reaches the multiplicity $e(Z)$, at which it stabilizes, and the regularity
index $\operatorname{reg}(Z)$ is the least integer $t$ such that
$H_{R/I}(t)=e(Z)$. So, if $t\geq\operatorname{reg}(Z)$ then
$H_{Z}(t)=e(Z)=\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+n-1}{n}$.

For fat points $Z_{\alpha}$ in $\mathbb{P}^{r}$, we also get: if
$t\geq\operatorname{reg}(Z_{\alpha})$, then
$H_{Z_{\alpha}}(t)=\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+r-1}{r}$. By
Theorem 3.5, $\operatorname{reg}(Z)=\operatorname{reg}(Z_{\alpha})$.
Therefore, if $t\geq\operatorname{reg}(Z)=\operatorname{reg}(Z_{\alpha})$ then

$H_{Z_{\alpha}}(t)=\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+r-1}{r}\leq\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+n-1}{n}=H_{Z}(t).$

This result implies that if $m_{1}=\cdots=m_{s}=1$ then
$H_{Z_{\alpha}}(t)=H_{Z}(t)$ for $t\geq\operatorname{reg}(Z)$, and if there
exists $m_{i}\geq 2$ and $r<n$, then $H_{Z_{\alpha}}(t)<H_{Z}(t)$ for
$t\geq\operatorname{reg}(Z)$.

∎

######  Corollary 3.7.

Let $Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$ be fat points in $\mathbb{P}^{n}$ with
the support in a $r$-dimensional linear subspace $\alpha$. If
$t\geq\operatorname{reg}(Z)$, then we have the relation between dimension of
the $t$-th graded part $I_{t}$ of ideal $I$ consisting all homogeneous
polynomials in $K[X_{0},\ldots,X_{n}]$ that vanish at $Pi$ to order $m_{i}$;
$i=1,\ldots,s$; and dimension of the $t$-th graded part $(I_{\alpha})_{t}$ of
ideal $I_{\alpha}$ consisting all homogeneous polynomials in
$K[X_{0},\ldots,X_{r}]$ that vanish at $Pi$ to order $m_{i}$; $i=1,\ldots,s$;
as following:

$\displaystyle(t+r+1)\cdots(t+n)\left[\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+r-1}{r}+\dim_{K}(I_{\alpha})_{t}\right]$
$\displaystyle=(r+1)\cdots(n-1)n\left[\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+n-1}{n}+\dim_{K}I_{t}\right].$

###### Proof.

We may consider $\alpha$ as a $r$-dimensional projective space
$\mathbb{P}^{r}$ containing the points
$P_{1\alpha}:=P_{1},\ldots,P_{s\alpha}:=P_{s}$, and consider
$Z_{\alpha}=m_{1}P_{1\alpha}+\cdots+m_{s}P_{s\alpha}$ as a set of fat points
in $\mathbb{P}^{r}$. Since $I=\underset{t\geq 0}{\oplus}I_{t}$ is the ideal
consists all homogeneous polynomials in $K[X_{0},\ldots,X_{n}]$ that vanish at
$Pi$ to order $m_{i}$, $i=1,\ldots,s$; the Hilbert function of $Z$ is

$H_{Z}(t)=\dim_{K}(K[X_{0},\ldots,X_{n}]/I)_{t}=\dim_{K}K[X_{0},\ldots,X_{n}]_{t}-\dim_{K}I_{t}=\binom{t+n}{n}-\dim_{K}I_{t}.$

Since $I_{\alpha}=\underset{t\geq 0}{\oplus}(I_{\alpha})_{t}$ is the ideal
consists all homogeneous polynomials in $K[X_{0},\ldots,X_{r}]$ that vanish at
$P_{i\alpha}$ to order $m_{i}$, $i=1,\ldots,s$; the Hilbert function of
$Z_{\alpha}$ is

$H_{Z_{\alpha}}(t)=\dim_{K}(K[X_{0},\ldots,X_{r}]/I_{\alpha})_{t}=\dim_{K}K[X_{0},\ldots,X_{r}]_{t}-\dim_{K}(I_{\alpha})_{t}=\binom{t+r}{r}-\dim_{K}(I_{\alpha})_{t}.$

By Theorem 3.5 we have $\operatorname{reg}(Z)=\operatorname{reg}(Z_{\alpha})$.
If $t\geq\operatorname{reg}(Z)=\operatorname{reg}(Z_{\alpha})$, then
$H_{Z}(t)$ reaches the multiplicity
$e(Z)=\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+n-1}{n}$, and
$H_{Z_{\alpha}}(t)$ reaches the multiplicity
$e(Z_{\alpha})=\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+r-1}{r}$, and so
we have

