# Statistical Analysis of Nearest Neighbor Methods for Anomaly Detection

Xiaoyi Gu
Department of Statistics
Carnegie Mellon Univeristy
Pittsburgh, PA 15213
<EMAIL_ADDRESS>
&Leman Akoglu
Heinz College of Information Systems and Public Policy
Carnegie Mellon Univeristy
Pittsburgh, PA 15213
<EMAIL_ADDRESS>
Alessandro Rinaldo
Department of Statistics
Carnegie Mellon Univeristy
Pittsburgh, PA 15213
<EMAIL_ADDRESS>

###### Abstract

Nearest-neighbor (NN) procedures are well studied and widely used in both
supervised and unsupervised learning problems. In this paper we are concerned
with investigating the performance of NN-based methods for anomaly detection.
We first show through extensive simulations that NN methods compare favorably
to some of the other state-of-the-art algorithms for anomaly detection based
on a set of benchmark synthetic datasets. We further consider the performance
of NN methods on real datasets, and relate it to the dimensionality of the
problem. Next, we analyze the theoretical properties of NN-methods for anomaly
detection by studying a more general quantity called distance-to-measure
(DTM), originally developed in the literature on robust geometric and
topological inference. We provide finite-sample uniform guarantees for the
empirical DTM and use them to derive misclassification rates for anomalous
observations under various settings. In our analysis we rely on Huber’s
contamination model and formulate mild geometric regularity assumptions on the
underlying distribution of the data.

##  1 Introduction

Anomaly detection is the process of detecting instances that deviate
significantly from the other sample members. The problem of detecting
anomalies can arise in many different applications, such as fraud detection in
financial transactions, intrusion detection for security systems, and various
medical examinations.

Depending on the availability of data labels, there are multiple setups for
anomaly detection. The first is the supervised setup, where labels are
available for both normal and anomalous instances during the training stage.
Because of its similarity to the standard classification setup, numerous
classification methods with good empirical performance and well-studied
theoretical properties can be adopted. The second setup is the semi-supervised
setup, where training data only comprise normal instances and no anomalies.
Well-known methods with theoretical guarantees include $k$NNG [1], BP-$k$NNG
[2] and BCOPS [3], with the first two methods developed based on the geometric
entropy minimization (GEM) principle proposed in [1], and the third on
conformal prediction. The third setup is the unsupervised setup, which is the
most flexible yet challenging setup. For the rest of the paper, we will only
focus on this setup and do not assume any prior knowledge on data labels.

Many empirical methods have been developed in the unsupervised setup, which
can be roughly classified into four categories: density based methods such as
the Robust KDE ($\mathrm{RKDE}$) [4], Local Outlier Factor ($\mathrm{LOF}$)
[5], and mixture models ($\mathrm{EGMM}$); distance based methods such as
$k\mathrm{NN}$ [6] and Angle-based Outlier Detection ($\mathrm{ABOD}$) [7];
model based methods such as the one-class SVM ($\mathrm{OCSVM}$) [8],
$\mathrm{SVDD}$ [9], and autoencoders [10]; ensemble methods such as Isolation
Forest ($\mathrm{IForest}$) [11] and $\mathrm{LODA}$ [12]. In practice,
ensemble methods are often times favored for their computational efficiency
and robustness to tuning parameters, yet there is little theoretical
understanding of how and why these algorithms work.

In this paper, we focus on studying NN-methods in the unsupervised setting. We
begin with an empirical analysis of NN-methods on a set of synthetic benchmark
datasets and show that they compare favorably to the other state-of-the-art
algorithms. We further discuss their performance on real datasets and relate
it to the dimensionality of the problem. Next, we provide statistical analysis
of NN-methods by analyzing the distance-to-a-measure (DTM) [13], a
generalization to the NN scheme. The quantity was initially raised in the
robust topological inference literature, in which DTM proves to be an
effective distance-like function for shape reconstruction in the presence of
outliers [14]. We give finite sample uniform guarantees on the empirical DTM,
and also demonstrate how DTM classifies the anomalies, under suitable
assumptions on the underlying distribution of the data.

##  2 Empirical Performance of NN-methods

Two versions of the NN anomaly detection algorithms have been proposed:
$k^{\mathrm{th}}\mathrm{NN}$ [15] and $k\mathrm{NN}$ [6].
$k^{\mathrm{th}}\mathrm{NN}$ assigns anomaly score of an instance by computing
the distance to its $k^{\text{th}}$-nearest-neighbor, whereas $k\mathrm{NN}$
takes the average distance over all $k$-nearest-neighbors. Both methods are
shown to have competitive performance in various comparative studies [16, 17,
12, 18]. In particular, the comparative study developed by Goldstein and
Uchida [16] is the one of most comprehensive analysis to date that includes
the discussion of NN-methods and, at the same time, aligns with the
unsupervised anomaly detection setup. However, the authors omit the analysis
of ensemble methods, some of which are considered as state-of-the-art
algorithms (e.g., $\mathrm{IForest}$ and $\mathrm{LODA}$). Emmott et al. [19]
constructed a large corpus (over 20,000) of synthetic benchmark datasets that
vary across multiple aspects (e.g., clusteredness, separability, difficulty,
etc). The authors evaluate the performance of eight top-performing algorithms,
including $\mathrm{IForest}$ and $\mathrm{LODA}$, but omit the analysis of NN-
methods. In this section, we provide a comprehensive empirical analysis of NN-
methods by comparing $k\mathrm{NN}$, $k^{\mathrm{th}}\mathrm{NN}$, and
$\mathrm{DTM}_{2}$111$\mathrm{DTM}_{2}$ stands for the empirical DTM (see
Section 3) with $q=2$. We include its empirical analysis here for comparison
purposes. to $\mathrm{IForest}$, $\mathrm{LOF}$ and $\mathrm{LODA}$ on (1) the
corpus of synthetic datasets developed in [19], (2) 23 real datasets from the
ODDS library [20], and (3) 6 high dimensional datasets from the UCI library
[21]. The code for all our experiments will be made publicly available. In
general, no one methodology should be expected to performs well in all
possible scenarios. In Section 4 of the Appendix we present different examples
in which IForest, LODA, LOF and $\mathrm{DTM}_{2}$ perform very differently.

###  2.1 Comparison on Benchmark Datasets

First, we complement Emmott et al.’s study [19] by extending it to NN-based
detectors. First, we calculate the ROC-AUC (AUC) and Average Precision (AP)
scores for each method on each benchmark, and compute their respective
quantiles on the empirical distributions for AUC and AP scores (refer to
Appendix E in [19] for more details on treating AUC and AP as random
variables). We say that an algorithm fails on a benchmark with metric AUC (or
AP) at significance level $\alpha$ if the computed AUC (or AP) quantiles are
less than $(1-\alpha)$. Then, the failure rate for each algorithm is found as
the percentage of failures over the entire benchmark corpus. The results are
shown in Table 1, where the top section is copied from [19] and the bottom
section shows the failure rates we obtained for $k\mathrm{NN}$,
$k^{\mathrm{th}}\mathrm{NN}$, and $\mathrm{DTM}_{2}$. The "Either" column
indicates that the benchmarks fail under at least one of the two metrics. For
all three methods, $k$ is set as $0.03\times($sample size), same as the
parameter the authors used for $\mathrm{LOF}$. Among all methods,
$\mathrm{IForest}$ gives the lowest failure rates (boldfaced) for all three
metrics. $k\mathrm{NN}$ and $\mathrm{DTM}_{2}$ turn out to be next-best top
performers, falling marginally behind $\mathrm{IForest}$.

Table 1: Algorithm Failure Rate with Significance Level $\alpha=0.001$. | AUC | AP | Either
---|---|---|---
$\mathrm{ABOD}$ | 0.5898 | 0.6784 | 0.7000
$\mathrm{IForest}$ | 0.5520 | 0.6514 | 0.6741
$\mathrm{LODA}$ | 0.6187 | 0.6955 | 0.7194
$\mathrm{LOF}$ | 0.6016 | 0.7071 | 0.7331
$\mathrm{RKDE}$ | 0.6122 | 0.7030 | 0.7194
$\mathrm{OCSVM}$ | 0.7218 | 0.7342 | 0.7969
$\mathrm{SVDD}$ | 0.8482 | 0.8868 | 0.9080
$\mathrm{EGMM}$ | 0.6188 | 0.7146 | 0.7303
$k\mathrm{NN}$ | 0.5646 | 0.6744 | 0.6960
$k^{\mathrm{th}}\mathrm{NN}$ | 0.5831 | 0.6886 | 0.7100
$\mathrm{DTM}_{2}$ | 0.5669 | 0.6761 | 0.6977

###  2.2 Comparison on Datasets from the ODDS library

Next, we compare the performance of $\mathrm{IForest}$, $\mathrm{LODA}$,
$\mathrm{LOF}$, $\mathrm{DTM}_{2}$, $k\mathrm{NN}$ and
$k^{\mathrm{th}}\mathrm{NN}$ on 23 real datasets from the ODDS library [20].
Figure 1 presents the overall distributions of AUC and AP scores of the five
methods as boxplots. It appears that all methods except for $\mathrm{LOF}$
have comparable performance, and we further verified this claim via pairwise
Wilcoxon signed-rank tests between methods, which showed no statistically
significant difference at level 0.05. The exact AUC and AP scores for each
dataset are given in the Appendix.

(a) AUC

(b) AP

Figure 1: Boxplots for AUC and AP scores on 23 real datasets.

###  2.3 Effect of the dimension

We then take a closer look at the performance of $\mathrm{IForest}$,
$\mathrm{LODA}$, $\mathrm{LOF}$, $\mathrm{DTM}_{2}$, $k\mathrm{NN}$ and
$k^{\mathrm{th}}\mathrm{NN}$ when the data is high dimensional. Additionally,
we include the analysis of $\mathrm{DTMF}_{2}$ in our experiments, a quantity
defined as the inverse ratio of the $\mathrm{DTM}_{2}$ of a point and the
average $\mathrm{DTM}_{2}$ of its $k$-nearest neighbors. $\mathrm{DTMF}_{2}$
can be interpreted as a $\mathrm{LOF}$ version of $\mathrm{DTM}_{2}$ and is
described in the Appendix. We consider six high dimensional real datasets from
the UCI library [21] (see [12] for details) and compute the AUC and AP scores
for each algorithm. The results are presented in Table 2(b). The $n$ and $d$
columns stand for the number of samples and dimension of the datasets. On
datasets gisette, isolet and letter, the performance of $\mathrm{IForest}$ and
$\mathrm{LODA}$ have significantly downgraded; the NN-methods give somewhat
better performance, whereas $\mathrm{LOF}$ and $\mathrm{DTMF}_{2}$ are showing
significantly stronger performance. However, on datasets cancer and
ionoshphere, where dimensions are slightly lower, the situations are reversed,
with $\mathrm{LOF}$ and $\mathrm{DTMF}_{2}$ giving significantly worse
performance than the others. This is consistent with our findings in Section
2.2. The deficiency of $\mathrm{IForest}$ in high dimensions is expected, as
the $\mathrm{IForest}$ trees are generated by random partitioning along a
randomly selected feature. However, in high dimensions, there is a high
probability that a large number of features are neglected in the process.

Table 2: AUC and AP performance on high dimensional datasets

AUC | $n$ | $d$ | $\mathrm{IForest}$ | $\mathrm{LODA}$ | $\mathrm{LOF}$ | $k\mathrm{NN}$ | $k^{\mathrm{th}}\mathrm{NN}$ | $\mathrm{DTM}_{2}$ | $\mathrm{DTMF}_{2}$
---|---|---|---|---|---|---|---|---|---
gisette | 3850 | 4970 | 0.5023 | 0.5176 | 0.6753 | 0.5696 | 0.5429 | 0.5692 | 0.7051
isolet | 4886 | 617 | 0.5485 | 0.5421 | 0.7330 | 0.6810 | 0.6480 | 0.6796 | 0.7645
letter | 4586 | 617 | 0.5600 | 0.5459 | 0.7846 | 0.7162 | 0.6826 | 0.7149 | 0.8096
madelon | 1430 | 500 | 0.5327 | 0.5427 | 0.5450 | 0.5608 | 0.5552 | 0.5607 | 0.5546
cancer | 385 | 30 | 0.9528 | 0.9626 | 0.8097 | 0.9780 | 0.9756 | 0.9773 | 0.6937
ionosphere | 242 | 33 | 0.9265 | 0.9118 | 0.9450 | 0.9832 | 0.9803 | 0.9824 | 0.9372
(a) AUC

AP | $n$ | $d$ | $\mathrm{IForest}$ | $\mathrm{LODA}$ | $\mathrm{LOF}$ | $k\mathrm{NN}$ | $k^{\mathrm{th}}\mathrm{NN}$ | $\mathrm{DTM}_{2}$ | $\mathrm{DTMF}_{2}$
---|---|---|---|---|---|---|---|---|---
gisette | 3850 | 4970 | 0.0877 | 0.0907 | 0.1628 | 0.1093 | 0.1015 | 0.1092 | 0.1723
isolet | 4886 | 617 | 0.1005 | 0.1003 | 0.2343 | 0.2074 | 0.1846 | 0.2070 | 0.2458
letter | 4586 | 617 | 0.0956 | 0.0980 | 0.2921 | 0.2328 | 0.2054 | 0.2319 | 0.3010
madelon | 1430 | 500 | 0.1067 | 0.0974 | 0.1171 | 0.1209 | 0.1181 | 0.1209 | 0.1166
cancer | 385 | 30 | 0.6274 | 0.8277 | 0.3121 | 0.8813 | 0.8840 | 0.8864 | 0.2800
ionosphere | 242 | 33 | 0.7222 | 0.7438 | 0.6058 | 0.8903 | 0.8801 | 0.8868 | 0.6105
(b) AP

Overall, our experiments show that $\mathrm{IForest}$ and NN-methods are the
top two methods with excellent overall performance on both low dimensional
synthetic and real datasets. However, NN-methods exhibit better performance
than $\mathrm{IForest}$ when the data is high dimensional. In the following
sections, we provide a theoretical understanding of how the NN-methods work
under the anomaly detection framework.

