# K3 surfaces with action of the group $M_{20}$

Paola Comparin, Romain Demelle

###### Abstract.

It was shown by Mukai that the maximum order of a finite group acting
faithfully and symplectically on a K3 surface is 960 and if such a group has
order 960, then it is isomorphic to the Mathieu group $M_{20}$. In this paper,
we are interested in projective K3 surfaces admitting a faithful symplectic
action of the group $M_{20}$. We show that there are infinitely many K3
surfaces with this action and we describe them and their projective models,
giving some explicit examples.

## Introduction

A K3 surface $X$ is a compact complex smooth surface which is simply connected
and admits a nowhere vanishing holomorphic 2-form $\omega_{X}$, unique up to
scalar multiplication. The study of finite groups acting on K3 surfaces and
their action in cohomology goes back to the first works by Nikulin and Mukai.
In particular, given a finite group $G$ acting on a K3 surface $X$, one can
study the action of $G$ according to the action on the 2-form $\omega_{X}$: an
automorphism $\sigma\in\mathrm{Aut}(X)$ is said to be symplectic if it acts as
the identity on $\omega_{X}$ and non-symplectic otherwise. In this last case
$\sigma^{*}$ acts as multiplication by some root of unity
$\zeta_{m}\in\mathbb{C}^{\times}$. Given a finite group of automorphisms $G$,
let $\alpha:G\rightarrow\mathbb{C}^{\times}$ corresponding to the action on
$\omega_{X}$. The following short exact sequence holds for some positive
integer $m$:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\mathbb{Z}/m\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

where $G_{0}=\ker\alpha$ corresponds to the automorphisms acting
symplectically on $X$. A first result in the classification of possible groups
$G$ acting on a K3 surface is given by Nikulin in [14], where the author
classified abelian finite groups acting symplectically and showed that there
is only a finite number of them.

Without the assumption of being abelian, Mukai in [13] studied finite groups
acting faithfully and symplectically on K3 surfaces and showed that every such
group is a subgroup of the Mathieu group $M_{23}$. Xiao in [21] gave an
explicit list of all possible finite groups acting symplectically, finding 81.
Mukai had also shown that the order of such a group is at most 960, being
equal to 960 if and only if $G$ is the Mathieu group $M_{20}$, that one can
describe also as $A_{5}\ltimes(\mathbb{Z}/2\mathbb{Z})^{4}$.

In this paper we are concerned with K3 surfaces admitting a faithful and
symplectic action of $M_{20}$. We first remark that in the two papers [3] and
[5], the authors classify possibilities for maximal groups $\Gamma$ acting on
a K3 surface $X$ and such that $\Gamma$ contains $M_{20}$ strictly. There are
three such groups and their order is $3840=4\cdot|M_{20}|$ in one case and
$1920=2\cdot|M_{20}|$ in the two other cases. The K3 surfaces where these
groups act first appeared in [3, 5, 12, 13].

Let $X$ be a K3 surface that admits a faithful and symplectic action of the
Mathieu group $M_{20}$. The surface $X$ has necessarily Picard number
$\rho(X):=\mathrm{rk}\,\mathrm{Pic}(X)$ equal to 20 by Xiao’s classification
[21]. A first question arising is how many K3 surfaces admit such an action of
$M_{20}$ and in Section 6 we prove the following:

######  Theorem 1.

There exists an infinite number of K3 surfaces admitting a faithful and
symplectic action of $M_{20}$.

Let $\mathbb{L}_{20}$ be the lattice defined by the following matrix:

$\mathbb{L}_{20}=\left(\begin{array}[]{ccc}4&0&-2\\\ 0&4&-2\\\
-2&-2&12\end{array}\right).$

We show in Section 1 some properties of this lattice and its link with the
invariant lattice $\mathrm{H}^{2}(X,\mathbb{Z})^{M_{20}}$ and the
transcendental lattice
$T_{X}=\mathrm{Pic}(X)^{\perp_{\mathrm{H}^{2}(X,\mathbb{Z})}}$. We consider
$L$ a primitive element of $\mathbb{L}_{20}$ invariant by the action of
$M_{20}$. As the square of all element in $\mathbb{L}_{20}$ is a multiple of
4, we automatically get $L^{2}=4n$, for some $n\in\mathbb{Z}_{\geq 1}$. The
second main result of the paper is the following:

######  Theorem 2.

There exists an embedding $\langle 4n\rangle\hookrightarrow\mathbb{L}_{20}$ if
and only if $n$ is not of the form $4^{i}(16j+6)$, with $i,j$ some non-
negative integers. If it is the case, we can construct a projective model of
the surface $X$ in $\mathbb{P}^{2n+1}$ and we have the following results:

  1. 1.