${\displaystyle\begin{cases}\binom{t+n}{n}=\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+n-1}{n}+\dim_{K}I_{t},\\\
\binom{t+r}{r}=\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+r-1}{r}+\dim_{K}(I_{\alpha})_{t}.\end{cases}}$

Hence if $t\geq\operatorname{reg}(Z)$, then

$\displaystyle(t+r+1)\cdots(t+n)\left[\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+r-1}{r}+\dim_{K}(I_{\alpha})_{t}\right]=(t+r+1)\cdots(t+n)\binom{t+r}{r}$
$\displaystyle=(r+1)\cdots(n-1)n\binom{t+n}{n}=(r+1)\cdots(n-1)n\left[\underset{i=1}{\overset{s}{\sum}}\binom{m_{i}+n-1}{n}+\dim_{K}I_{t}\right].$

∎

Let $Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$ be a set of fat points in
$\mathbb{P}^{n}$, and let $\alpha$ be a $r$-dimensional linear subspace. We
put

$\alpha\cap Z:=\underset{P_{i}\in\alpha}{\sum}m_{i}P_{i}$

and

$w_{\alpha\cap Z}:=\underset{P_{i}\in\alpha}{\sum}m_{i}.$

######  Corollary 3.8.

Let $Z$ be a set of fat points in $\mathbb{P}^{n}$. Assume that $\alpha$ is a
$r$-dimensional linear subspace such that
${\displaystyle\left[\frac{w_{\alpha\cap Z}-2}{r}\right]=T(Z)}$. Consider
$(\alpha\cap Z)_{\alpha}$ as a set of fat points in $\mathbb{P}^{r}$. If the
support of $(\alpha\cap Z)_{\alpha}$ lies on a line or on a rational normal
curve of $\mathbb{P}^{r}$, then $\operatorname{reg}(Z)=T(Z)$.

###### Proof.

By Theorem 3.5 we have

$\operatorname{reg}(\alpha\cap Z)=\operatorname{reg}((\alpha\cap
Z)_{\alpha}).$

By using Lemma 2.6 and [7, Theorem 5.3] we get

$\operatorname{reg}(\alpha\cap Z)\leq\operatorname{reg}(Z)\leq T(Z).$

Since the support of $(\alpha\cap Z)_{\alpha}$ lies on a line or on a rational
normal curve of $\mathbb{P}^{r}$, by using [5, Corollary 2.3] and [4,
Proposition 7] we get

$\operatorname{reg}((\alpha\cap Z)_{\alpha})\geq\left[\frac{w_{\alpha\cap
Z}-2}{r}\right].$

Note that ${\displaystyle\left[\frac{w_{\alpha\cap Z}-2}{r}\right]=T(Z)}$ by
the hypothesis. Therefore,

$\operatorname{reg}(Z)=\operatorname{reg}((\alpha\cap Z)_{\alpha})=T(Z).$

∎

Let $Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$ be a set of fat points in
$\mathbb{P}^{n}$. If $\alpha$ is a linear subspace of $\mathbb{P}^{n}$ such
that $P_{1},\ldots,P_{s}$ span $\alpha$, then we say that $Z$ is non-
degenerate in $\alpha$. From Theorem 3.5, we have the following remark.

######  Remark 3.9.

To calculate the regularity index of a set of fat points $Z$ in
$\mathbb{P}^{n}$, we find a linear subspace $\alpha\cong\mathbb{P}^{r}$ such
that $Z$ is non-degenerate on $\alpha$. Then we calculate the regularity index
of the non-degenerate fat points $Z_{\alpha}$ in $\mathbb{P}^{r}$.

##  4 Segre’s bound and the regularity index of non-degenerate equimultiple
fat points

Recall that for a set of fat points $Z=m_{1}P_{1}+\cdots+m_{s}P_{s}$ in
$\mathbb{P}^{n}$, the Segre’s bound for the regularity index of $Z$ is the
number

$T(Z)=\max\\{T_{j}(Z)|\ j=1,\ldots,n\\},$

where

$T_{j}(Z):=\max\left\\{\left[\frac{\sum_{l=1}^{q}m_{i_{l}}+j-2}{j}\right]|\
P_{i_{1}},\ldots,P_{i_{q}}\text{ on a }j\text{-dimensional linear
subspace}\right\\}.$