##  3 Theoretical Analysis

In this section we formalize the settings for a simple yet natural anomaly
detection problem based on the classic Huber-contamination model [22, 23],
whereby a target distribution generating normal observations is corrupted by a
distribution from which anomalous observations are drawn. We introduce the
notion of distance-to-a-measure (DTM) [13], as an overall functional of the
data based on nearest neighbors statistics and provide finite sample bounds on
the empirical nearest neighbor radii and on the rates of consistency of the
DTM in the supremum norm. These theoretical guarantees are novel and may be of
independent interest. Finally, we derive conditions under which DTM-based
methods provably separate normal and anomalous points, as a function of the
level of contamination and the separation between the normal distribution and
the anomalous distribution. All the proofs are given in the Appendix.

###  3.1 Problem Setup

We assume we observe $n$ i.i.d. realization
$\mathbb{X}_{n}=(X_{1},\ldots,X_{n})$ from a distribution $P$ on
$\mathbb{R}^{d}$ that follows the Huber contamination model [22, 23]

$\displaystyle P=(1-\varepsilon)P_{0}+\varepsilon P_{1},$

where $P_{0}$ and $P_{1}$ are, respectively, the underlying distribution for
the normal and anomalous instances, and $\varepsilon\in[0,1)$ is the
proportion of contamination. Letting $S_{0}$ and $S_{1}$ be the support of
$P_{0}$ and $P_{1}$, respectively, we further assume that $S_{0}\cap
S_{1}=\emptyset$. The distributions $P_{0}$ and $P_{1}$, their support and the
level of contamination $\varepsilon$ are unknown.

Our goal is devise a procedure that is able to discriminate the normal
observations $X_{i}$’s belonging to $S_{0}$, from the anomalous one, falling
in the set $S_{1}$. Since we will be focusing exclusively on NN methods, we
will begin by introducing a population counterpart to the notion of $k$th
nearest neighbor. Throughout the article, for any $x\in\mathbb{R}^{d}$ and
$r>0$, $B(x,r)$ denotes the closed Euclidean ball of radius $r$ centered at
$x$.

######  Definition 3.1 ($p$-NN radius).

Let $p\in(0,1)$. For any $x$, define $r_{p}(x)$ to be the radius of the
smallest ball centered at $x$ with $P$-probability mass at least $p$.
Formally,

$r_{p}(x)=\inf\\{r>0:P(B(x,r))\geq p\\}.$

Setting, for a non-negative integer $k\leq n$, $p=\frac{k}{n}$, the $k$th-
nearest neighbor radius of a point $x\in\mathbb{R}^{d}$ with respect to the
sample $(X_{1},\ldots,X_{n})$ is simply the $p$-NN radius $\hat{r}_{p}(x)$ of
the corresponding empirical measure $P_{n}$ – the probability measure that
puts mass $1/n$ on each $X_{i}$. Thus,

$P_{n}(B(x,\hat{r}_{p}(x))=\frac{1}{n}\left|\\{X_{1},\ldots,X_{n}\\}\cap
B(x,\hat{r}_{p}(x))\right|\geq\frac{k}{n}.$

Throughout the rest of the article, we take $p=\frac{k}{n}$.

We will impose the following, mild regularity assumptions on the distribution
$P$:

  * •

Assumption (A0):
$S_{0}$ is compact, and $S_{0}$ and $S_{1}$ are disjoint.

  * •

Assumption (A1):
There exists positive constants $C=C(P)$ and
$\varepsilon_{0}=\varepsilon_{0}(P)$ such that for all
$0<\varepsilon<\varepsilon_{0}$ and $\eta\in\mathbb{R}$,

$|P(B(x,r_{p}(x)+\eta))-P(B(x,r_{p}(x)))|\leq\varepsilon\Rightarrow|\eta|<C\varepsilon,$

for $P$-almost every $x$.

  * •

Assumption (A2):
$P_{0}$ satisfies the (a,b)-condition: For $b>0$, for any $x\in S_{0}$, there
exists $a=a(x)>0$, and $r>0$ such that $P_{0}(B(x,r))\geq\min\\{1,ar^{b}\\}$.

Intuitively, assumption (A1) implies that $P$ has non-zero probability content
around the boundary of $B(x,r_{p}(x))$. Observing further that the function
$r\in\mathbb{R}_{+}\mapsto F_{x}(r)=P(B(x,r))$ is the c.d.f. of the random
variable $\|X-x\|$, where $X\sim P$, then a sufficient condition for (A1) to
hold is that, uniformly over all $x$, $F_{x}$ has its derivative uniformly
bounded away from zero in a fixed neighborhood of $r_{p}(x)$. This condition,
originally formulated in [14] to derive bootstrap-based confidence bands for
the DTM function, appears to be a natural regularity assumption in the
analysis of NN-type methods. When $a(x)=a$ for all $x\in S_{0}$, assumption
(A2) reduces to a widely used condition in the literature on statistical
inference for geometric and topological data analysis [24, 25]. Such condition
requires the support of $P_{0}$ to not locally resemble a lower dimensional
set; in particular, it prevents $S_{0}$ from having thin ridges or outward
cusps. When (A2) is violated, it becomes impossible to estimate $S_{0}$, no
matter the size of the sample. The parameter $b$ can be interpreted as the
intrinsic dimension of $P$. In particular, if $P$ admits a strictly positive
density on a $D$-dimensional smooth manifold, then it can be shown that $b=D$.

######  Definition 3.2 (DTM [13]).

The distance-to-a-measure (DTM) with respect to a probability distribution $P$
with parameter $m\in(0,1)$ and power $q\geq 1$ is defined as

$\displaystyle
d(x)=d_{P,m,q}(x)=\left(\frac{1}{m}\int_{0}^{m}r_{p}(x)^{q}\,dp\right)^{1/q}.$
(1)

When $q=\infty$, we set $d(x)=d_{P,m,\infty}(x)=r_{m}(x)$.

It is immediate from the definition that a point $x\in\mathbb{R}^{d}$ has a
small DTM value $d(x)$ if its $p$-NN radii, when averaged across all
$p\in(0,m)$ are small. Intuitively, $d(x)$ can be thought of as a measure of
the distance of $x$ from the bulk of the mass of the probability distribution
$P$ at level of accuracy specified by the parameter $m$. The choice of the
parameter $q$ allows to weight differently the impact of large versus small
$p$-NN radii.

By substituting $r_{p}(x)$ with $\hat{r}_{p}(x)$ in (1), the empirical DTM can
be seen to be

$\displaystyle\hat{d}(x)=d_{P_{n},m,q}(x)=\left(\frac{1}{k}\sum_{X_{i}\in
N_{k}(x)}\|X_{i}-x\|^{q}\right)^{1/q},$

where $k=\lceil mn\rceil$ and $N_{k}(x)$ denotes the set of $k$-nearest
neighbors to $x$ among the sample. Different values of $q\geq 1$ yield
different NN-functionals. In particular, the empirical DTM with $q=1$ is
equivalent to the $k\mathrm{NN}$ method, and the empirical DTM with $q=\infty$
is equivalent to $k^{\mathrm{th}}\mathrm{NN}$. The notion of DTM was initially
introduced in the geometric inference literature [14], where DTM was developed
for shape reconstruction under the presence of outliers. The DTM is known to
have several nice properties: it is 1-Lipschitz and it is robust with respect
to perturbations of the original distributions with respect to the Wasserstein
distance. The case of $q=2$ is special: the corresponding DTM, denoted below
as $\mathrm{DTM}_{2}$, is also semi-concave and distance-like, and admits
strong regularity conditions on its sub-level sets. Chazal et al. [14] have
also derived the limiting distribution and confidence set for DTM.

###  3.2 Uniform bounds for $\hat{r}_{p}$ and $\hat{d}$

In this section we derive finite sample bounds on the deviation of
$\hat{r}_{p}$ and $\hat{d}$ from $r_{p}$ and $d_{P,m,q}$, respectively, that
hold uniformly over all $x\in\mathbb{R}^{d}$ or only over the sample points.
These theoretical guarantees are, to the best of our knowledge, novel and may
be of independent interest.

######  Theorem 3.3.

Let $\delta\in(0,1)$, denote
$\beta_{n}=\sqrt{(4/n)((d+1)\log{2n}+\log{(8/\delta)})}$. Under assumption
(A1), the following bound is satisfied with probability at least $1-\delta$:

$\displaystyle\sup_{x}|\hat{r}_{p}(x)-r_{p}(x)|\leq
C(\beta_{n}^{2}+\beta_{n}\sqrt{p})$

The dimension $d$ enters in the previous bound in such a way that, for fixed
$p$, $\sup_{x}|\hat{r}_{p}(x)-r_{p}(x)|\to 0$ with high probability provided
that $\frac{d}{n}\to 0$. If we limit the supremum only to the sample points,
then the dependence on the dimension disappears altogether and we can instead
achieve a nearly-parametric rate of $\sqrt{\frac{\log n}{n}}$.

######  Theorem 3.4.

Let $\delta\in(0,1)$, $p=k/n$, and $p^{\prime}=(k-1)/(n-1)$.
Denote $\alpha_{n}=\sqrt{(4/(n-1))(\log{2(n-1)}+\log{(8n/\delta)})}$. Under
assumption A1, the following bound is satisfied with probability at least
$1-\delta$:

$\displaystyle\max_{i=1,\dots,n}|\hat{r}_{p}(X_{i})-r_{p}(X_{i})|\leq
C(\alpha_{n}^{2}+\alpha_{n}\sqrt{p^{\prime}}+\frac{1}{n})$

The results in Theorem 3.3 and Theorem 3.4 yield the following uniform bounds
for the DTM of all order.

######  Theorem 3.5.

Under assumption (A1), with probability at least $1-\delta$,

$\displaystyle\sup_{x}|d(x)-\hat{d}(x)|\leq C\beta_{n}(\beta_{n}+\sqrt{m}),$
(2)

and

$\displaystyle\max_{i=1,\dots,n}|d(X_{i})-\hat{d}(X_{i})|\leq
C\alpha_{n}(\alpha_{n}+\sqrt{m}).$ (3)

where $\beta_{n}$ and $\alpha_{n}$ are defined in Theorem 3.3 and Theorem 3.4.

######  Remark.

The bound in Theorem 3.5 holds for all choices of $q\geq 1$, including the
case of $q=\infty$. Evaluating explicitly the integral
$\int_{0}^{m}(\beta_{n}+\sqrt{p})^{q}\,dp$ will bring out an explicit
dependence on $q$ but will not lead to better rates.

###  3.3 DTM for anomaly detection: theoretical guarantees

We are now ready to derive some theoretical guarantees on the performance of
DTM-based methods for discriminating normal and anomalous points in the sample
$(X_{1},\ldots,X_{n})$ according to the Huber-contamination model described
above in Section 3.1. We recall that in our setting, a sample point $X_{i}$ is
normal if it belongs to the support $S_{0}$ of $P_{0}$, and is otherwise
deemed an anomaly if it lies in $S_{1}$, the support of $P_{1}$, where
$S_{1}\cap S_{0}=\emptyset$.

The methodology we consider is quite simple, and it is consistent with the
prevailing practice of assigning to each sample point a score that expresses
its degree of being anomalous compared to the other points. In detail, we rank
the sample points based on their empirical DTM values, and we declare the
points with largest empirical DTM values as anomalies. This simple procedure
will work perfectly well if

$\max_{X_{i}\colon X_{i}\in S_{0}}\hat{d}(X_{i})<\min_{x_{I}\in
S_{1}}\hat{d}(X_{i})$

and if the difference between the two quantities is large. In general, of
course, one would expect that some sample points in $S_{0}$ may have smaller
empirical DTMs of some of the points in $S_{1}$. The extent to which such
incorrect labeling occurs depends on two key factors: how closely the
empirical DTM tracks the true DTM and whether the population DTM could itself
discriminate normal points versus anomalous ones. The former issue can be
handled using the high probability bounds on the stochastic fluctuations of
the empirical DTM obtained in the previous section. The latter issue will
instead require to specify some degree of separation between the mixture
components $P_{0}$ and $P_{1}$, both in terms the distance between their
supports but also in terms of how their probability mass gets distributed.
There is more than one way to formalize this setting. Here we choose to remain
completely agnostic to the form of the contaminating distribution $P_{1}$, for
which we impose virtually no constraint. On the other hand, we require the
normal distribution $P_{0}$ to satisfy condition (A2) above in such a way that
point inside the support will have larger values of $a(x)$ than points near
the boundary of $S_{0}$. This condition, which is satisfied if for example
$P_{0}$ admits a Lebesgue density whose values increase as a function of the
distance from the boundary of $S_{0}$, ensures that the population DTM will be
large near the boundary of $S_{0}$ and small everywhere else. As a result,
incorrect labeling of normal points will only occur around the boundary of
$S_{0}$ but not inside the bulk the distribution $P_{0}$. We formalize this
intuition in our next result, which is purely deterministic.