$L$ is an ample class;

  2. 2.

the linear system $|L|$ is not hyperelliptic and is base point free;

  3. 3.

the projective model is only defined by quadrics.

Morevover, the number of quadrics $Q_{4n}$ defining $X$ is

$Q_{4n}=2n^{2}-3n+1.$

The paper is organized as follows: in Section 1, we recall some notations and
results we use throughout the paper. In order to illustrate Theorem 2, we
start with some particular cases. The cases $L^{2}=4n$ with $n=1,2,10$ have
been already studied in [3, 12, 5], thus in Section 2 we begin by studying the
first new case which is $n=3$, that is $L^{2}=12$. We prove the existence of
the embedding and we construct explicitly the projective model. In Section 3
we cover the cases $L^{2}=4n$ with $4\leq n\leq 10$: we show the cases we
cannot construct the embedding for and see possible behaviours such as the
existence of different K3 surfaces for the same $n$ or multiple embeddings on
a same K3 surface. All the results we obtain are summarized in Appendix A.
Section 4 is dedicated to the proof of Theorem 2. Thanks to the results in
[17] we show that the linear system $|L|$ is base point free, is not
hyperelliptic and the projective model is defined only by quadrics. In Section
5 we are interested in some non-primitive embeddings, where the non-
primitiveness allows to find explicit equations for the surfaces. Finally, in
Section 6 we prove Theorem 1.

## Acknowledgments

We would like to thank Cédric Bonnafé, Xavier Roulleau and Alessandra Sarti
for very useful insights and discussions. The authors have been partially
supported by Fondecyt Iniciación en la investigación N.11190428, Programa de
cooperación científica ECOS-ANID C19E06 and Math AmSud-ANID 21 Math 02. The
first author has been partially suported by Fondecyt Regular N.1200608.

##  1\. Notations and preliminaries

Let $X$ be a K3 surface and denote by $\omega_{X}$ the nowhere vanishing
holomorphic 2-form. We have
$\mathrm{H}^{2,0}(X):=\mathrm{H}^{0}(X,\Omega_{X}^{2})=\mathbb{C}\cdot\omega_{X}$
and the cohomology group $\mathrm{H}^{2}(X,\mathbb{Z})$ is isometric to the K3
lattice $\Lambda_{K3}$:

$\Lambda_{K3}:=U^{\oplus 3}\oplus E_{8}(-1)^{\oplus 2}$

where $U$ is the hyperbolic plane and $E_{8}(-1)$ is the negative definite
lattice associated to the root system with the same name.

Let $G_{0}$ be a group of symplectic automorphisms of $X$, that is the
automorphisms acting as the identity on the 2-form $\omega_{X}$. We are
interested in the case where $G_{0}$ is the Mathieu group $M_{20}$. In this
paper, we want to describe all projective K3 surfaces admitting a symplectic
and faithful action of $M_{20}$.

The lattice $\mathbb{L}_{20}$ we defined in the Introduction has rank $3$ and
signature $(3,0)$. Denoting by $\mathrm{H}^{2}(X,\mathbb{Z})^{M_{20}}$ the
invariant lattice by the action of $M_{20}$ on the K3 lattice, we recall the
following result:

######  Proposition 1.1.

([12, proof of Proposition 2.1]) Let $X$ be a K3 surface with a faithful
symplectic action by $M_{20}$. Then the invariant lattice
$\mathrm{H}^{2}(X,\mathbb{Z})^{M_{20}}$ is isometric to $\mathbb{L}_{20}$.