######  Lemma 4.1.

Let $Z=2P_{1}+\cdots+2P_{7}$ be a set of seven non-degenerate double points in
$\mathbb{P}^{4}$. Then

$\operatorname{reg}(Z)\geq 4.$

###### Proof.

Let $U=2Q_{1}+\cdots+2Q_{7}$ be a set of seven generic double points in
$\mathbb{P}^{4}$. Denote by $H_{Z}(t)$ the Hilbert funtion of $Z$ and by
$H_{U}(t)$ the Hilbert function of $U$. By the maximality of the Hilbert
function of generic fat points we have

$H_{Z}(t)\leq H_{U}(t).$

Note that $Z$ and $U$ have the same multiplicity

$e(Z)=e(U)=7\binom{2+4-1}{4}=35.$

By the Alexander-Hirschowits Theorem [1] about the Hilbert funtion of generic
double points in $\mathbb{P}^{4}$ we have

$H_{U}(3)=34.$

Since $H_{Z}(3)\leq H_{U}(3)=34<35=e(Z)$, by the definition of
$\operatorname{reg}(Z)$ we get

$\operatorname{reg}(Z)\geq 4.$

∎

The above lemma and Theorem 3.5 help us to calculate the regularity index of a
set of $n+3$ non-degenerate double points in $\mathbb{P}^{n}$. In this case we
see that the regularity index reaches the Segre’s bound.

######  Proposition 4.2.

Let $Z=2P_{1}+\cdots+2P_{n+3}$ be a set of $n+3$ non-degenerate double points
in $\mathbb{P}^{n}$. Then

$\operatorname{reg}(Z)=T(Z),$

where

$T(Z)=\max\\{T_{j}(Z)|\ j=1,\ldots,n\\},$
$T_{j}(Z):=\max\left\\{\left[\frac{2q+j-2}{j}\right]|\
P_{i_{1}},\ldots,P_{i_{q}}\text{ on a }j\text{-dimensional linear
subspace}\right\\}.$

###### Proof.

Consider the fat points in the assumption of Lemma 2.5, we see that

$\displaystyle T(Z,L)$
$\displaystyle=\max\left\\{\left[\frac{\sum_{l=1}^{q}m_{i_{l}}+j-2}{j}\right]|\
P_{i_{1}},\ldots,P_{i_{q}}\in L\text{ is a $j$-dimensional linear
subspace}\right\\}$ $\displaystyle=T_{j}(Z).$

Hence

$\max\\{T(Z,L):L\subseteq\mathbb{P}^{n}\\}=T(Z).$

By using Lemma 2.5 we get $\operatorname{reg}(Z)\leq T(Z)$. So, it is
sufficient to prove

$\operatorname{reg}(Z)\geq T(Z).$

We argue by induction on $n$. For $n=2$, then $n+3=5$. By Lemma 2.9 we get
$\operatorname{reg}(Z)=T(Z)$. For $n\geq 3$, we assume that the proposition is
true for any $m+3$ non-degenerate double points $2P_{1}+\cdots+2P_{m+3}$ in
$\mathbb{P}^{m}$, $m<s$. We now consider $n+3$ non-degenerate double points
$Z=2P_{1}+\cdots+2P_{n+3}$ in $\mathbb{P}^{n}$.

Since $P_{1},\ldots,P_{n+3}$ are not not on any $(n-1)$-dimensional linear
subspace, there are at most $j+3$ points of $\\{P_{1},\ldots,P_{n+3}\\}$ on a
$j$-dimensional linear subspace of $\mathbb{P}^{n}$; $j=1,\ldots,n-1$. Put

$p=\min\\{j\ |\ T_{j}(Z)=T(Z)\\}.$

and denote by $\alpha$ the $p$-dimensional linear subspace containing points
$P_{i_{1}},\ldots,P_{i_{r}}$ of $\\{P_{1},\ldots,P_{n+3}\\}$ such that

$\left[\frac{2r+p-2}{p}\right]=T_{p}(Z)=T(Z).$

Then the points $P_{i_{1}},\ldots,P_{i_{r}}$ are not on any
$(p-1)$-dimensional linear subspace. So, $r\geq p+1$. Since
$P_{1},\ldots,P_{n+3}$ are non-degenerate in $\mathbb{P}^{n}$, there are at
most $p+3$ points of $\\{P_{1},\ldots,P_{n+3}\\}$ on a $p$-dimensional linear
subspace. Hence $r\leq p+3$.