######  Proposition 3.6.

Under assumptions (A0) and (A2), suppose that $a(x)=g(d(x,\partial S_{0}))$,
where $g(z)$ is a non-decreasing function on $[0,z_{0})$ for some $z_{0}$, and
$g(z)\geq g(z_{0})$ for all $z\geq z_{0}$. Let

$\eta=\min_{x\in S_{0},y\in S_{1}}\|x-y\|$ (4)

be the distance between $S_{0}$ and $S_{1}$ and $h>0$ be a given threshold
parameter. For any $m>\varepsilon$, additionally assume that

$g(z_{0})\geq
g_{0}:=\begin{cases}\frac{m}{1-\varepsilon}\left(\frac{b+q}{b}\left(\frac{m-\varepsilon}{m}\eta^{q}-h\right)\right)^{-b/q}&1\leq
q<\infty\\\ \frac{m}{1-\varepsilon}(\eta-h)^{-b}&q=\infty.\end{cases}$ (5)

Next, define the "safety zone" $A_{\eta}$ as

$A_{\eta}=\left\\{x\in S_{0}:d(x,\partial S_{0})\geq g^{-1}(g_{0})\right\\}$
(6)

Then, we have

$\displaystyle\sup_{x\in A_{\eta}}d_{P,m,q}(x)+h<\inf_{y\in
S_{1}}d_{P,m,q}(y).$ (7)

The main message from the previous result is that there exists a subset
$A_{\eta}$ of the support of the normal distribution, which intuitively
corresponds to a region deep inside the support of $P_{0}$ of high density,
over which the population DTM will be smaller than at any point in the support
$S_{1}$ of the contaminating distribution. Thus, the true DTM is guaranteed to
perfectly separate $A_{\eta}$ from $S_{1}$, making mistakes (possibly) only
for the normal points in $S_{0}\setminus A_{\eta}$.

Notice that the definition of $A_{\eta}$ depends on all the relevant
quantities, namely the contamination parameter $\varepsilon$, the probability
parameter $m$, the dimension $b$ of $P_{0}$ and the order $q$ of the DTM
through the expression (5). Importantly, it is necessary that $m>\varepsilon$,
otherwise inequality (7) maybe not be satisfied. For example, we can take
$P_{1}$ to have point mass at a single point $y$; then $r_{P,t}(y)=0$ for all
$t\leq m$, and the right hand side of (7) is zero.

When $g(0)=a_{0}>0$, which occurs, e.g., if $P_{0}$ has a density bounded away
from $0$ over its support, implies that $A_{\eta}=S_{0}$ if

$\displaystyle\eta>\left(\frac{m}{m-\varepsilon}\left(\frac{b}{b+q}\left(\frac{m}{a_{0}(1-\varepsilon)}\right)^{q/b}+h\right)\right)^{-1/q}.$

That is, when $S_{0}$ and $S_{1}$ are sufficiently well-separated, the DTM
will classify all the points in $S_{0}$ as normals.

The parameter $h$ serves as a buffer that allows one to replace the DTM
function $d(x)$ with any estimator that is close to it in the supremum norm by
no more than $h$. Thus, we may plug-in the high-probability bounds of Theorem
3.4 and Theorem 3.3 to conclude that the empirical DTM will will identify all
normal instances within $A_{\eta}$ correctly, with high probability.

######  Corollary 3.6.1.

Taking $h$ to be twice the upper bound in (3), we get, with probability at
least $1-\delta$,

$\displaystyle\max_{X_{i}\in A_{\eta}}\hat{d}_{P,m,q}(X_{i})<\min_{X_{i}\in
S_{1}}\hat{d}_{P,m,q}(X_{i}).$

Similarly, if $h$ is twice the upper bound in (2), we have that

$\sup_{x\in A_{\eta}}\hat{d}_{P,m,q}(x)<\inf_{y\in S_{1}}\hat{d}_{P,m,q}(y).$
(8)

The guarantee in (8) calls for a higher sample complexity that depends on the
dimension $d$. At the same time, it extends to all the points in $A_{\eta}$
and not just the sample points. Thus the DTM can accurately identify not only
the normal instance in the sample but any other normal instance, such as
future observations.

###  3.4 Illustrative examples

We illustrate the separation condition in Proposition 3.6 with the following
example. Consider a collection of normal instances generated from a standard
normal distribution. Figure 2 shows the mis-classification rates for
$\mathrm{DTM_{2}}$ as a cluster of 5 anomalies approaches the normal
instances. The color of each point represents its class, with black being the
normal instances and red being anomalies. The radius of the circle around each
point represents its empirical DTM score, and the color of the circle
represents its predicted class from $\mathrm{DTM_{2}}$. As we see, as the
anomalies approach the normal instances, more and more data around the
boundaries of the normal distribution get mis-classified as anomalies.

(a) High Separation

(b) Medium Separation

(c) Low Separation

Figure 2: Performance of DTM when the separation distance between the normal
instances and anomalies gradually decreases.

##  4 Conclusions

In this paper we have presented empirical evidence, based on simulated and
real-life benchmark datasets, that NN-based methods show very good performance
at identifying anomalous instances in an unsupervised anomaly detection set-
up. We have introduced a simple but natural framework for anomaly detection
based on the Huber contamination model and have used it to characterize the
performance of a class of NN methods for anomaly detection that are based on
the distance-to-a-measure (DTM) functional. In our results we rely on various
geometric and analytic properties of the underlying distribution to the
accuracy of DTM-methods for anomaly detection. We are able to demonstrate
that, under mild conditions, NN methods will mis-classify normal points only
around the boundary of the support of the distribution generating normal
instances and have quantified this phenomenon rigorously. Finally, we have
derived novel finite sample bounds on the nearest neighbor radii and on the
rate of convergence of the empirical DTM to the true DTM that may be of
independent interest.

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##  Appendix A Definition for $\mathrm{DTMF}_{2}$

######  Definition A.1.

The $\mathrm{DTM}_{2}$ and $\mathrm{DTMF}_{2}$ scores are defined as:

$\displaystyle\mathrm{DTM}_{2}(x)=\left(\frac{1}{k}\sum_{X_{i}\in
N_{k}(x)}\|X_{i}-x\|^{2}\right)^{1/2},$
$\displaystyle\mathrm{DTMF}_{2}(x)=\frac{1}{|N_{k}(x)|}\sum_{y\in
N_{k}(x)}\frac{\mathrm{DTM}_{2}(y)}{\mathrm{DTM}_{2}(x)}.$

##  Appendix B Proof of Theorems

###  B.1 Proof of Theorem 3.3

###### Proof.

By standard VC theory [26, 27], for any ball $B\subset\mathbb{R}^{d}$, we have

$\displaystyle P(B)\geq p+\beta_{n}^{2}+\beta_{n}\sqrt{p}\Rightarrow
P_{n}(B)\geq p.$ (9) $\displaystyle
P(B)<p-\beta_{n}^{2}-\beta_{n}\sqrt{p}\Rightarrow P_{n}(B)<p$ (10)

with probability at least $1-\delta$.

Step1: First, we want to show that

$\displaystyle\hat{r}_{p}(x)\leq r_{p}(x)+C(\beta_{n}^{2}+\beta_{n}\sqrt{p})$
(11)

for all $x$. By definition of $r_{p}(x)$, we have $P(B(x,r_{p}(x)))\geq p$.
Define $r^{+}=\inf\\{r:P(B(x,r))\geq p+\beta_{n}^{2}+\beta_{n}\sqrt{p}\\}$.
Then, we have

$\displaystyle P(B(x,r^{+}))\geq p+\beta_{n}^{2}+\beta_{n}\sqrt{p}\Rightarrow
P_{n}(B(x,r^{+}))\geq p$

by (9). Therefore, $r^{+}\geq\hat{r}_{p}(x)$. Next, note that $r_{p}(x)\leq
r^{+}$. If $r_{p}(x)=r^{+}$, (11) holds trivially. If $r_{p}(x)<r^{+}$, then
for all $s$ such that $r_{p}(x)<s<r^{+}$, we have

$\displaystyle p\leq P(B(x,r_{p}(x)))\leq P(B(x,s))\leq
p+\beta_{n}^{2}+\beta_{n}\sqrt{p}.$

Then by assumption (A1),

$\displaystyle s\leq r_{p}(x)+C(\beta_{n}^{2}+\beta_{n}\sqrt{p}).$

Taking $s\uparrow r^{+}$, we get $\hat{r}_{p}(x)\leq r^{+}\leq
r_{p}(x)+C(\beta_{n}^{2}+\beta_{n}\sqrt{p})$ as desired.

Step 2: Next, we want to show the reverse direction:

$\displaystyle r_{p}(x)\leq\hat{r}_{p}(x)+C(\beta_{n}^{2}+\beta_{n}\sqrt{p}).$
(12)

Let $r^{-}=\inf\\{r:P(B(x,r))\geq p-\beta_{n}^{2}-\beta_{n}\sqrt{p}\\}$. Then,
clearly $r^{-}\leq r_{p}(x)$ and $P(B(x,r^{-}))\geq
p-\beta_{n}^{2}-\beta_{n}\sqrt{p}$. For all $s<r^{-}$, we have

$\displaystyle P(B(x,s))<p-\beta_{n}^{2}-\beta_{n}\sqrt{p}$
$\displaystyle\Rightarrow P_{n}(B(x,s))<p$ $\displaystyle\Rightarrow
s<\hat{r}_{p}(x)$

where the first implication follows from (10). Taking $s\uparrow r^{-}$, we
get $r^{-}\leq\hat{r}_{p}(x)$. If $r^{-}=r_{p}(x)$, (12) holds trivially. If
$r^{-}<r_{p}(x)$, then for any $u$ satisfying $r^{-}<u<r_{p}(x)$, we have

$\displaystyle p-\beta_{n}^{2}-\beta_{n}\sqrt{p}\leq P(B(x,r^{-}))\leq
P(B(x,u))\leq p$ $\displaystyle\Rightarrow u\leq
r^{-}+C(\beta_{n}^{2}+\beta_{n}\sqrt{p}).$

Taking $u\uparrow r_{p}(x)$, we get

$\displaystyle r_{p}(x)\leq
r^{-}+C(\beta_{n}^{2}+\beta_{n}\sqrt{p})\leq\hat{r}_{p}(x)+C(\beta_{n}^{2}+\beta_{n}\sqrt{p})$

as desired. ∎

###  B.2 Proof of Theorem 3.4

###### Proof.

By assumption (A1), it suffices to show that

$\displaystyle\max_{i=1,\dots,n}|P(B(x_{i},r_{p}(x_{i})))-P(B(x_{i},\hat{r}_{p}(x_{i})))|\leq\alpha_{n}^{2}+\alpha_{n}\sqrt{p^{\prime}}+\frac{1}{n}$

with probability higher than $1-\delta$.

Fix $X_{i}$, and let $P_{i,n-1}$ be the marginal distribution of
$\mathbb{X}_{n}\backslash\\{X_{i}\\}$. Define the event

$A_{i,n}=\\{|P(B(X_{i},\hat{r}_{p}(X_{i})))-P_{i,n-1}(B(X_{i},\hat{r}_{p}(X_{i})))|>\alpha_{n}^{2}+\alpha_{n}\sqrt{p^{\prime}}\\},$

and thus

$\displaystyle\max_{i=1,\dots,n}|P(B(X_{i},\hat{r}_{p}(X_{i})))-P_{i,n-1}(B(X_{i},\hat{r}_{p}(X_{i})))|>\alpha_{n}^{2}+\alpha_{n}\sqrt{p^{\prime}}=\bigcup_{i=1}^{n}A_{i,n}.$

Then, $A_{i,n}$ is contained in the event

$B_{i,n}=\\{|P(B(X_{i},r))-P_{i,n-1}(B(X_{i},r))|>\alpha_{n}^{2}+\alpha_{n}\sqrt{p^{\prime}},\,\forall
r>0\\}.$

We have,

$\displaystyle P(\bigcup_{i=1}^{n}A_{i,n})\leq
P(\bigcup_{i=1}^{n}B_{i,n})\leq\sum_{i=1}^{n}\mathbb{E}_{i}[\mathbb{E}_{-i}[1_{B_{i,n}(X_{1},\dots,X_{n})}]]$

where $1_{B_{i,n}}(\cdot)$ is the indicator function of $B_{i,n}$,
$\mathbb{E}_{i}$ and $\mathbb{E}_{-i}$ are respectively the expectation with
respect to the marginal distribution of $X_{i}$ and
$\mathbb{X}_{n}\backslash\\{X_{i}\\}$.