As $\mathbb{L}_{20}$ has signature $(3,0)$, its isometry group
$\mathcal{O}\left(\mathbb{L}_{20}\right)$ is finite, see [12, proof of Prop.
2.1]. We describe it completely thanks to the following result coming from [3,
Remark 2.4]:

######  Proposition 1.2.

Denote by $\rho_{1}$ and $\rho_{2}$ the isometries of $\mathbb{L}_{20}$
defined by the following matrices:

$\rho_{1}=\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\
0&0&1\end{array}\right)\quad\mathrm{et}\quad\rho_{2}=\left(\begin{array}[]{ccc}1&0&-1\\\
0&-1&0\\\ 0&0&-1\end{array}\right)\,.$

Then the isometry group $\mathcal{O}\left(\mathbb{L}_{20}\right)$ of
$\mathbb{L}_{20}$ is spanned by $-\mathrm{id}_{\mathbb{L}_{20}}$, $\rho_{1}$
and $\rho_{2}$. Moreover, $\mathcal{O}\left(\mathbb{L}_{20}\right)$ has order
$16$.

By [21, Nr. 81, Table 2], as the K3 surface $X$ admits a faithful symplectic
action of $M_{20}$, the minimal resolution of the quotient of $X$ by $M_{20}$
is a K3 surface with Picard number 20. By a result of Inose [11, Corollary
1.2], this also means that $X$ has Picard number 20. Denoting by
$\mathrm{T}_{X}$ the transcendental lattice of $X$, i.e. the orthogonal
complement of $\mathrm{Pic}(X)$ in $\mathrm{H}^{2}(X,\mathbb{Z})$, we have
$\mathrm{rk}\,\mathrm{T}_{X}=2$. This lattice is even, with signature $(2,0)$.
In order to have an explicit description of $T_{X}$, we recall some facts
about K3 surfaces with Picard number $20$, which are called singular K3
surfaces (cf. [19, Section 4]). Denote by $\mathcal{Q}$ the set of $2\times 2$
positive-definite even integral matrices. For $Q\in\mathcal{Q}$ we have:

$Q=\left(\begin{array}[]{cc}2a&b\\\
b&2c\end{array}\right),\,a,b,c\in\mathbb{Z}$

with $a,c>0$ and $d:=4ac-b^{2}>0$. We define an equivalence relation $\sim$ on
$\mathcal{Q}$ by:

$\forall\ Q_{1},Q_{2}\in\mathcal{Q},\,Q_{1}\sim
Q_{2}\Longleftrightarrow\exists\gamma\in\mathrm{SL}_{2}(\mathbb{Z})\
,\,Q_{1}=\,^{t}\gamma Q_{2}\gamma.$

Let $\left[Q\right]$ be the equivalence class of a matrix $Q\in\mathcal{Q}$
and
${\raisebox{1.99997pt}{$\mathcal{Q}$}\left/\raisebox{-1.99997pt}{$\mathrm{SL}_{2}(\mathbb{Z})$}\right.}$
the set of equivalence classes.

######  Theorem 1.3.

([19, Theorem 4]) The map $X\mapsto\left[T_{X}\right]$ establishes a bijective
correspondence from the set of singular K3 surfaces onto
${\raisebox{1.99997pt}{$\mathcal{Q}$}\left/\raisebox{-1.99997pt}{$\mathrm{SL}_{2}(\mathbb{Z})$}\right.}$.

In particular, K3 surfaces with Picard number $20$ are classified in terms of
their transcendental lattice. Moreover, we can choose a representative $Q$ in
reduced form, that is $-a\leq b\leq a\leq c$ and $b^{2}\leq ac\leq\frac{d}{3}$
and we have the following:

######  Theorem 1.4.

([6, Theorem 2.4]) No distinct reduced quadratic forms are equivalent, except
for the following:

$\left(\begin{array}[]{cc}2a&a\\\
a&2c\end{array}\right)\sim\left(\begin{array}[]{cc}2a&-a\\\
-a&2c\end{array}\right)\quad\mathrm{and}\quad\left(\begin{array}[]{cc}2a&b\\\
b&2a\end{array}\right)\sim\left(\begin{array}[]{cc}2a&-b\\\
-b&2a\end{array}\right).$