Consider the set of $r$ double points

$Y=2P_{i_{1}}+\cdots+2P_{i_{r}}.$

in $\mathbb{P}^{n}$. Put $P_{i_{j}\alpha}=P_{j}$, $j=1,\ldots,r$. Consider the
set of of $r$ non-degenerate double points

$Y_{\alpha}=2P_{i_{1}\alpha}+\cdots+2P_{i_{r}\alpha}$

in $\mathbb{P}^{p}$. If $r=p+1$ or $r=p+2$, then by Lemma 2.7 we get

$\operatorname{reg}(Y_{\alpha})=T(Y_{\alpha})=\left[\frac{2r+p-2}{p}\right].$

So, $\operatorname{reg}(Y_{\alpha})=T(Z).$ By Theorem 3.5 we get
$\operatorname{reg}(Y)=\operatorname{reg}(Y_{\alpha})$. By use Lemma 2.6, we
obtain

$\operatorname{reg}(Z)\geq\operatorname{reg}(Y)=T(Z).$

If $r=p+3$, then we consider two following cases for $p$:

Case $p<n$: Since $p+3$ points $P_{i_{1}},\ldots,P_{i_{p+3}}$ lie on the
$p$-dimensional linear subspace $\alpha$ and do not lie on a
$(p-1)$-dimensional linear subspace, by the inductive assumption we have

$\operatorname{reg}(Y_{\alpha})=T(Y_{\alpha}).$

By using Theorem 3.5 we get
$\operatorname{reg}(Y)=\operatorname{reg}(Y_{\alpha})$. Now use Lemma 2.6 and
obtain $\operatorname{reg}(Z)\geq\operatorname{reg}(Y)$. Since
$\operatorname{reg}(Y)=\operatorname{reg}(Y_{\alpha})=T(Y_{\alpha})=\left[\frac{2r+p-2}{p}\right]=T(Z)$,
we get

$\operatorname{reg}(Z)\geq T_{p}(Z)=T(Z).$

Case $p=n$: Then $T_{n}(Z)>T_{j}(Z)$; $j=1,\ldots,n-1$. By defining we have

$T_{n}(Z)=\left[\frac{(n+3)2+n-2}{n}\right]=3+\left[\frac{4}{n}\right],\text{
and }T_{1}(Z)\geq 2.2-1=3.$

Since $T_{n}(Z)>T_{1}(Z)$, we have $n\leq 4$. But we are considering $n\geq
3$, so $n=3$ or $n=4$.

For $n=3$, then $T_{3}(Z)=4=T(Z)$. Consider the set of $6$ non-degenerate
double points

$Z=2P_{1}+\cdots+2P_{6}$

in $\mathbb{P}^{3}$. If $P_{1},\ldots,P_{6}$ are not in general position in
$\mathbb{P}^{3}$, then there are $3$ points of $\\{P_{1},\ldots,P_{6}\\}$ on a
line or there are $4$ points of $\\{P_{1},\ldots,P_{6}\\}$ on a
$2$-dimensional linear subspace. So, $T_{1}(Z)\geq 5$ or $T_{2}(Z)\geq 4$,
this is a contradiction with $T(Z)=T_{p}(Z)=T_{3}(Z)=4>T_{j}(Z);j=1,2$. So,
$P_{1},\ldots,P_{6}$ are in general position in $\mathbb{P}^{3}$. By using the
Lemma 2.8 we get

$\operatorname{reg}(Z)=T(Z).$

For $n=4$, then $T_{4}(Z)=4=T(Z)$. Consider the set of seven non-degenerate
double points

$Z=2P_{1}+\cdots+2P_{7}$

in $\mathbb{P}^{4}.$ By Lemma 4.1 we get

$\operatorname{reg}(Z)\geq 4=T(Z).$

The proof of Proposition 4.2 is completed. ∎

The Proposition 4.2 help us to prove the following theorem which shows that
the Segre’s bound is attained for the regularity index of any $s+3$ non-
degenerate equimultiple fat points in $\mathbb{P}^{n}$, $s\leq n+3$.

######  Theorem 4.3.

Let $m$ be a positive integer and $Z=mP_{1}+\cdots+mP_{s}$ be a set of $s$
non-degenerate equimultiple points in $\mathbb{P}^{n}$. If $s\leq n+3$, then

$\operatorname{reg}(Z)=T(Z),$

where

$T(Z)=\max\\{T_{j}(Z)|\ j=1,\ldots,n\\},$
$T_{j}(Z):=\max\left\\{\left[\frac{mq+j-2}{j}\right]|\
P_{i_{1}},\ldots,P_{i_{q}}\text{ on a }j\text{-dimensional linear
subspace}\right\\}.$

###### Proof.

Since $P_{1},\ldots,P_{s}$ are not on any $(n-1)$-dimensional linear subspace,
we have $s\geq n+1$. We consider the three following cases for $s$.