Since for each fixed realization of $X_{i}$, the class of balls
$B(x_{i},\cdot)$ centered at $x_{i}$ with arbitrary radius has VC dimension 1,
by standard VC theory [26, 27], we have

$\displaystyle\mathbb{E}_{-i}[1_{B_{i,n}(X_{1},\dots,X_{i-1},x_{i},X_{i+1},\dots,X_{n})}]\leq\frac{\delta}{n}.$

Therefore,

$\displaystyle\max_{i=1,\dots,n}|P(B(X_{i},\hat{r}_{p}(X_{i})))-P_{i,n-1}(B(X_{i},\hat{r}_{p}(X_{i})))|\leq\alpha_{n}^{2}+\alpha_{n}\sqrt{p^{\prime}}$

with probability at least $1-\delta$. Next, since

$\displaystyle|P_{i,n-1}(B(X_{i},\hat{r}_{p}(X_{i})))-P_{n}(B(X_{i},\hat{r}_{p}(X_{i})))|=|p^{\prime}-p|\leq\frac{1}{n},$
$\displaystyle\max_{i=1,\dots,n}|P(B(X_{i},\hat{r}_{p}(X_{i})))-P_{n}(B(X_{i},\hat{r}_{p}(X_{i})))|\leq\alpha_{n}^{2}+\alpha_{n}\sqrt{p^{\prime}}+\frac{1}{n}$

with probability at least $1-\delta$. Finally, notice that assumption (A1)
ensures that the c.d.f of the random variable $\|X-x\|$ is almost surely
continuous at $r_{p}(x)$. Thus, we have
$P_{n}(B(X_{i},\hat{r}_{p}(X_{i})))=p=P(B(X_{i},r_{p}(X_{i})))$. ∎

###  B.3 Proof of Theorem 3.5

###### Proof.

With probability at least $1-\delta$,

$\displaystyle\sup_{x}|d(x)-\hat{d}(x)|$
$\displaystyle=\sup_{x}\left|\left(\frac{1}{m}\int_{0}^{m}r_{p}(x)^{q}\,dp\right)^{1/q}-\left(\frac{1}{m}\int_{0}^{m}\hat{r}_{p}(x)^{q}\,dp\right)^{1/q}\right|$
$\displaystyle\leq\sup_{x}\left(\frac{1}{m}\int_{0}^{m}(r_{p}(x)-\hat{r}_{p}(x))^{q}\,dp\right)^{1/q}$
By Minkowski Inequality $\displaystyle\leq
C\beta_{n}(\frac{1}{m}\int_{0}^{m}(\beta_{n}+\sqrt{p})^{q}\,dp)^{1/q}$ By
Theorem 3.3 $\displaystyle\leq
C\beta_{n}(\beta_{n}+\sqrt{m})\left(\frac{1}{m}\int_{0}^{m}dp\right)^{1/q}$
$\displaystyle=C\beta_{n}(\beta_{n}+\sqrt{m}).$

∎

###  B.4 Proof of Proposition 3.6

###### Proof.

Equivalently, by the definition of DTM, we will need to show that

$\displaystyle\sup_{x\in
A_{\eta}}\frac{1}{m}\int_{0}^{m}r_{P,t}(x)^{q}\,dt<\inf_{y\in
S_{1}}\frac{1}{m}\int_{0}^{m}r_{P,t}(y)^{q}\,dt-h.$ (13)

Since for any $x\in A_{\eta}$, $P(B(x,r))\geq(1-\varepsilon)P_{0}(B(x,r))$, we
have $r_{P,t}(x)\leq
r_{(1-\varepsilon)P_{0},t}(x)=r_{P_{0},t/1-\varepsilon}(x)$. When
$r=(\frac{t}{a(x)(1-\varepsilon)})^{1/b}$, by the $(a,b)$-condition of
$P_{0}$, we have $P_{0}(B(x,r))\geq a(x)r^{b}=\frac{t}{1-\varepsilon}$. Hence,
$r_{P_{0},t/1-\varepsilon}(x)\leq(\frac{t}{a(x)(1-\varepsilon)})^{1/b}$.
Putting the inequalities together, the LHS of (13) gives

$\displaystyle\frac{1}{m}\int_{0}^{m}r_{P,t}(x)^{q}\,dt$
$\displaystyle\leq\frac{1}{m}\int_{0}^{m}(\frac{t}{a(x)(1-\varepsilon)})^{q/b}\,dt$
$\displaystyle=\frac{b}{b+q}(\frac{m}{a(x)(1-\varepsilon)})^{q/b}$

Next, consider the right hand side of (13). We have that

$\displaystyle\inf_{y\in S_{1}}\frac{1}{m}\int_{0}^{m}r_{P,t}(y)^{q}\,dt$
$\displaystyle=\inf_{y\in
S_{1}}\frac{1}{m}\int_{0}^{\varepsilon}r_{P,t}(y)^{q}\,dt+\frac{1}{m}\int_{\varepsilon}^{m}r_{P,t}(y)^{q}\,dt$
$\displaystyle\geq 0+\frac{1}{m}\int_{\varepsilon}^{m}\eta^{q}\,dt$
$\displaystyle=\frac{m-\varepsilon}{m}\eta^{q}$

Hence, (13) holds if

$\displaystyle\frac{b}{b+q}\left(\frac{m}{a(x)(1-\varepsilon)}\right)^{q/b}<\frac{m-\varepsilon}{m}\eta^{q}-h$
(14) $\displaystyle\Leftrightarrow
a(x)>\frac{m}{1-\varepsilon}\left(\frac{b+q}{b}\left(\frac{m-\varepsilon}{m}\eta^{q}-h\right)\right)^{-b/q}$
(15) $\displaystyle\Leftrightarrow x\in A_{\eta}$ (16)

∎

##  Appendix C Simulation Results on 23 Real Datasets from ODDS

Table 3 gives the exact AUC and AP scores of $\mathrm{IForest}$,
$\mathrm{LODA}$, $\mathrm{LOF}$, $\mathrm{DTM}_{2}$, $k\mathrm{NN}$, and
$k^{\mathrm{th}}\mathrm{NN}$ on 23 real datasets from the ODDS library.

Table 3: Performance of $\mathrm{IForest}$, $\mathrm{LODA}$, $\mathrm{LOF}$,
$\mathrm{DTM}_{2}$, $k\mathrm{NN}$, and $k^{\mathrm{th}}\mathrm{NN}$ on 23
real datasets from the ODDS library.

AUC | $\mathrm{IForest}$ | $\mathrm{LODA}$ | $\mathrm{LOF}$ | $\mathrm{DTM}_{2}$ | $k\mathrm{NN}$ | $k^{\mathrm{th}}\mathrm{NN}$
---|---|---|---|---|---|---
annthyroid | 0.846217 | 0.711716 | 0.688763 | 0.677126 | 0.681196 | 0.662250
arrhythmia | 0.774180 | 0.789645 | 0.763778 | 0.807466 | 0.806681 | 0.815473
breastw | 0.988089 | 0.987891 | 0.376371 | 0.980041 | 0.979805 | 0.982081
cardio | 0.925666 | 0.904219 | 0.705637 | 0.831097 | 0.820695 | 0.880306
glass | 0.706775 | 0.771816 | 0.737669 | 0.867751 | 0.867209 | 0.869106
ionosphere | 0.842363 | 0.853369 | 0.899506 | 0.928007 | 0.928148 | 0.920141
letter | 0.600280 | 0.622487 | 0.842000 | 0.856193 | 0.861893 | 0.809837
lympho | 1.000000 | 0.992958 | 0.981221 | 0.977700 | 0.977700 | 0.978286
mammography | 0.853864 | 0.866368 | 0.819344 | 0.850100 | 0.850604 | 0.849169
mnist | 0.792829 | 0.595506 | 0.839678 | 0.862295 | 0.861369 | 0.861813
musk | 0.999944 | 0.994193 | 0.286222 | 0.957031 | 0.936976 | 1.000000
optdigits | 0.714978 | 0.714282 | 0.612373 | 0.560559 | 0.537313 | 0.842404
pendigits | 0.961689 | 0.950902 | 0.850733 | 0.958278 | 0.950210 | 0.970528
pima | 0.675037 | 0.618657 | 0.557993 | 0.636045 | 0.634418 | 0.639545
satellite | 0.686132 | 0.725766 | 0.578879 | 0.768331 | 0.764688 | 0.795738
satimage-2 | 0.993326 | 0.994631 | 0.991675 | 0.999054 | 0.999079 | 0.998954
shuttle | 0.997529 | 0.992264 | 0.522135 | 0.989215 | 0.984996 | 0.993954
speech | 0.441678 | 0.441248 | 0.478689 | 0.482781 | 0.483310 | 0.478594
thyroid | 0.978939 | 0.954587 | 0.963042 | 0.946970 | 0.947420 | 0.943083
vertebral | 0.359048 | 0.338889 | 0.495714 | 0.331746 | 0.330794 | 0.323968
vowels | 0.739488 | 0.757411 | 0.937155 | 0.961067 | 0.963144 | 0.946216
wbc | 0.943177 | 0.958517 | 0.910764 | 0.948113 | 0.946379 | 0.949980
wine | 0.776471 | 0.963025 | 0.428151 | 0.994958 | 0.993277 | 0.996218
(a) AUC

AUC | $\mathrm{IForest}$ | $\mathrm{LODA}$ | $\mathrm{LOF}$ | $\mathrm{DTM}_{2}$ | $k\mathrm{NN}$ | $k^{\mathrm{th}}\mathrm{NN}$
---|---|---|---|---|---|---
annthyroid | 0.336719 | 0.221278 | 0.252282 | 0.201405 | 0.203313 | 0.191132
arrhythmia | 0.422741 | 0.479021 | 0.372709 | 0.491718 | 0.489785 | 0.511596
breastw | 0.972689 | 0.970735 | 0.272824 | 0.945230 | 0.944475 | 0.951773
cardio | 0.577570 | 0.579294 | 0.202455 | 0.404516 | 0.399020 | 0.450174
glass | 0.096007 | 0.140315 | 0.193538 | 0.162208 | 0.162824 | 0.155266
ionosphere | 0.794018 | 0.794706 | 0.866694 | 0.928604 | 0.928993 | 0.911973
letter | 0.089123 | 0.090754 | 0.334208 | 0.260399 | 0.268795 | 0.200453
lympho | 1.000000 | 0.835714 | 0.655556 | 0.723611 | 0.723611 | 0.695202
nmammography | 0.193517 | 0.264330 | 0.130258 | 0.167475 | 0.169236 | 0.161568
mnist | 0.267129 | 0.143844 | 0.397000 | 0.404172 | 0.403502 | 0.387391
musk | 0.998328 | 0.881940 | 0.021850 | 0.618577 | 0.496054 | 1.000000
noptdigits | 0.051332 | 0.047997 | 0.033582 | 0.032489 | 0.031128 | 0.081907
npendigits | 0.328479 | 0.263207 | 0.077311 | 0.217698 | 0.193527 | 0.315036
npima | 0.506879 | 0.491468 | 0.391361 | 0.486558 | 0.485157 | 0.492184
satellite | 0.659824 | 0.693257 | 0.406871 | 0.639164 | 0.634576 | 0.680913
satimage-2 | 0.936035 | 0.911899 | 0.516222 | 0.972246 | 0.972113 | 0.972834
shuttle | 0.983694 | 0.825359 | 0.261309 | 0.746971 | 0.694083 | 0.818767
speech | 0.016421 | 0.015298 | 0.018916 | 0.019062 | 0.019088 | 0.018781
thyroid | 0.595528 | 0.276667 | 0.397719 | 0.297644 | 0.296979 | 0.285007
vertebral | 0.094209 | 0.090683 | 0.121829 | 0.089739 | 0.089664 | 0.088901
vowels | 0.179951 | 0.181478 | 0.396334 | 0.484752 | 0.501906 | 0.403366
wbc | 0.588631 | 0.640832 | 0.279934 | 0.495254 | 0.488687 | 0.554438
wine | 0.192461 | 0.544417 | 0.072027 | 0.941540 | 0.928312 | 0.954040
(b) AP

##  Appendix D Performance on the Difficult Examples

Figure 3 gives three examples of difficult situations where some algorithms
will very likely fail. The black dots represent the normal instances, and the
two red dots represent anomalies. In Figure 3(a) where the anomalies are
located in the center of a circle of normal points, $\mathrm{IForest}$ and
$\mathrm{LODA}$ will have a hard time detecting the anomalies, whereas
$\mathrm{LOF}$ and NN-methods have no trouble. In Figure 3(b), if the
anomalies are locally relatively far away from a group of normal points, NN-
methods, $\mathrm{IForest}$, and $\mathrm{LODA}$ won’t be able to pick them
up, whereas $\mathrm{LOF}$ is designed to handle this specific case. However,
we observed through extensive simulations that $\mathrm{LOF}$ can easily make
mistakes on global anomalies, and Figure 3(c) gives one such example. If we
have a cluster of anomalies located at some distance from a collection of
normal points, $\mathrm{LOF}$ tends to mis-identify some of the anomalies as
normal points, whereas the other methods have no such problem.

(a) Ring

(b) Local Anomalies

(c) Clustered Anomalies

Figure 3: Examples of difficult datasets.

Figure 4 5 6 show the performance of $\mathrm{IForest}$, $\mathrm{LODA}$,
$\mathrm{LOF}$, and $\mathrm{DTM}_{2}$ in each of the difficult examples. The
radius of the circle around each point gives the anomaly score of each
algorithm, and the color of the circle represents the predicted class by the
algorithm.

(a) $\mathrm{IForest}$

(b) $\mathrm{LODA}$

(c) $\mathrm{DTM}_{2}$

(d) $\mathrm{LOF}$

Figure 4: Performance on the difficult datasets. Case: ring

(a) $\mathrm{IForest}$

(b) $\mathrm{LODA}$

(c) $\mathrm{DTM}_{2}$

(d) $\mathrm{LOF}$

Figure 5: Performance on the difficult datasets. Case: local anomalies

(a) $\mathrm{IForest}$

(b) $\mathrm{LODA}$

(c) $\mathrm{DTM}_{2}$

(d) $\mathrm{LOF}$

Figure 6: Performance on the difficult datasets. Case: clustered anomalies


# Indecomposable motivic cycles on $K3$ surfaces of degree $2$

Ramesh Sreekantan  Ramesh Sreekantan

###### Abstract

In this paper we construct new indecomposable motivic cycles in the group
$H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))$ where $X$ is a degree $2$ $K3$
surface. This generalizes our construction in [Sre22] for Kummer surfaces of
Abelian surfaces as well as the recent work of Ma and Sato [MS23] on degree
$2$ $K3$ surfaces.