Since the action is symplectic one can show that
$\mathrm{T}_{X}\subset\mathrm{H}^{2}(X,\mathbb{Z})^{M_{20}}\simeq\mathbb{L}_{20}$.
In fact, consider $x\in T_{X}\subset\mathrm{H}^{2}(X,\mathbb{Z})$ and $g\in
M_{20}$. We have

$\langle\omega_{X},x\rangle=\langle
g^{*}\omega_{X},g^{*}x\rangle=\langle\omega_{X},g^{*}x\rangle$

so that $\langle\omega_{X},x-g^{*}x\rangle=0$. Then $x-g^{*}x$ is in
$(\mathbb{C}\omega_{X})^{\perp}\cap\mathrm{H}^{2}(X,\mathbb{Z})$ which is
$\mathrm{Pic}(X)$. Moreover, $x\in T_{X}$ and $g^{*}x\in T_{X}$ because
$g^{*}$ is a Hodge isometry, so it preserves the lattice $T_{X}$. Hence
$x-g^{*}x\in\mathrm{Pic}(X)\cap T_{X}=\\{0\\}$ by [14, Sec. 3.2] which implies
that $g^{*}x=x$. So $T_{X}$ is included in the invariant lattice
$\mathrm{H}^{2}(X,\mathbb{Z})^{M_{20}}$ which is isomorphic to
$\mathbb{L}_{20}$.

If $(u,v)$ is a $\mathbb{Z}$-basis of $T_{X}\subset\mathbb{L}_{20}$, then
$u^{2},v^{2}\in 4\mathbb{Z}$ and $u\cdot v\in 2\mathbb{Z}$. Hence

$\mathrm{T}_{X}\simeq\left(\begin{array}[]{cc}4a&2b\\\
2b&4c\end{array}\right)$

where $a,b,c\in\mathbb{Z}$ satisfy the following conditions:

($\star$) $\begin{cases}d:=4ac-b^{2}>0;\\\ b^{2}\leq
ac\leq\textstyle\frac{d}{3};\\\ -a\leq b\leq a\leq c.\end{cases}$

Finally, observe that $X$ contains an ample class $L\in\mathrm{Pic}(X)$ which
is $M_{20}$-invariant and such that $L^{2}=4n$ for some $n$ in
$\mathbb{Z}_{\geq 1}$. One of our main goals is to describe the embedding
$\langle 4n\rangle\hookrightarrow\mathbb{L}_{20}$ in order to construct a
projective model of $X$. To do this, the following result is a direct
consequence of [17, Section 4.1].

######  Proposition 1.5.

If the linear system $|L|$ is not hyperelliptic, then it defines a map:
$\varphi_{L}:X\rightarrow\mathbb{P}^{p_{a}(L)}$, where
$p_{a}(L)=\frac{1}{2}L^{2}+1$.

As we will see in Section 4, the linear system $|L|$ is never hyperelliptic.
As $L^{2}=4n$, the projective models of the K3 surfaces that we consider will
be in $\mathbb{P}^{2n+1}$.

The cases $n=1,2,10$ were already studied by C. Bonnafé et A. Sarti [3], S.
Brandhorst and K. Hashimoto [5] and S. Kondō [12], when the surface $X$ admits
the symplectic action of a maximal group $\Gamma$ containing $M_{20}$
properly. We recall their results and notations, as well as explicit equations
for the K3 surfaces, in the following proposition. Trascendental lattices of
these three surfaces are recalled in Appendix A.

######  Proposition 1.6.

(see [3, 5, 12]) Let $G$ be a maximal group acting faithfully on a K3 surface
$X$ which strictly contain the Mathieu group $M_{20}$. Thus there only are
three possible cases:

  1. 1.

$|G|=3840$ and $L^{2}=40$: the K3 surface $X_{\mathrm{Ko}}$ is the minimal
resolution of the quotient of the Fermat quartic
$\left\\{x^{4}+y^{4}+z^{4}+t^{4}=0\right\\}$ in $\mathbb{P}^{3}$ by the
symplectic involution $(x:y:z:t:)\mapsto(x:y:-z:-t)$. It corresponds to the
Kummer surface $\mathrm{Km}\left(E_{i}\times E_{i}\right)$ (see [12], [3,
Section 3]).