Case $s=n+1$: Then $P_{1},\ldots,P_{n+1}$ are in general position in
$\mathbb{P}^{n}$. By Lemma 2.4 we get

$\operatorname{reg}(Z)=2m-1=T_{1}(Z)=T(Z).$

Case $s=n+2$: Then $P_{1},\ldots,P_{n+2}$ are not on a $(n-1)$-dimensional
linear subspace in $\mathbb{P}^{n}$. By Lemma 2.7 we get

$\operatorname{reg}(Z)=T(Z).$

Case $s=n+3$: If $m\neq 2$, then by using Lemma 2.10 we get

$\operatorname{reg}(mP_{1}+\cdots+mP_{n+3})=T(Z).$

If $m=2$, then by Proposition 4.2 we get

$\operatorname{reg}(2P_{1}+\cdots+2P_{n+3})=T(Z).$

∎

The following simple example shows that Theorem 3.5 is not true for $n=2$,
$s=n+4$.

From now on, if $H$ is a hyperplane in $\mathbb{P}^{n}$, we coincide $H$ with
the linear form in $R$ defining it. So, if $H_{j}$ is a hyperplane passing
through points $P_{j}$ and $m_{j}$ is a positive integer; $j=1,\ldots,r$; then
we get $H_{1}^{m_{1}}\cdots
H_{r}^{m_{r}}\in\wp_{1}^{m_{1}}\cap\cdots\cap\wp_{r}^{m_{r}}$.

######  Example 4.4.

Let $l_{1}$, $l_{2}$ be two distinct lines in $\mathbb{P}^{2}$ and
$P_{1},\ldots,P_{6}$ be six distinct points such that $P_{1},P_{2},P_{3}\in
l_{1}\setminus l_{2}$; $P_{4},P_{5}\in l_{2}\setminus l_{1}$ and $P_{6}\notin
l_{1}\cup l_{2}$.

$\text{\large P}_{1}$$\text{\large P}_{2}$$\text{\large P}_{3}$$\text{\large
P}_{4}$$\text{\large P}_{5}$$\text{\large P}_{6}$$\text{\large
l}_{1}$$\text{\large l}_{2}$

Let $m$ be a positive integer, then $Z=mP_{1}+\cdots+mP_{6}$ is a set of $6$
non-degenerate equimultiple fat points in $\mathbb{P}^{2}$. We have

$T_{1}(Z)=3m-1,\ T_{2}(Z)=\left[\frac{6m}{2}\right]=3m=T(Z).$

Let $\wp_{j}$ be the homogeneous prime ideal in
$\mathbb{R}=K[X_{0},X_{1},X_{2}]$ corresponding to the point $P_{j}$;
$j=1,\ldots,6$. Choose $P_{6}=(1,0,0)$, then $\wp_{6}=(X_{1},X_{2})$. Put
$J=\wp_{1}^{m}\cap\cdots\cap\wp_{5}^{m}$, $I=J\cap\wp_{6}^{m}$. Then $R/I$ is
the coordinate ring of $Z$, so
$\operatorname{reg}(Z)=\operatorname{reg}(R/I)$, the Castelnuovo-Mumford
regularity index of the ring $R/I$. By Lemma 2.1 we get

$\operatorname{reg}(R/I)=\max\left\\{m-1,\operatorname{reg}(R/J),\operatorname{reg}(R/(J+\wp_{6}^{m}))\right\\}.$

Since $l_{1}$ is a line passing through three points $P_{1},P_{2},P_{3}$; we
have $l_{1}^{m}\in\wp_{1}^{m}\cap\wp_{2}^{m}\cap\wp_{3}^{m}$. Since $l_{2}$ is
a line passing through two points $P_{4},P_{5}$; we have
$l_{2}^{m}\in\wp_{4}^{m}\cap\wp_{5}^{m}$. Hence, $l_{1}^{m}l_{2}^{m}\in J$.
This implies that for every monomial $M$ of degree $i$ in $X_{1},X_{2}$;
$i=0,\ldots,m-1$; we have

$l_{1}^{m}l_{2}^{m}M\in J.$

Since $l_{j}$ is a line avoiding $P_{6}=(1,0,0)$, we can write
$l_{j}=X_{0}+g_{j}$ for some $g_{j}\in\wp_{6}=(X_{1},X_{2})$; $j=1,2$. Thus we
have