###### Contents

  1. 1 Introduction
    1. 1.1 Degree 2 $K3$ surfaces
    2. 1.2 Motivic cycles
    3. 1.3 Acknowledgements
  2. 2 $K3$ surfaces of degree 2
    1. 2.1 $K3$ surfaces as double covers of ${\mathds{P}}^{2}$ ramified at a sextic
    2. 2.2 Lattices and the Moduli of $K3$ surfaces
    3. 2.3 $K3$ surfaces with larger Picard number
  3. 3 Motivic cycles
    1. 3.1 Presentation of cycles in $H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))$.
    2. 3.2 The localization sequence
  4. 4 Construction of the cycles on the generic $K3$ surface of degree $2$.
    1. 4.1 A Theorem in Enumerative Geometry
    2. 4.2 Indecomposable Motivic cycles
      1. 4.2.1 Construction
      2. 4.2.2 Indecomposability
  5. 5 Applications
    1. 5.1 Mumford’s conjecture.
    2. 5.2 Hodge-${\mathcal{D}}$-conjecture for $K3$ surfaces over a local field
    3. 5.3 Relations with modular forms

##  1 Introduction

###  1.1 Degree 2 $K3$ surfaces

A degree $2$ $K3$ surface $X$ is a $K3$ surface on which there is a line
bundle ${\mathcal{L}}$ such that
${\mathcal{L}}^{2}=({\mathcal{L}}.{\mathcal{L}})=2$. This is equivalent to the
$K3$ surface being a double cover of ${\mathds{P}}^{2}$ ramified at a sextic.
The linear system $|{\mathcal{L}}|$ gives a map to ${\mathds{P}}^{2}$ which is
ramified at a sextic.

Conversely if $S$ is a sextic in ${\mathds{P}}^{2}$ with at most isolated
rational double point singularities, then the desingularisation of the double
cover ramified at $S$ is a $K3$ surface of degree $2$.

Generically the rank of the Neron-Severi group (the Picard number $\rho(X)$)
is $1$ \- though there are some interesting special cases when the rank is
much larger. If the sextic is a product of six lines, then the rank is $16$
and the moduli of such $K3$ surfaces is four dimensional.

On this $4$-fold there are two notable subvarieties. If the six lines are
futher tangent to a conic, then the $K3$ surface is a Kummer surface of a
principally polarised Abelian surface and the Picard number is $17$ [BW03]. If
the six lines degenerate such that three of them are concurrent then Clingher
and Malmendier show that the $K3$ surface is the Kummer surface of an Abelian
surface with a polarization of type $(1,2)$ [CM17].

###  1.2 Motivic cycles

In this paper we construct elements in the motivic cohomology group
$H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))$ where $X$ is a degree $2$ $K3$
surface. The group $H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))$ has many avatars
- it is usually defined as a graded piece of $K_{1}(X)$ with respect to the
Adams filtration, but for us we will use the fact that it is the same as the
Bloch higher Chow group $CH^{2}(X,1)$.

The cycles we construct are indecomposable – namely they are non-trivial
elements of the quotient of the motivic cohomology group by subgroup of cycles
coming from lower graded pieces of the filtration. In a sense they are ‘new’
cycles.

In general it is not so easy to find indecomposable cycles and the discovery
of them is a minor cause for celebration. They have a number of applications -
from algebraicity of values of Greens functions [Mel08],[Sre22], to the
Hodge-${\mathcal{D}}$-conjecture [CL05], [Sre08],[Sre14] to torsion in co-
dimension $2$ [Mil92],[Spi99]. For varieties over number fields, the existence
of indecomposable cycles is predicted by the Beilinson conjectures though
there are few cases when these have been realised.

The method of construction is similar to that in [Sre22] where we studied the
case of Abelian surfaces in some detail. The idea is to show the existence of
certain rational curves on ${\mathds{P}}^{2}$. In this paper we show that the
same construction can be generalised to this much larger class of varieties,
The Kummer surfaces of principally polarised Abelian surfaces are special
cases of $K3$ surfaces of degree $2$.

As this paper was nearing completion we became aware of the preprint of Ma and
Sato [MS23] where they also construct indecomposable motovic cycles in this
case. Their methods of proof are quite different though. Their construction
can be viewed as a special case when the rational curve is a line.

A corollary of the construction is another proof of a conjecture of Mumford’s
on the existence of infiniitely many rational curves on a generic $K3$ surface
of degree $2$.

###  1.3 Acknowledgements

The history of this article is linked with the online seminar on $K3$ surfaces
which has resulted in these proceedings. I spoke in the seminar on the
construction in the article [Sre22] which was about Kummer surfaces of
principally polarised Abelian surfaces, Prof. Armand Brumer was in the
audience and he asked about the non-principally polarised case. When I dug
into the literature I became aware of the work of Clingher and Malmendier
[CM17] on the $(1,2)$ case and it was clear that the argument extends to that
case. It soon became clear that the argument extends to the general degree $2$
case as well. So I would first like to thank Devendra Tiwari and the rest of
the organisers for inviting me to speak and second would like to thank Armand
Brumer for his comment. I would also like to thank Shouhei Ma and Ken Sato for
their comments. I would like to thank Subham Sarkar for his comments and
making me aware of the preprint of Ma and Sato. I would like to thank Arvind
Nair for his remarks on orthogonal Shimura varieties. Finally I would like to
thank the Indian Statistical Institute for their support.

##  2 $K3$ surfaces of degree 2

###  2.1 $K3$ surfaces as double covers of ${\mathds{P}}^{2}$ ramified at a
sextic

Let $X$ be a $K3$ surface with a line bundle ${\mathcal{L}}$ such that
${\mathcal{L}}^{2}=2$. The linear system $|{\mathcal{L}}|$ has no fixed points
and maps to ${\mathds{P}}^{2}$. This is a double cover ramified at a sextic
$S=S_{X}$. We call $S_{X}$ the associated sextic of $X$. The hyperplane
section on ${\mathds{P}}^{2}$ pulls back to a double cover of a line ramified
at six points - hence is a genus $2$ curve in $|{\mathcal{L}}|$ [May72]. On
the complement of an infinite union of closed subvarieties, the Picard number
$\rho(X)$ of the such $K3$ surfaces is $1$ generated by this curve [Sha80].
Such $K3$ surfaces are called generic.

Conversely suppose $S$ is a sextic in ${\mathds{P}}^{2}$ with only isolated
double point singularities. Then the minimal desingularisation of the double
cover ramified at $S$ is a $K3$ surface of degree $2$. If there are double
points on the sextic the Picard number of the $K3$ surfaces is larger as the
exceptional cycles in the desingularization give new elements of the Neron-
Severi group.

The compactification of the moduli of smooth sextics was first studied by Shah
[Sha80] and more recently Yu and Zheng [YZ23] have studied the singular sextic
cases.

###  2.2 Lattices and the Moduli of $K3$ surfaces

In this section we state some results about the moduli of $K3$ surfaces of
degree $2$. These results can be found in Bruinier [Bru08]. We have followed
Peterson [Pet15].

If $X$ is a $K3$ surface, the group $H^{2}(X,{\mathds{Z}})$ is of rank $22$.
As a lattice it is

$H^{2}(X,{\mathds{Z}})=U^{\oplus 3}\bigoplus E_{8}(-1)^{\oplus 2}$

where $U$ is a unimodular lattice of rank $2$ and signature $(1,1)$ and
$E_{8}(-1)$ is the root lattice of the Lie algebra $E_{8}$ with the $(-1)$
indicating that we take the negative of the standard positive definite
quadratic form. Hence this is a lattice of signature $(3,19)$

In $H^{2}(X,{\mathds{Z}})$ the class of ${\mathcal{L}}$ gives us an element
${\mathcal{L}}$ of square $2$. Let $L_{2}$ be the orthogonal complement
${\mathcal{L}}^{\perp}$ of ${\mathcal{L}}$ in $H^{2}(X,{\mathds{Z}})$. This is
a lattice of signature $(2,19)$

$L_{2}=<-2>\bigoplus U^{\oplus 2}\bigoplus E_{8}(-1)^{\oplus 3}$

where $<-2>$ is the rank 1 lattice generated by an element $w$ such that
$(w,w)=-2$.

The moduli of $K3$ surfaces of degree $2$ can be realised as an orthogonal
Shimura variety associated to the lattice $L_{2}$ as follows.

For a lattice $L$ let $O(L)$ denote the group of isometries of $L$. Let
$L^{\wedge}$ denote the dual lattice

${\hat{L}}=\\{m\in L\otimes{\mathds{Q}}|(m,n)\in{\mathds{Z}}\text{ for all
}n\in L\\}$

The discriminant group is $D_{L}={\hat{L}}/L$. Let
${\tilde{O}}(L)=\operatorname{Ker}\\{O(L)\rightarrow O(D_{L})\\}$. Let
${O^{+}}(L)$ denote the subgroup of $O(L)$ of spinor norm $1$ and
${\tilde{O}^{+}}(L)={\tilde{O}}(L)\cap{O^{+}}$.

Let

${\mathcal{D}}\cup\overline{{\mathcal{D}}}=\\{{\mathds{C}}z|\;(z,z)=0,(z,\bar{z})>0\\}\subset{\mathds{P}}(L\otimes{\mathds{C}})$

Choose a component ${\mathcal{D}}$. The moduli space of $K3$ surfaces of
degree $2$ with transcendental lattice $L$ is

${\mathcal{M}}_{L}={\tilde{O}^{+}}(L)\backslash{\mathcal{D}}$

An alternative description is as a quotient of the Hermitian symmetric space
of $O(L)$. If $L$ is of signature $(2,n)$ this is

${\mathds{H}}=\frac{O(2,n)}{O(2)\times O(n)}$

The moduli space ${\mathcal{M}}$ is

${\mathcal{M}}={\tilde{O}^{+}}(L)\backslash{\mathds{H}}$

When $L$ is the lattice $L_{2}$ this moduli ${\mathcal{M}}_{2}$ is a $19$
dimensional variety. If one considers singular sextics, the transcendental
lattice becomes smaller as the exceptional fibres are elements of the Neron-
Severi group. One interesting submoduli is when the sextic $S$ degenerates in
to six lines. Here the Picard number is $16$ as the Neron-Severi is generated
by the class of ${\mathcal{L}}$ and the $15$ exceptional fibres over the nodes
of $S$. The transcendental lattice is hence of signature $(2,4)$ and the
moduli space is $4$ dimensional.

In this there is a particularly interesting divisor – if the six lines are
tangent to a conic. Such $K3$ surfaces have Picard number $17$ and the
transcendental lattice is of signature $(2,3)$. The new cycle is obtained as a
component of the double cover of the conic. These $K3$ surfaces are the Kummer
surfaces of principaly polarised abelian surfaces and the moduli is nothing
but an alternative description of the Siegel modular threefold.

In between there are several other moduli corresponding to singular sextics.
These are described in Yu and Zheng. [YZ23]

###  2.3 $K3$ surfaces with larger Picard number

As mentioned above, when the sextic has nodal singulaties, the corresponding
$K3$ surface has additional elements in the Neron-Severi group coming from the
exceptional fibres. In this section we will determine another condition under
which there are new elements in the Neron-Severi group. In what follows we use
${\mathcal{L}}$ to denote the class of the line bundle ${\mathcal{L}}$ in the
Neron-Severi group. Let $D$ be an element of $NS(X)$ which is not a multiple
of ${\mathcal{L}}$. The moduli of those $K3$ surfaces of degree $2$ where the
Neron-Severi group is generated by ${\mathcal{L}}$ and $D$ is a divisor on
${\mathcal{M}}_{2}$.

Recall that $({\mathcal{L}},{\mathcal{L}})=2$. Consider the intersection
matrix of the lattice generated by ${\mathcal{L}}$ and $D$,

$A=\begin{pmatrix}2&({\mathcal{L}}.D)\\\ ({\mathcal{L}}.D)&(D.D)\end{pmatrix}$

Let $\Delta=\Delta(D)=-\det(A)=({\mathcal{L}}.D)^{2}-2D^{2}$ and
$\delta=(D.{\mathcal{L}})\operatorname{mod}2$, so $\delta=\\{0,1\\}$. From the
Hodge Index theorem we have that $\Delta(D)\geq 0$ and $\Delta>0$ if and only
if $D$ is not a multiple of ${\mathcal{L}}$.

The irreducible Noether-Lefschetz divisor ${{\mathcal{P}}}_{\Delta,d}$ is the
closure of the set of polarised $K3$ surfaces where the Picard lattice is a
rank $2$ lattice with discriminant $\Delta(D)=\Delta$ and class $\delta$.

For number theoretic applications one considers Heegner divisors which are
sums of these ${{\mathcal{P}}}_{\Delta,\delta}$ for certain choices of
$\Delta$. [Pet15]. Numbers determined by them are known to appear as
coefficients of modular forms.

It is known that the Picard number of a Abelian surface with real
multiplication is at least $2$ which translates to the corresponding Kummer
surface having Picard number at least $18$. Birkenhake-Wilhelm [BW03] showed,
generalizing Humbert, that this corresponds to certain special rational curves
on the associated ${\mathds{P}}^{2}$. Conversely, if one has special rational
curves on ${\mathds{P}}^{2}$ there are additional cycles in the corresponding
$K3$ surface.

Along the lines of their argument, we have the following. Recall that a $K3$
surface $X$ is called generic if the Picard number $\rho(X)=1$.

######  Proposition 2.1.

Let $X$ be a generic $K3$ surface of degree $2$ and $S_{X}$ the associated
sextic in ${\mathds{P}}^{2}$. Let $\pi:X\rightarrow{\mathds{P}}^{2}$ denote
the double cover of ${\mathds{P}}^{2}$ ramified at $S_{X}$ induced by
${\mathcal{L}}$. Suppose there exists a rational curve $Q$ of degree $d$ in
${\mathds{P}}^{2}$ such that it meets $S_{X}$ only at points of even
multiplicity. Then $Q$ determines an element in the Neron-Severi group of $X$
whose class is not a multiple of the class of $L$. In particular, the moduli
point of $X$ lies on ${{\mathcal{P}}}_{\Delta,\delta}$ for some $\Delta$ and
$\delta$.