  2. 2.

$|G|=1920$ and $L^{2}=4$: the K3 surface $X_{\mathrm{Mu}}$ is defined as the
following quartic in $\mathbb{P}^{3}$:

(1)
$\\{x^{4}+y^{4}+z^{4}+t^{4}-6\left(x^{2}y^{2}+x^{2}z^{2}+x^{2}t^{2}+y^{2}z^{2}+y^{2}t^{2}+z^{2}t^{2}\right)=0\\}.$

It corresponds to the Kummer surface $\mathrm{Km}\left(E_{i\sqrt{10}}\times
E_{i\sqrt{10}}\right)$ (see [13, p. 190] and [3, Section 4]).

  3. 3.

$|G|=1920$ and $L^{2}=8$: the K3 surface $X_{\mathrm{BH}}$ is defined as the
complete intersection of the three quadrics in $\mathbb{P}^{5}$:

$\begin{cases}x_{0}^{2}+x_{3}^{2}-\phi x_{4}^{2}+\phi x_{5}^{2}=0\\\
x_{1}^{2}-\phi x_{3}^{2}+x_{4}^{2}-\phi x_{5}^{2}=0\\\ x_{2}^{2}+\phi
x_{3}^{2}-\phi x_{4}^{2}+x_{5}^{2}=0\end{cases}$

where $\phi=\frac{1+\sqrt{5}}{2}$ (see [3, Section 5]).

##  2\. The case $L^{2}=12$

We now start by studying the first new case, i.e. the polarization with
$L^{2}=4n$ with $n=3$.

###  2.1. Existence of the projective model

Suppose that $L^{2}=12$. We denote by $(e,f,h)$ the standard basis of
$\mathbb{L}_{20}$, that is the vectors in $\mathbb{L}_{20}$ which give the
matrix of the bilinear form written before. We search
$\lambda,\mu,\delta\in\mathbb{Z}$ such that

$L=\lambda e+\mu f+\delta h.$

Thanks to [3, Lemma 2.8] we have

(2) $L^{2}=(2\lambda-\delta)^{2}+(2\mu-\delta)^{2}+10\delta^{2}.$

If $L^{2}=12$, then possible solutions for $\delta$ are
$\delta\in\\{-1,0,1\\}$ and

  * •

if $\delta=0$, then $2\lambda^{2}+2\mu^{2}=12$, that is
$\lambda^{2}+\mu^{2}=6$. This is not possible since $6$ is not the sum of two
squares;

  * •

if $\delta=1$, then $(2\lambda-1)^{2}+(2\mu-1)^{2}=2$, that is
$(2\lambda-1)^{2}=(2\mu-1)^{2}=1$. Hence $\lambda,\mu\in\\{0,1\\}$ and then

$L\in\\{h;e+h;f+h;e+f+h\\};$

  * •

if $\delta=-1$, as before one has $\lambda,\mu\in\\{0,-1\\}$, then

$L\in\\{-h;-e-h;-f-h;-e-f-h\\}.$

Hence there are 8 possibilities for $L$ and they are all in the same orbit, up
to isometry. In fact observe that the possibilities for $\delta=-1$ are the
same as in the case $\delta=1$ up to the isometry
$-\mathrm{id}_{\mathbb{L}_{20}}$. Moreover we have:

$-\mathrm{id}\circ\rho_{2}(e+f+h)=f+h,\quad\rho_{1}(f+h)=e+h,\quad-\mathrm{id}\circ\rho_{2}(e+h)=h.$

Therefore, up to isometry, there exists a unique embedding of lattices
$\langle 12\rangle\hookrightarrow\mathbb{L}_{20}$ which maps $L$ on $h$.