$(X_{0}+g_{1})^{m}(X_{0}+g_{2})^{m}M\in J.$

Moreover, since $g_{1},g_{2}\in\wp_{6}$ and $M\in\wp_{6}^{i}$, we get
$X_{0}^{2m}M\in J+\wp_{6}^{i+1}$ for $i=0,\ldots,m-1$. This implies that

$X_{0}^{3m-1-i}M\in J+\wp_{6}^{i+1}$

for $i=0,\ldots,m-1$. By Lemma 2.2 we get

$\operatorname{reg}(R/(J+\wp_{6}^{2}))\leq 3m-1.$

Let $U=mP_{1}+\cdots+mP_{5}$, we have

$T_{1}(U)=3m-1\geq
T_{2}(U)=\left[\frac{5m}{2}\right]=2m+\left[\frac{m}{2}\right].$

So $T(U)=T_{1}(U)=3m-1$. By Lemma 2.9 we get $\operatorname{reg}(U)=3m-1.$
Since $R/J$ is the homogeneous coordinate ring of $U$. Thus

$\operatorname{reg}(R/J)=\operatorname{reg}(U)=3m-1.$

Therefore,

$\operatorname{reg}(Z)=\operatorname{reg}(R/I)=\max\\{m-1,3m-1,\operatorname{reg}(R/(J+\wp_{6}^{m}))\\}=3m-1<T(Z).$

By using Theorem 3.5 and Example 4.4 we show that Theorem 4.3 is not true for
$s=n+4$. The Segre’s bound is not attained for this case.

######  Corollary 4.5.

Let $m,n$ be positive integers, $n\geq 2$. There always exists a set of $n+4$
non-degenerate equimultiple fat points $Z=mP_{1}+\cdots+mP_{n+4}$ in
$\mathbb{P}^{n}$ such that

$\operatorname{reg}(Z)=3m-1<T(Z)=3m,$

where

$T(Z)=\max\\{T_{j}(Z)|\ j=1,\ldots,n\\},$
$T_{j}(Z):=\max\left\\{\left[\frac{mq+j-2}{j}\right]|\
P_{i_{1}},\ldots,P_{i_{q}}\text{ on a }j\text{-dimensional linear
subspace}\right\\}.$

###### Proof.

We consider the set of fat points $Z$ with the configuration as following.

Let $\alpha_{1}$ be the $2$-dimensional linear subspace in $\mathbb{P}^{n}$
defining by $n-2$ linear forms $X_{3},\ldots,X_{n}$; and $l_{1}$, $l_{2}$ be
two distinct lines in $\alpha_{1}\cong\mathbb{P}^{2}$. Let
$P_{1},\ldots,P_{3}$ be three distinct points on $l_{1}\setminus\l_{2}$; and
$P_{4},P_{5}$ be two distinct points on $l_{2}\setminus l_{1}$. Let $P_{6}$ be
a points in $\alpha_{1}$ and not on $l_{1}\cup l_{2}$. Consider the set of $6$
double points $Z_{\alpha_{1}}=mP_{1}+\cdots+mP_{6}$ in $\mathbb{P}^{2}$, by
Example 4.4 we get

$\operatorname{reg}(Z_{\alpha_{1}})=3m-1\text{ and
}T(Z_{\alpha_{1}})=T_{2}(Z_{\alpha_{1}})=3m.$

Since $P_{4},P_{5},P_{6}$ do not lie on a line, we can choose
$P_{7},\ldots,P_{n+4}$ be distinct points in $\mathbb{P}^{n}$ such that $n+1$
points $P_{4},P_{5},P_{6},P_{7},\ldots,P_{n+4}$ span $\mathbb{P}^{n}$. Then
$P_{4},\ldots,P_{n+4}$ are in linearly general position in $\mathbb{P}^{n}$
and

$Z=mP_{1}+\cdots+mP_{n+4}$

is the set of $n+4$ non-degenerate equimultiple fat points in
$\mathbb{P}^{n}$. We have

$\displaystyle T_{1}(Z)$ $\displaystyle=3m-1,T_{2}(Z)=3m,$ $\displaystyle
T_{j}(Z)$
$\displaystyle=\left[\frac{6m+(j-2)m+j-1}{j}\right]=m+\left[\frac{4m+j-2}{j}\right];j=3,\ldots,n-1.$