###### Proof.

The Picard number of ${\mathds{P}}^{2}$ is $1$ so the class of $Q$ is a
multiple of the class of the hyperplane section $H$. Since it is of degree
$d$, $Q\in|dH|$. $\pi^{*}(H)={\mathcal{L}}$ hence
$\pi^{-1}(Q)\in|{\mathcal{L}}^{d}|$. In particular, it is not a new element.
So we cannot simply use the preimage of $Q$.

Let $C=\pi^{-1}(Q)$ be the double cover of $Q$ induced by the map
$\pi:X\rightarrow{\mathds{P}}^{2}$. Let $\eta:\tilde{C}\longrightarrow C$
denote its normalization. The ramification points of $\pi:C\longrightarrow Q$
are the points $S_{X}\cap Q$. By assumption $S_{X}$ and $Q$ meet with even
multiplicity. In the double cover $C$ these are nodal points. In the
normalization $\tilde{C}$, these are no longer ramified. Since $Q$ is rational
the normalization $\tilde{C}$ is then an unramified double cover of
${\mathds{P}}^{1}$. This cannot be irreducible as ${\mathds{P}}^{1}$ is simply
connected and does not have an irreducible unramified double cover. Since it
is a double cover it has two components, say $\tilde{C}_{1}$ and
$\tilde{C}_{2}$.

Hence $C=C_{1}\cup C_{2}$ for some possibly singular rational curves $C_{1}$
and $C_{2}$. To show that the classes of $C_{1}$ and $C_{2}$ are not multiples
of the class of ${\mathcal{L}}$ we show that $\Delta(C_{1})>0$.

Let $\iota$ denote the involution determined by the double cover. One has

$\iota(C_{1})=C_{2}$

The involution extends to the Neron-Severi group and acts trivially on
${\mathcal{L}}$. If $C_{1}$ is a multiple of ${\mathcal{L}}$ then
$\iota(C_{1})=C_{1}$. Hence $C_{1}=C_{2}$ and the whole curve is ramified.

Further this implies

$C_{1}^{2}=(C_{1},C_{2})=3d$

as they meet at the double points. Since $\pi$ is a double cover we have

$(C,{\mathcal{L}})=2(Q,H)=2d$

where $H$ is the hyperplane section in ${\mathds{P}}^{2}$. Since
$C=C_{1}+C_{2}=2C_{1}$ this implies

$(C_{1},{\mathcal{L}})=d$

Therefore

$\Delta(C_{1})=({\mathcal{L}},C_{1})^{2}-2(C_{1}^{2})=d^{2}-6d=d(d-6)$

This cannot be $0$ unless $d=0$, in which case $Q$ is of degree $0$, or $d=6$.
If $d=0$ then $Q$ is a point. Hence the only possibility is $d=6$ and $Q$ is
the ramified sextic. Mayer [May72], Propostion 3(d), shows that the ramified
sextic is irreducible, so $\pi^{-1}(Q)$ cannot split in to two components. ∎

In fact the theorem likely holds for higher genus curves as well [BW03],
Proposition 6.4. We will only be concerned with the case when the curve is
rational in what follows and hence will restrict ourselves to that case.

The converse need not be true. Namely there exists
${{\mathcal{P}}}_{\Delta,\delta}$ such that the extra cycle need not be
represented by a rational curve.

For a rational curve $Q$ which meets the sextic $S$ at points of even
multiplicity, let ${{\mathcal{P}}}_{Q}$ denote the copy of
${{\mathcal{P}}}_{\Delta(Q),\delta}$ where the lattice is determined by $Q$.

In the case when the sextic is a product of six lines tangent to a conic it is
known that the $K3$ surface $X$ is the Kummer surface of a principally
polarized Abelian surface. In that case $\Delta$ is called the Humbert
invariant and corresponds to the moduli of Abelian surfaces with endomorphism
ring containing ${\mathds{Z}}[\sqrt{\Delta}]$. For a generic Kummer surface
the Picard number is $17$ and on these Humbert surfaces the Picard number is
at least $18$. This special case was studied in some detail in [Sre22].

In the case when the sextic degenerates to six lines three of which are
concurrent the $K3$ surface is the Kummer surface of an Abelian surface with
polarization of type $(1,2)$ [CM17]. Once again the
${{\mathcal{P}}}_{\Delta,\delta}$ will correspond to Abelian surfaces with
extra endomorphism.

##  3 Motivic cycles

###  3.1 Presentation of cycles in $H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))$.

Let $X$ be a surface over a field $K$. The group
$H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))$ has the following presentation. It
is generated by sums of the form

$\sum(C_{i},f_{i})$

where $C_{i}$ are curves on $X$ and $f_{i}$ functions on them satisfying the
co-cycle condition

$\sum\operatorname{div}(f_{i})=0$

Equivalently these can be though of as codimensional $2$ subvarieties of
$X\times{\mathds{P}}^{1}$ which do not meet $X\times 0$ and $X\times\infty$.
The cycle given by the sum of the graphs of $f_{i}$ in
$X\times{\mathds{P}}^{1}$ has this property.

Relations are given by the Tame symbol of a pair of functions. If $f$ and $g$
are two functions on $X$, the Tame symbol of the pair

$\tau(f,g)=\sum_{Z\in
X^{1}}\left(Z,\frac{f^{\operatorname{ord}_{g}(Z)}}{g^{\operatorname{ord}_{f}(Z)}}\right)$

satisfies the co-cycle condition and such cycles are defined to be $0$ is the
group.

One example of an element of this group is a cycle of the form $(C,a)$ where
$a$ is a nonzero constant function and $C$ is a curve on $X$. This trivially
satisfies the cocycle condition. More generally, if $L/K$ is a finite
extension and $X_{L}=X\times_{\operatorname{Spec}(K)}\operatorname{Spec}(L)$
then one has a product map

$\bigoplus_{L/K}H^{2}_{{\mathcal{M}}}(X_{L},{\mathds{Q}}(1))\otimes
H^{1}_{{\mathcal{M}}}(X_{L},{\mathds{Q}}(1))\longrightarrow\bigoplus_{L/K}H^{3}_{{\mathcal{M}}}(X_{L},{\mathds{Q}}(2))\stackrel{{\scriptstyle
Nm_{L:K}}}{{\longrightarrow}}H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))$

The image of this is the group of decomposable cyles
$H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))_{dec}$. The group of indecomposable
cycles is the quotient group

$H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))_{indec}=H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))/H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))_{dec}$

In general it is not clear how to construct non-trivial elements of this
group.

A way of constructing cycles in this group is as follows. Since we use it in
the main construction we label it a proposition.

######  Proposition 3.1.

Let $Q$ be a nodal rational curve on $X$ with node $P$. Let
$\nu:\tilde{Q}\rightarrow Q$ be its normalization in the blow up
$\pi:\tilde{X}\longrightarrow X$ at $P$ The strict transform $\tilde{Q}$ meets
the exceptional fibre $E_{P}$ at two points $P_{1}$ and $P_{2}$. Both
$\tilde{Q}$ and $E_{P}$ are rational curves. Let $f_{P}$ be the function with
$\operatorname{div}(f_{P})=P_{1}-P_{2}$ on $\tilde{Q}$ and similarly let
$g_{P}$ be the function with $\operatorname{div}(g_{P})=P_{2}-P_{1}$ on
$E_{P}$. Then

$\tilde{Z}_{P}=(\tilde{Q},f_{P})+(E_{P},g_{P})$

is an element of $H^{3}_{{\mathcal{M}}}(\tilde{X},{\mathds{Q}}(2))$ and its
pushforward $Z_{P}=\pi_{*}(\tilde{Z}_{P})$ is an element of
$H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))$.

A priori it is not clear that if this cycle is indecomposable. We will show
that is the case in many instances. For this we need to use the localization
sequence in motivic cohomology.

###  3.2 The localization sequence

If ${\mathcal{X}}\rightarrow S$ is a family of varieties over a base $S$ with
$X_{\eta}$ the generic fibre and $X_{s}$ the fibre over a closed subvariety
$s$ then one has a localization sequence relating the three.

$\cdots\rightarrow
H^{3}_{{\mathcal{M}}}(X_{\eta},{\mathds{Q}}(2)\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}\bigoplus_{s\in
S^{1}}H^{2}_{{\mathcal{M}}}(X_{s},{\mathds{Q}}(1))\longrightarrow
H^{4}_{{\mathcal{M}}}({\mathcal{X}},{\mathds{Q}}(2))\rightarrow\cdots$

where $S^{1}$ denotes the set of closed subvarieties of codimension $1$. The
boundary map of an element $Z=\sum_{i}(C_{\eta,i},f_{\eta,i})$ is given as
follows. Let ${\mathcal{C}}$ denote the closure of $C_{\eta}$ in
${\mathcal{X}}$. Then

$\partial(Z)=\sum_{i}\operatorname{div}_{{\mathcal{C}}_{i}}(f_{\eta,i})$

The cocycle condition $\sum_{i}\operatorname{div}_{C_{\eta,i}}(f_{\eta,i})=0$
implies that the ‘horizontal’ divisors cancel out and only the components in
some of the fibres survive.

Let ${\mathcal{X}}$ be a family of $K3$ surfaces. To check for
indecomposability one can use the boundary map $\partial$. Over the algebraic
closure of the base field, a decomposable cycle in
$H^{3}_{{\mathcal{M}}}(X_{\eta},{\mathds{Q}}(2))$ is given by
$(C_{\eta},a_{\eta})$ where $C_{\eta}$ is a curve on $X_{\eta})$ and
$a_{\eta}$ a constant (that is, a function on the base $S$).

The boundary map is particularly simple to compute in the case of a
decomposable element

$\partial((C_{\eta},a_{\eta}))=\sum\operatorname{ord}_{s}(a_{\eta})C_{s}$

where $C_{s}$ is the restriction of the closure ${\mathcal{C}}$ of $C_{\eta}$
to the fibre over $s$. Hence in the boundary of decomposable elements one can
obtain only those cycles in the special fibres which are the restrictions of
cycles in the generic fibre. In particular, if $Z_{\eta}$ is a motivic cycle
and the boundary contains cycles which are not the restriction of cyces in the
generic fibre then the motivic cycle is necessarily indecomposable. We will
show that that is the case for our cycles.

An alternate way, as is done in Chen-Lewis [CL05], Ma-Sato [MS23] or Sato
[Sat23], is to compute the Beilinson regulator and show that, when evaluated
against a particular transcendental $(1,1)$ form, this is non-zero. In the
case when the base is the ring of integers of a glabal field or a local field,
the boundary map can be thought of as a non-Archimedean regulator map and the
idea is essentially the same. The transcendental form is represented by a new
algebraic cycle in the special fibre. Evaluation against this form corresponds
to intersecting with the new cycle. That being non-zero implies that the
component of the boundary in the direction of the new cycle is non-zero. This
also fits in to the philosophy espoused by Manin [Man91] that the Archimedean
component of an arithmetic variety can be viewed as the ‘special fibre at
$\infty$’.

##  4 Construction of the cycles on the generic $K3$ surface of degree $2$.

###  4.1 A Theorem in Enumerative Geometry

We use Proposition 3.1 to construct infinitely many distinct cycles in the
generic $K3$ surface of degree $2$. The cycles we construct will be defined on
those $K3$ surfaces whose moduli point lies in the complement of the special
subvarieties ${{\mathcal{P}}}_{Q}$.

To apply Proposition 3.1 we need a suitable rational curve in the generic $K3$
surface. We find that by ‘deforming’ the curve $Q$ to the generic degree 2
$K3$ surface.

There is a well known theorem in enumerative geometry [Zin05] which states the
following:

######  Theorem 4.1.

Let $N_{d}$ denote the number of rational curves of degree $d$ passing through
$3d-1$ points in ${\mathds{P}}^{2}$ in general position. Then $N_{d}$ is
finite and non-zero.

###### Proof.

A proof of this can be found in the book by Dusa McDuff and Dieter Salamon
[MS12], page 213. ∎

The numbers $N_{1}=1$, $N_{2}=1$, $N_{3}=12$ are classical. The precise number
$N_{d}$ is the celebrated theorem of Kontsevich-Manin [KM94] and Ruan-Tian
[RT95].

Recall that ${{\mathcal{P}}}_{Q}$ is the submoduli of the moduli of $K3$
surfaces $X$ of degree $2$ where there is a rational curve $Q$ of degree $d$
which meets the sextic $S_{X}$ at points of even multiplicity. Since $Q$ is of
degree $d$ it meets $S_{X}$ at $6d$ points. However, since they meet at either
ordinary double points $q$ of the sextic or points of even multiplicity, there
are at most $3d$ distinct points of intersection.

If we discard one of the double points of $Q\cap S_{X}$, we determine $3d-1$
points. Hence the theorem above suggests that perhaps for any $S_{X}$ we can
find a rational curve of degree $d$ passing through these $3d-1$ points.

We have the following theorem, the proof of which was suggested to me by Tom
Graber. The proof is easier as we assume that for some $X$ there exists a
rational curve meeting $S_{X}$ at the correct number of points.

######  Theorem 4.2 ((Graber,S.)).

Consider pairs $(S,Q)$ where $S$ is a sextic in ${\mathds{P}}^{2}$ and $Q$ is
a rational curve of degree $d$. In general they will meet at $6d$ points.
Suppose there exists a sextic $S_{0}$ and a nodal rational curve $Q_{0}$ of
degree $d$ such that they meet at $\leq 3d+1$ points of the following type

  1. 1.

$k$ nodes of $S_{0}$, if $S_{0}$ has ordinary nodes.