We assume now that $L=h$. As $\mathrm{T}_{X}=L^{\perp}\cap\mathbb{L}_{20}$,
the orthogonal complement of the lattice $\mathbb{Z}h$ in $\mathbb{L}_{20}$ is
given by vectors $\lambda e+\mu f+\delta h$ such that

$\langle\lambda e+\mu f+\delta
h,h\rangle=0,\quad\lambda,\mu,\delta\in\mathbb{Z}.$

This equality gives $\lambda+\mu-6\delta=0$ and then

$\begin{array}[]{ccc}\lambda e+\mu f+\delta h&=&(6\delta-\mu)e+\mu f+\delta
h\\\ &=&\mu(f-e)+\delta(h+6e).\end{array}$

Let $u_{1}=f-e$ and $u_{2}=h+6e$. Then $u_{1}$ and $u_{2}$ span the
transcendental lattice $\mathrm{T}_{X}$. Moreover $u_{1}^{2}=8$,
$u_{2}^{2}=132$ and $\langle u_{1},u_{2}\rangle=-24$. Using the notations
introduced before, one has $a=2$, $b=-12$ and $c=33$, but these integers do
not satisfy the conditions ($\star$ ‣ 1). We then set
$\overset{\sim}{u_{1}}:=u_{1}$ and $\overset{\sim}{u_{2}}:=3u_{1}+u_{2}$. Now
$\overset{\sim}{u_{2}}^{2}=60$ and
$\langle\overset{\sim}{u_{1}},\overset{\sim}{u_{2}}\rangle=0$, so that

$\mathrm{T}_{X}=\left(\begin{array}[]{cc}8&0\\\ 0&60\end{array}\right)\,.$

In this case $a=2$, $b=0$ and $c=15$. The conditions ($\star$ ‣ 1) hold for
these integers, hence this is the matrix we are looking for.

###  2.2. Construction of the K3 surface

The aim of this section is to find the K3 surface in $\mathbb{P}^{7}$ with an
action of $M_{20}$. To understand the problem, we have the following diagram:

$\textstyle{M_{20}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{GL}_{8}(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{PGL}_{8}(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

In general, the arrow $M_{20}\longrightarrow\mathrm{GL}_{8}(\mathbb{C})$ does
not exist. So we must consider a group
$\overset{\sim}{M_{20}}\subset\mathrm{GL}_{8}(\mathbb{C})$ such that:

  * •

$Z(\overset{\sim}{M_{20}})\simeq\mu_{d}$;

  * •

${\raisebox{1.99997pt}{$\overset{\sim}{M_{20}}$}\left/\raisebox{-1.99997pt}{$\mu_{d}$}\right.}\simeq
M_{20}$.

Here $\mu_{d}$ denotes the group of primitive $d$-th roots of unity and
$Z(\overset{\sim}{M_{20}})$ is the center of the group
$\overset{\sim}{M_{20}}$ in $\mathrm{GL}_{8}(\mathbb{C})$. Then we have the
following diagram:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mu_{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overset{\sim}{M_{20}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{20}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{GL}_{8}(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{PGL}_{8}(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

We want to understand the non-trivial central extensions of $M_{20}$ by a
cyclic group and, thanks to the web version of the Atlas of finite groups [1],
we know there are $6$ such extensions:

  1. •

$M_{20}$;

  2. •

$H_{1}$, $H_{2}$ and $H_{3}$, with $Z\left(H_{k}\right)\simeq\mu_{2}$,
$k=1,2,3$;

  3. •

$G_{1}$ and $G_{2}$, with $Z\left(G_{k}\right)\simeq\mu_{4}$, $k=1,2$.

Denote by $G$ one of these groups. We are looking for the irreducible
representations of $G$ in $\mathrm{GL}_{8}(\mathbb{C})$. In our case, by using
the computer algebra system MAGMA [4], we obtain that only the group $G_{1}$
gives such a representation. Let
$\rho:G\rightarrow\mathrm{GL}_{8}(\mathbb{C})$ be this representation. By
abuse of notation, we identify $G$ with the image $\rho(G)$. The following
holds:

$Z(G)\simeq\mu_{4}\quad\mathrm{and}\quad{\raisebox{1.99997pt}{$G$}\left/\raisebox{-1.99997pt}{$Z(G)$}\right.}\simeq
M_{20}\,.$