Thus,

$T(Z)=T_{2}(Z)=3m.$

For $j=2,\ldots,n-1$; the points $P_{4},P_{5},P_{6},\ldots,P_{j+5}$ span a
$(j+1)$-dimensional linear subspace, say $\alpha_{j}$. Put
$P_{i\alpha_{j}}=P_{i}$; $i=1,\ldots,j+5$; then

$Z_{\alpha_{j}}=mP_{1\alpha_{j}}+\cdots+mP_{(j+5)\alpha_{j}}$

is the set of $j+5$ non-degenerate equimultiple fat points in
$\mathbb{P}^{j}$. Since $\alpha_{n-1}=\mathbb{P}^{n}$, we get

$\operatorname{reg}(Z)=\operatorname{reg}(Z_{\alpha_{n-1}}).$

We argue by induction on $j$ to prove that

$\operatorname{reg}(Z_{\alpha_{j}})=\operatorname{reg}(Z_{\alpha_{1}})=3m-1$

for $j=2,\ldots,n-1$.

For $j=2$, consider the set of $7$ non-degenerate equimultiple fat points

$Z_{\alpha_{2}}=m_{1}P_{1\alpha_{2}}+\cdots+mP_{7\alpha_{2}}$

in $\mathbb{P}^{3}$. Put
$J_{\alpha_{2}}=\wp_{1\alpha_{2}}^{m}\cap\cdots\cap\wp_{6\alpha_{2}}^{m}$ and
$I_{\alpha_{2}}=J_{\alpha_{2}}\cap\wp_{7\alpha_{2}}^{m}$,
$R_{\alpha_{2}}=K[X_{0},X_{1},X_{2},X_{3}]$. Then
$\operatorname{reg}(Z_{\alpha_{2}})=\operatorname{reg}(R_{\alpha_{2}}/I_{\alpha_{2}})$.
By Lemma 2.1 we get

$\operatorname{reg}(R_{\alpha_{2}}/I_{\alpha_{2}})=\max\\{m-1,\operatorname{reg}(R_{\alpha_{2}}/J_{\alpha_{2}}),\operatorname{reg}(R_{\alpha_{2}}/(J_{\alpha_{2}}+\wp_{7\alpha_{2}}^{m}))\\}.$

Since $P_{1\alpha_{2}}=P_{1},\ldots,P_{6\alpha_{2}}=P_{6}$ are on
$\alpha_{1}\cong\mathbb{P}^{2}$; by using Example 4.4 we get

$\operatorname{reg}(R_{\alpha_{2}}/J_{\alpha_{2}})=\operatorname{reg}(Z_{\alpha_{1}})=3m-1.$

Let $P_{7\alpha_{2}}=(1,0,0,0)$ in $\mathbb{P}^{3}$. Since
$P_{7\alpha_{2}}\notin\alpha_{1}$ and
$P_{1\alpha_{2}},\ldots,P_{6\alpha_{2}}\in\alpha_{1}$, we can choose a
$2$-dimensional linear subspace $H$ in $\mathbb{P}^{3}$ passing through
$P_{1\alpha_{2}},\ldots,P_{6\alpha_{2}}$ and avoiding $P_{7\alpha_{2}}$. Then
we get

$H^{m}\in\wp_{1\alpha_{2}}^{m}\cap\cdots\cap\wp_{6\alpha_{2}}^{m}=J_{\alpha_{2}}.$

It follows that

$H^{2m-i}M\in J_{\alpha_{2}}$

for every monomial $M$ of degree $i$ in $X_{1},X_{2},x_{3}$; $i=0,\ldots,m-1$.
Since $P_{7\alpha_{2}}\notin H$, we can write $H=X_{0}+G$ for some linear form
$G\in\wp_{7\alpha_{2}}=(X_{1},X_{2},X_{3})$. Thus we have

$(X_{0}+G)^{2m-i}M\in J_{\alpha_{2}}$

for every monomial $M$ of degree $i$ in $X_{1},X_{2},x_{3}$; $i=0,\ldots,m-1$.
Moreover, since $G\in\wp_{7\alpha_{2}}$ and $M\in\wp_{7\alpha_{2}}^{i}$, we
get

$X_{0}^{2m-i}M\in J_{\alpha_{2}}+\wp_{7\alpha_{2}}^{i+1}$

for every monomial $M$ of degree $i$ in $X_{1},X_{2},x_{3}$; $i=0,\ldots,m-1$.
By Lemma 2.2 we get