  2. 2.

$3d-1-k$ other points tangent to $S_{0}$ (which could coincide to be of even
multiplicity).

  3. 3.

$2$ other points (which could coincide and be of one of the above types).

Then, for any sextic $S$ with $m\geq k$ ordinary double points, the number
$N_{d,k}$ of nodal rational curves $Q$ of degree $d$ meeting the sextic at
$3d-1-k$ points of tangency and passing through $k$ nodes is finite and non-
zero.

###### Proof.

Let ${\mathcal{M}}_{d}$ be the space of degree $d$ nodal rational curves in
${\mathds{P}}^{2}$ and let ${{\mathcal{S}}}$ be the space of sextic curves. It
is known that the dimension $\operatorname{dim}{\mathcal{M}}_{d}=3d-1$. Let
$\operatorname{dim}({{\mathcal{S}}})=s$.

Let ${\mathcal{X}}$ be the locus in ${\mathcal{M}}\times{{\mathcal{S}}}$
parametrizing pairs $(Q,S)$ where $Q$ is tangent to $S$ at $3d-1$ points.
There is a morphism

$\pi:{\mathcal{M}}\times{{\mathcal{S}}}\longrightarrow{\mathcal{H}}_{6d}$
$\pi(Q,S)\longrightarrow(Q\cap S)$

where ${\mathcal{H}}_{6d}$ is the Hilbert scheme of $6d$ points in
${\mathds{P}}^{2}$.

Let $Z$ be the stratum of ${\mathcal{H}}_{6d}$ parametrizing subschemes of
$3d-1$ double points $\cup$ $2$ reduced points. Then ${\mathcal{X}}$ is just
$\pi^{-1}(Z)$. The closure $\bar{Z}$ is an irreducible codimension $(3d-1)$
subscheme of ${\mathcal{H}}_{6d}$ as it is determined by choosing $3d+1$
points - which is $6d+2$ dimensional - and then $3d-1$ tangents at $3d-1$ of
those points - which makes it $9d+1$ dimensional. Further ${\mathcal{H}}_{6d}$
is $12d$ dimensional. Since ${\mathcal{H}}_{6d}$ is smooth every irreducible
component of $\pi^{-1}(\bar{Z})$ has codimension at most $(3d-1)$.

From the Theorem of Birkenhake and Wilhelm, we know that for some $S_{0}$
there are only finitely many $Q_{0}$ \- namely when the corresponding Abelian
surface has extra endomorphisms and $Q_{0}\cap S_{0}$ has $3d$ points.
Equivalently, the local dimension of
${\mathcal{X}}\cap({\mathcal{M}}\times\\{S_{0}\\})$ is $0$. Looking at $\pi$
near $(Q_{0},S_{0})$, since
$\operatorname{dim}({\mathcal{X}})\geq\operatorname{dim}({{\mathcal{S}}})$ and
the fact that the local fibre dimension is $0$ at $(Q_{0},S_{0})$ implies all
nearby fibre dimensions are $0$. This is equivalent to the fact that there are
finitely many such $Q$ for every $S$.

∎

For instance, if $Q$ is a conic, there exists 70956 conics tangent to a sextic
at five points [Gat05] (Corollary 3.6).

###  4.2 Indecomposable Motivic cycles

####  4.2.1 Construction

We now use Theorem 4.2 to construct a motivic cycle.

######  Theorem 4.3.

Let $P_{Q}$ as above be the moduli of degree $2$ $K3$ surfaces $X$ for which
there is a rational curve $Q$ of degree $d$ meeting the associated sextic
$S_{X}$ at $3d$ points of even multiplicity. Assume ${{\mathcal{P}}}_{Q}$ is
non-empty. Then there exists a motivic cycle

$Z_{Q_{\eta}}\in H^{3}_{{\mathcal{M}}}(X_{\eta},{\mathds{Q}}(2))$

where $X_{\eta}$ is the generic $K3$ surface of degree $2$. This cycle is
defined in the fibres of the complement of ${{\mathcal{P}}}_{Q}$: If f
$z\notin{{\mathcal{P}}}_{Q}$ the cycle restricts to give a cycle $Z_{Q_{z}}\in
H^{3}_{{\mathcal{M}}}(X_{z},{\mathds{Q}}(2))$ where $X_{z}$ is the $K3$
surface corresponding to $z$.

###### Proof.

Let $X_{z}$ denote the $K3$ surface corresponding to the point $z$. Let
$S_{z}=S_{X_{z}}$ be the associated sextic. Let $\bar{X}_{z}$ denote the
surface which is obtained by blowing down the exceptional cycles of $X_{z}$
lying over the nodes of $S_{z}$ \- so $\bar{X}_{z}$ is a double cover of
${\mathds{P}}^{2}$ ramified at $S_{z}$.

If $z_{0}$ lies on ${{\mathcal{P}}}_{Q}$, the curve $Q$ determines $3d$ points
$Q\cap S_{z_{0}}$. If we discard one of them, from Theorem 4.2 above, if
$S_{\eta}$ denotes the generic sextic, then there exists a rational curve
$Q_{\eta}$ which meets $S_{\eta}$ at $3d-1$ points of the type determined by
$Q\cap S_{z_{0}}$. $Q_{\eta}$ will meet $S_{\eta}$ at two other points
$s_{\eta}$ and $t_{\eta}$ which are not points of even multiplicity.

Let $\pi:C_{\eta}\longrightarrow Q_{\eta}$ be the double cover of $Q_{\eta}$
in $\bar{X}_{\eta}$. The normalization of $C_{\eta}$ is a double cover of
${\mathds{P}}^{1}$ ramified at two points - namely the points $s_{\eta}$ and
$t_{\eta}$ and so is an irreducible conic. This has nodes at the points
$C_{\eta}\cap S_{\eta}$ and there are $3d-1$ such nodes.

Let $P_{\eta}$ be one of nodes. If it is a node of $S_{\eta}$ as well then
from Propostion 3.1, using the function $f_{P_{\eta}}$ on the strict transform
$\tilde{C}_{\eta}$ of $C_{\eta}$ along with the function $g_{P_{\eta}}$ on the
exceptional fibre $E_{P_{\eta}}$, we obtain a cycle

$Z_{Q_{\eta}}=(\tilde{C}_{P_{\eta}},f_{P_{\eta}})+(E_{P_{\eta}},g_{P_{\eta}})\in
H^{3}_{{\mathcal{M}}}(X_{\eta},{\mathds{Q}}(2))$

If $P_{\eta}$ is not a node then let $b:X_{P_{\eta}}\longrightarrow X_{\eta}$
be the blow up of $X_{\eta}$ at $P_{\eta}$. We can construct a function on the
strict transform of $C_{\eta}$ in $X_{P_{\eta}}$ as above to obtain a cycle in
$H^{3}_{{\mathcal{M}}}(X_{P_{\eta}},{\mathds{Q}}(2))$ and push that down using
$b_{*}$ to obtain a cycle $Z_{Q{\eta}}\in
H^{3}_{{\mathcal{M}}}(X_{\eta},{\mathds{Q}}(2))$.

We futher $Z_{Q_{\eta}}$ by requireing the function $f_{P_{\eta}}$ satisfy

$f_{P_{\eta}}(s_{\eta})=1$

If $z$ is a point on the moduli space, $Z_{Q_{\eta}}$ restricts to give a
cycle in the motivic cohomology of $X_{z}$ as long as the restriction of
$C_{\eta}$ is irreducible. This happens as long as
$z\notin{{\mathcal{P}}}_{Q}$, or equivalently as long as $s_{z}$ and $t_{z}$
are distinct. So it is defined in the complement of the fibres over
${{\mathcal{P}}}_{Q}$.

∎

For instance, consider the case of the conics meeting the sextic tangentially.
There always exists a conic meeting the sextic tangentially at five points.
This conic meets the sextic at two other points. The normalisation of the
double cover of this conic ramified at all the points of interesection is a
double cover of ${\mathds{P}}^{1}$ ramified at two points - namely $t_{z}$ and
$s_{z}$ \- hence is an irreducible smooth rational curve.

One can restrict to the case when the sextic degenerates to a product of six
lines - so the moduli space is $4$ dimensional. The $K3$ surface is obtained
by blowing up the nodes of the double cover of ${\mathds{P}}^{2}$ ramified at
these $6$ lines. There are $15$ nodes. The generic Picard number of the degree
$2$ $K3$ surface $X$ corresponding to this type of a sextic is $16$. $15$ of
those come from the exceptional fibres and one more coming from the hyperplane
section.

The conic above will be tangent to $5$ of the six lines and will meet the
remaining line at $2$ points. If these points coincide then the conic is
tangent to the six lines and then one knows that corresponding $K3$ surface is
the Kummer surface of an Abelian surface. And further, the Picard number is
$17$ \- one more cycle coming from a component of the double cover of the
conic.

This is an example of the above situation - on the $4$ dimensional moduli of
double convers of degenerate sextics, there is a motivic cycle supported in
the fibres over the complement of the moduli of Kummer surfaces. This is
distinct from the cycles constructed in [Sre22] or [MS23].

To apply the theorem we need that ${{\mathcal{P}}}_{Q}$ is non-empty. The
theorems (Theorem 7.1-7.4) of Birkenhake-Wilhelm [BW03] show that for certain
choices of $\Delta$ there exists such rational curves with $\Delta(Q)=\Delta$.
The case above is $\Delta=6$ since the component of the double cover of the
special conic is a $-1$ curve. In the Birkenhake-Wilhelm case one only obtains
$\Delta\equiv 0,1\operatorname{mod}4$ since they are discriminants of real
quadratic fields. Ma and Sato [MS23] consider the case when $Q$ is a line
meeting the sextic at $3$ points of tangency. This corresponds to the case
when $\Delta=4$.

In the construction we have made some choices. We chose a point to discard to
obtain $3d-1$ points and we also chose the node $P_{\eta}$. A different choice
of point to discard results in a different rational curve $Q_{\eta}$ but the
support of the boundary is the same. Another choice of $P_{\eta}$ will result
in a different function $f_{P_{\eta}}$.

In the next section, we will show that all these choices will result in cycles
with the same boundary. Interestingly, Sato [Sat23] shows that in the case he
consides of products of elliptic curves, the Archimedean regulator does depend
on the choice of $P_{\eta}$.

####  4.2.2 Indecomposability

We now claim that $Z_{Q_{\eta}}$ in
$H^{3}_{{\mathcal{M}}}(X_{\eta},{\mathds{Q}}(2))$ is in fact an indecomposable
cycle.

Recall that over ${{\mathcal{P}}}_{Q}$ the conic $C_{\eta}$ gains a new node
$R_{Q}$ since $s_{\eta}$ and $t_{\eta}$ coincide. As a result it breaks uo in
to two components.

There are cycles $\tilde{C}_{{{\mathcal{P}}}_{Q},1}$ and
$\tilde{C}_{{{\mathcal{P}}}_{Q},2}$ in the universal $K3$ surface over
${{\mathcal{P}}}_{Q}$ such that

$\tilde{\mathcal{C}}|_{{{\mathcal{P}}}_{Q}}=\tilde{C}_{{{\mathcal{P}}}_{Q},1}+\tilde{C}_{{{\mathcal{P}}}_{Q},2}+E_{R_{Q}}$

where $\tilde{\mathcal{C}}$ denotes the closure of $\tilde{C}_{\eta}$ in the
universal $K3$ surface over the moduli space (or its blow up at $P_{\eta}$)
and $E_{R_{Q}}$ the exceptional fibre over the node $R_{Q}$.

As explained in Section 3.2, if a cycle is decomposable, then the boundary
under the connecting homomorphism of the localization sequence is the
restriction of a generic cycle. If we show that the boundary is not the
restriction of a generic cycle then it is is necessarily indecomposable.

######  Theorem 4.4.

The boundary of the cycle $Z_{Q_{\eta}}$ is supported in the fibres over
${{\mathcal{P}}}_{Q}$ and is a non-zero multiple of
$\tilde{C}_{{{\mathcal{P}}}_{Q},1}-\tilde{C}_{{{\mathcal{P}}}_{Q},2}$.

$\partial(Z_{Q_{\eta}})=\tilde{C}_{{{\mathcal{P}}}_{Q},1}-\tilde{C}_{{{\mathcal{P}}}_{Q},2}$

This cycle is not the restriction of a cycle in the generic fibre and hence
the cycle $Z_{Q}$ is indecomposable

###### Proof.

Let $Z_{Q_{\eta}}=(\tilde{C}_{\eta},f_{P_{\eta}})+(E_{P_{\eta}},g_{P_{\eta}})$
be the cycle in the generic fibre constructed in Theorem 4.3.

$\operatorname{div}(f_{P_{\eta}})=P_{\eta,1}-P_{\eta,2}$

where $P_{1,\eta}$ and $P_{2,\eta}$ are the two points lying over $P_{\eta}$.
We have further assumed that $f_{P_{\eta}}(s_{\eta})=1$, where $s_{\eta}$ is
one of the non-singular ramified points.

The double cover $\pi:X_{\eta}\rightarrow{\mathds{P}}^{2}_{K_{\eta}}$ induces
an involution $\iota$ on $C_{\eta}$ which fixes the ramified points. This
involution lifts to the normalization $\tilde{C}_{\eta}$. If $P_{1,\eta}$ and
$P_{2,\eta}$ are two points lying over a singular ramified point $P_{\eta}$,
then $\iota(P_{1,\eta})=P_{2,\eta}$. Further
$\iota(E_{P_{\eta}})=E_{P_{\eta}}$ for any node $P_{\eta}$.