$\operatorname{reg}(R_{\alpha_{2}}/(J_{\alpha_{2}}+\wp_{7\alpha_{2}}^{m}))\leq
2m.$

Therefore,

$\operatorname{reg}(Z_{\alpha_{2}})=3m-1.$

Assume that $\operatorname{reg}(Z_{\alpha_{n-2}})=3m-1$. We need prove that
$\operatorname{reg}(Z_{\alpha_{n-1}})=3m-1$. Consider the set of $n+4$ non-
degenerate equimultiple fat points

$Z=mP_{1}+\cdots+mP_{n+4}$

in $\alpha_{n-1}=\mathbb{P}^{n}$. Put
$J=\wp_{1}^{m}\cap\cdots\cap\wp_{n_{3}}^{m}$ and $I=J\cap\wp_{n+4}^{m}$. Then
$\operatorname{reg}(Z)=\operatorname{reg}(R/I)$. By Lemma 2.1 we get

$\operatorname{reg}(R/I)=\max\\{m-1,\operatorname{reg}(R/J),\operatorname{reg}(R/(J+\wp_{n+4}^{m}))\\}.$

By Theorem 3.5 and inductive assumption we get $\operatorname{reg}(R/J)=3m-1$.
Choose $P_{n+4}=(1,0,\ldots,0)$. Since $\alpha_{n-2}$ is a $(n-1)$-dimensional
linear subspace passing through $P_{1},\ldots,P_{n+3}$, we have

$\alpha_{n-2}^{m}N\in J$

for every monomial $N$ of degree $i$ in $X_{1},\ldots,X_{n}$;
$i=0,\ldots,m-1$. By a similar argue as above, since
$P_{n+4}\notin\alpha_{2}$, we can write $\alpha_{n-2}=X_{0}+L$ for some linear
form $L\in\wp_{n+4}$. Thus, we have

$(X_{0}+L)^{2m-i}N\in J.$

Moreover, since $L\in\wp_{n+4}$ and $N\in\wp_{n+4}^{i}$, we get

$X_{0}^{2m-i}N\in J+\wp_{n+4}^{i+1}$

for every monomial $N$ of degree $i$ in $X_{1},\ldots,X_{n}$;
$i=0,\ldots,m-1$. By Lemma 2.2 we get

$\operatorname{reg}(R/(J+\wp_{n+4}^{m}))\leq 2m-1.$

Therefore,

$\operatorname{reg}(Z)=3m-1.$

∎

## References

  * [1] J. Alexander et A. Hirschowitz, La méthode d’Horace éclatée: application à I’interpolation en degré quatre, Invent. Math. 107 (1992), 585-602.
  * [2] E.Ballico, O. Dumitrescu and E. Postinghel, On Segre’s bound for fat points in $P^{n}$, J. Pure Appl. Algebra 220 (2016), 2307-2323.
  * [3] B. Benedetti, G. Fatabbi and A. Lorenzini, Segre’s Bound and the case of $n+2$ fat points of $P^{n}$, Comm. Algebra 40 (2012), 395-403.
  * [4] M.V. Catalisano, N.V. Trung and G. Valla, A sharp bound for the regularity index of fat points in general position, Proc. Amer. Math. Soc. 118 (1993), 717-724.
  * [5] E.D. Davis and A.V. Geramita, The Hilbert function of a special class of 1-dimensional Cohen-Macaulay graded algebras, The Curves Seminar at Queen’s, Queen’s Papers in Pure and Appl. Math. 67 (1984), 1-29.
  * [6] G. Fatabbi, A. Lorenzini On a sharp bound for the regularity index of any set of fat points, J. Pure and Appl. Algebra 161 (2001), 91-111.
  * [7] U. Nagel and B. Trok, Segre’s regularity bound for fat point schemes, Annalli della Scuola Normale Superiore, Vol. XX (2020), 217-237.
  * [8] B. Segre, Alcune questioni su insiemi finiti di punti in geometria algebrica, Atti. Convergno. Intern. di Torino 1961, 15-33.
  * [9] P.V. Thien, Segre bound for the regularity index of fat points in $\mathbb{P}^{3}$, J. Pure and Appl. Algebra 151 (2000), 197-214.
  * [10] P.V. Thien, Regularity index of $s+2$ fat points not on a linear $(s-1)$-space, Comm. Algebra 40 (2012), 3704-3715.
  * [11] P.V. Thien and T.N. Sinh, On the regularity index of s fat points not on a linear $(r-1)$-space, $s\leq r+3$, Comm. Algebra, 45 (2017), 4123-4138.