Let $f^{\iota}=f\circ\iota$. Then $f^{\iota}_{P_{\eta}}$ has divisor

$\operatorname{div}(f^{\iota}_{P_{\eta}})=P_{\eta,2}-P_{\eta,1}=-\operatorname{div}(f_{P_{\eta}})$

hence

$f^{\iota}_{P_{\eta}}=\frac{c_{\eta}}{f_{P_{\eta}}}$

for some constant (function on the base) $c_{\eta}$. Since we have further
assumed that $f(s_{\eta})=1$ and $\iota(s_{\eta})=s_{\eta}$ we have
$c_{\eta}\equiv 1$.

The closure of ${\mathcal{C}}_{Q}$ of $\tilde{C}_{\eta}$ has two components
over ${{\mathcal{P}}}_{Q}$, $\tilde{C}_{{{\mathcal{P}}}_{Q},1}$ and
$\tilde{C}_{{{\mathcal{P}}}_{Q},2}$ and the involution interchanges them. The
closure of one of the points lying over $P_{\eta}$ will lie on
$\tilde{C}_{{{\mathcal{P}}}_{Q},1}$ and the other on
$\tilde{C}_{{{\mathcal{P}}}_{Q},2}$. Let $P_{{{\mathcal{P}}}_{Q},i}$ denote
the closures of $P_{\eta,i}$ and we can assume $P_{{{\mathcal{P}}}_{Q},i}$
lies on $\tilde{C}_{{{\mathcal{P}}}_{Q},i}$.

The closure of points $s_{\eta}$ and $t_{\eta}$ meet over
${{\mathcal{P}}}_{Q}$ at a point $R_{Q}$ hence $\tilde{\mathcal{C}}_{Q}$ has a
node at $R_{Q}$. Let $\tilde{\mathcal{C}}_{Q}$ be the blow up of
${\mathcal{C}}_{Q}$ at $R_{Q}$. This has fibre

$\tilde{C}_{{{\mathcal{P}}}_{Q},1}+\tilde{C}_{{{\mathcal{P}}}_{Q},2}+E_{R_{Q}}$

over ${{\mathcal{P}}}_{Q}$. The involution fixes $E_{R_{Q}}$.

The boundary map in the localization sequence is

$\partial(Z_{Q_{\eta}})=\operatorname{div}_{\tilde{\mathcal{C}}_{Q}}(f_{P_{\eta}})+\operatorname{div}_{E_{P_{\eta}}}(g_{P_{\eta}}).$

The first term is

$\operatorname{div}_{\tilde{\mathcal{C}}_{Q}}(f_{P_{\eta}})={\mathcal{H}}+a_{1}\tilde{C}_{{{\mathcal{P}}}_{Q},1}+a_{2}\tilde{C}_{{{\mathcal{P}}}_{Q},2}+a_{3}E_{R_{Q}}$

We then observe that

$\operatorname{div}_{\tilde{\mathcal{C}}_{Q}}(f^{\iota}_{P_{\eta}})=\iota(\operatorname{div}(f_{P_{\eta}}))\text{
and }f^{\iota}_{P_{\eta}}=\frac{1}{f_{P_{\eta}}}$

Therefore, on one hand

$\operatorname{div}_{\tilde{\mathcal{C}}_{Q}}(f^{\iota}_{P_{\eta}})=\iota({\mathcal{H}}+a_{1}\tilde{C}_{{{\mathcal{P}}}_{Q},1}+a_{2}\tilde{C}_{{{\mathcal{P}}}_{Q},2}+a_{3}E_{R_{Q}})=-{\mathcal{H}}+a_{1}\tilde{C}_{{{\mathcal{P}}}_{Q},2}+a_{2}\tilde{C}_{{{\mathcal{P}}}_{Q},1}+a_{3}E_{R_{Q}}$

and on the other

$\operatorname{div}_{\tilde{\mathcal{C}}_{Q}}(f^{\iota}_{P_{\eta}})=-\operatorname{div}_{\overline{\tilde{C}}_{P_{\eta}}}(f_{P_{\eta}})=-{\mathcal{H}}-a_{1}\tilde{C}_{{{\mathcal{P}}}_{Q},1}-a_{2}\tilde{C}_{{{\mathcal{P}}}_{Q},2}-\sum
a_{c}E_{c}.$

This implies $a_{1}=-a_{2}$ and $a_{3}=-a_{3}$. Hence $a_{3}=0$. Let
$a=a_{1}$. We have

$\operatorname{div}_{\tilde{\mathcal{C}}_{Q}}(f_{P_{\eta}})={\mathcal{H}}+a(\tilde{C}_{{{\mathcal{P}}}_{Q},1}\tilde{C}_{{{\mathcal{P}}}_{Q},2}).$

To show $a\neq 0$ we suppose $a=0$. Then

$\operatorname{div}_{\tilde{\mathcal{C}}_{Q}}(f_{P_{\eta}})={\mathcal{H}}$

Since $\operatorname{div}(f_{P_{\eta}})$ does not contain
$\tilde{C}_{{{\mathcal{P}}}_{Q},1}$ it restricts to a function on
$\tilde{C}_{{{\mathcal{P}}}_{Q},1}$ and has degree 0. However,
${\mathcal{H}}\cap\tilde{C}_{{{\mathcal{P}}}_{Q},1}=P_{1,{{\mathcal{P}}}_{Q}}$
so
$\operatorname{deg}(\operatorname{div}_{\tilde{\mathcal{C}}_{Q}}(f_{P_{\eta}}))=1$
which is a contradiction. Hence $a\neq 0$.

Finally, $\operatorname{div}_{E_{P}}(g_{P})=-{\mathcal{H}}$ as
$E_{{{\mathcal{P}}}_{Q}}$ remains irreducible when $z$ lies on
${{\mathcal{P}}}_{Q}$. Hence

$\partial(Z_{Q})=a\left(\tilde{C}_{{{\mathcal{P}}}_{Q},1}-\tilde{C}_{{{\mathcal{P}}}_{Q},2}\right).$

for some $a\neq 0$. In particular, it is indecomposable.

To compute $a$ we do the following. Let $h$ be a function on the base with
divisor $\operatorname{div}(h)={{\mathcal{P}}}_{Q}-\sum a_{D}D$ and such that
$Supp(D)\cap{{\mathcal{P}}}_{Q}=\emptyset$. Such a function exists as $H^{1}$
of the moduli spaces is $0$ so homological and rational equivalence are the
same. Let $(\tilde{C}_{\eta},h)$ be the decomposable cycle in
$H^{3}_{{\mathcal{M}}}(X_{\eta},{\mathds{Q}}(2))$. Then

$\partial((\tilde{C}_{\eta},h))=\tilde{C}_{{{\mathcal{P}}}_{Q},1}+\tilde{C}_{{{\mathcal{P}}}_{Q},2}+E_{R_{Q}}-\sum
a_{D}\tilde{\mathcal{C}}_{Q,D}$

where $\tilde{\mathcal{C}}_{Q,D}$ is the $\tilde{\mathcal{C}}_{Q}|_{D}$. These
fibres are irreducible and have no intersection with cycles in the fibres over
${{\mathcal{P}}}_{Q}$.

The function $h^{-a}f_{P_{\eta}}$ has divisor

$\operatorname{div}_{\tilde{\mathcal{C}}_{Q}}(h^{-a}f_{P_{\eta}})={\mathcal{H}}+a\left(\tilde{C}_{{{\mathcal{P}}}_{Q},1}-\tilde{C}_{{{\mathcal{P}}}_{Q},2}\right)-a\left(\tilde{C}_{{{\mathcal{P}}}_{Q},1}+\tilde{C}_{{{\mathcal{P}}}_{Q},2}+E_{R_{Q}}-\sum
a_{D}\tilde{\mathcal{C}}_{Q,D}\right)$
$={\mathcal{H}}-2a\tilde{C}_{{{\mathcal{P}}}_{Q},2}-aE_{R_{Q}}+a\sum
a_{D}\tilde{\mathcal{C}}_{Q,D}$

so does not have $\tilde{C}_{{{\mathcal{P}}}_{Q},1}$ in its support. We can
therefore restrict the function $h^{-a}f_{P_{\eta}}$ to it and

$\operatorname{deg}(\operatorname{div}(h^{-a}f_{P_{\eta}}))=0$

On the other hand one has

$\operatorname{deg}(\operatorname{div}(h^{-a}f_{P_{\eta}}))=(\operatorname{div}_{\tilde{\mathcal{C}}_{Q}}(h^{-a}f_{P_{\eta}}),\tilde{C}_{{{\mathcal{P}}}_{Q},1})$

Since $({\mathcal{H}},\tilde{C}_{{{\mathcal{P}}}_{Q},1})=1$,
$(\tilde{C}_{{{\mathcal{P}}}_{Q},2},\tilde{C}_{{{\mathcal{P}}}_{Q},1})=0$,
$(E_{R_{Q}},\tilde{C}_{{{\mathcal{P}}}_{Q},1})=1$ and
$(\tilde{\mathcal{C}}_{Q,D},\tilde{C}_{{{\mathcal{P}}}_{Q},1})=0$ for all $D$,
this implies

$0=1-a$

Therefore $a=1$.

∎

##  5 Applications

There are some immediate conseqences of the existence of such indecomposable
motivic cycles. In the case of the base being a local ring, Spiess [Spi99]
shows that it has some consequences for torsion in codimension $2$. Here we
give a rough outline of possible other applications.

###  5.1 Mumford’s conjecture.

Mumford conjectured that there are infinitely many rational cures on a $K3$
surface. Mori and Mukai [MM83] showed this in the complex case and Bogomolov,
Hassett and Tschinkel [BHT11] showed this in the case of mixed characteristics
using a delicate deformation argument deforming rational curves in special
fibres to the generic fibre.

A byproduct of the construction above is that there are infinitely many
rational curves on the general degree $2$ $K3$ surface. The motivic cycles we
have are made up of rational curves and there are infinitely many distict ones
as they degenerate on different moduli.

###  5.2 Hodge-${\mathcal{D}}$-conjecture for $K3$ surfaces over a local field

Let $X$ is a variety over a local field $K$ with residue field $k$. Let
$X_{k}$ denote the special fibre of a regular proper model ${\mathcal{X}}$
over ${\mathcal{O}}_{K}$. The Hodge ${\mathcal{D}}$-conjecture asserts that
the map

$H^{3}_{{\mathcal{M}}}(X,{\mathds{Q}}(2))\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}H^{2}_{{\mathcal{M}}}({\mathcal{X}}_{k},{\mathds{Q}}(1))$

is surjective. This is equivalent to showing that for any cycle in the Neron-
Severi group there is a motivic cycle bounding it.

In several cases, if $X$ is a surface, the rank of the Neron-Severi of the
special fibre is greater than that of the general fibre. In these cases, For
instance, in the case of a product of elliptic curves without $CM$, in the
special fibre the Frobenius induces complex multiplication and the graph of
the Frobenius is a new cycle.

In some instances, the enumerative geoemetry argument might work directly in
mixed characteristic (at least for odd primes). For instance, the statement
that there is a conic tangent to $5$ lines general position is purely
algebraic and extends to the mixed charateristic case.

###  5.3 Relations with modular forms

The Heegner divisors on the moduli space of Abelian surfaces are intimately
connected with modular forms, as in the case of the theorems of Gross-Zagier
[GZ86], Hirzebruch-Zagier [HZ76], van der Geer [vdG82] and Hermann [Her95].

The theorems state that Heegner divisors give rise to coefficients of modular
forms. Borcherds showed that all these theorems can be proved in a similar
manner. The idea is that to show a certain power series is a modular form of a
certain weight $k$, it suffies to show its coefficients satisfy relations
between coefficients of modular forms of that weight. The space of ‘relations
between coefficients of modular forms of weight $k$’ is the space of weakly
holomorphic modular forms of weight $2-k$.

So an approach to this theorem is to relate weakly holomorphic modular forms
of weight $2-k$ with ‘relations between Heegner divisors’. Borcherds did so by
showing that his lift of weakly holomorphic modular forms gave rise to
functions with divisoral support on Heegner divisors - or in other words a
relation of rational equivalence among Heegner divisors.

There are similar theorems expected for higher codimension Heegner cycles in
the universal families over these Shimura varieties. Gross and Zagier [GZ86]
that Heegner divisors on a modular curve are related to modular forms of
weight $\frac{3}{2}$. Zhang [Zha97] showed that Heegner cycles on self product
of the universal family are related to higher weight modular forms. For
instance if one consideres the threefold which is compactification of self-
product the universal elliptic curve, then the Heegner cycles are related to
coefficients of weight $\frac{5}{2}$.

It it intriguing to speculate if these can be proved along the lines of
Borcherds thereom. The modular form formalism extends without any problem.
Relations among Heegner cycles are given by elements of the motivic cohomology
group so the question becomes ‘Can one construct motivic cycles from weakly
holomorphic modular forms?’. Borcherds theorem is an example of this as
functions are elements of the motivic cohomology group
$H^{1}_{{\mathcal{M}}}(S_{\eta},{\mathds{Q}}(1))$ where $S_{\eta}$ is generic
point of the modular variety.

Borcherds lifts are functions, so one cannot expect it to directly work for
higher codimensions. However, given a motivic cycle in the generic fibre of
the the universal family one can consider its regulator. This is a current on
certain forms and for a suitable choice of form varying over the family
computing the regulator on it gives a function on the base. Hence one can ask
whether a certain Borcherds lift of weakly holomorphic forms is the regulator
of a motivic cycle. In [Sre22] and some work in progress on products of
elliptic curves we provide some evidence of this for codimension $2$.

In this paper what we construct is relations between codimension two cycles in
the universal family of $K3$ surfaces of degree $2$ and one might expect that
once again there is a relation between the regulator of these cycles and
Borcherds lifts of modular forms.

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