Let $X$ be the K3 surface and assume for the moment that $X$ is defined by
quadrics111In Section 4.4 we will show that the model is not hyperelliptic and
$X$ is only defined by quadrics. in $\mathbb{P}^{7}$. Denote
$F:=\mathbb{C}\left[x_{1},\dots,x_{8}\right]_{2}$ the vector space of
homogeneous polynomials of degree $2$ in $8$ variables and let
$f_{1},\dots,f_{n}\in F$ be the equations of the quadrics defining $X$. Then
$\mathbb{C}f_{1}+\dots+\mathbb{C}f_{n}\subset F$ must be stable by the action
of $G$. We want to find the stable subspaces of $F$. By the theory of linear
representations [18, Proposition 8], we have the following decomposition:

$F=\bigoplus\limits_{S\in\mathrm{Irr}(G)}\mathrm{Im}\left(p_{S}^{F}\right)$

where $p_{S}^{F}\in\mathrm{end}_{\mathbb{C}}(F)$ is the projection on the
isotypic component associated with $S$. Thanks to MAGMA, we obtain
$F=F_{6}\oplus F_{10}\oplus F_{20}$, where the $F_{i}$ are $G$-stable,
irreducible and of dimension $i$. Let $X_{k}$ be the subvariety in
$\mathbb{P}^{7}$ defined by the elements of $F_{k}$. By using MAGMA, we can
compute the dimension of each space: $\dim\left(X_{6}\right)=4$,
$\dim\left(X_{10}\right)=2$ and $\dim\left(X_{20}\right)=-1$. This means that
$X_{10}$ is a good candidate for the surface we are looking for. Thanks to
standard commands of MAGMA, we check the following conditions:

  1. •

the canonical divisor is trivial;

  2. •

the surface is smooth.

At this point, it remains two possibilities: either $X_{10}$ is an abelian
surface or it is a K3 surface. To conclude, we will compute the Euler
characteristic $\chi(X_{10})$ of $X_{10}$ and therefore distinguish if
$\chi(X_{10})=0$ and $X_{10}$ is abelian or $\chi(X_{10})=24$ and $X_{10}$ is
a K3 surface. We consider

$\mathcal{L}:g\in M_{20}\longmapsto\sum\limits_{i\geq
0}(-1)^{i}\mathrm{Tr}\left(g^{\ast}|\mathrm{H}^{i}\left(X_{10}\right)\right).$

On the one hand, by the Lefschetz fixed point formula [8, Chapter 3.4], we
have $\mathcal{L}(1)=\chi\left(X_{10}\right)$. On the other hand, we can
compute $\mathcal{L}(1)$ using the following lemma:

######  Lemma 2.1.

Let $\Gamma$ be a finite group and $f:\Gamma\rightarrow\mathbb{C}$ be a
function which is the difference of two characters of representation. Then

$f(1)\equiv-\sum\limits_{\underset{\gamma\neq
1}{\gamma\in\Gamma}}f(\gamma)\mod|\Gamma|\,.$

Before proving this result, we need to recall some definitions and results
coming from representation theory [18, Chapters 1 and 2].
Let $\left(E,\rho_{E}\right)$ be a linear representation of the group
$\Gamma$. This means that $E$ is a complex vector space and
$\rho_{E}:\Gamma\rightarrow\mathrm{GL}(E)$ is a group morphism. In particular,
$\rho_{\mathbb{C}}$ is the so-called trivial representation, that is for all
$\gamma\in\Gamma$, $\rho_{\mathbb{C}}(\gamma)=\mathrm{Id}_{\mathbb{C}}$. We
call a character of the representation $\rho_{E}$ the morphism
$\chi_{E}:\Gamma\rightarrow\mathbb{C}$ defined by

$\chi_{E}(\gamma)=\mathrm{Tr}\ \rho_{E}(\gamma),\ \gamma\in\Gamma.$

Finally, we denote by $p_{S}^{E}$ the projection on the isotypic component
associated with $S$. If $S$ is an irreducible representation of $\Gamma$, we
have a characterization of this endomorphism which is

$p_{S}^{E}=\frac{\dim
S}{|\Gamma|}\sum\limits_{\gamma\in\Gamma}\overline{\chi_{S}(\gamma)}\rho_{E}(\gamma).$

###### Proof of Lemma 2.1.

Let $S=\mathbb{C}$. For the endomorphism $p_{\mathbb{C}}^{E}$ we have

$p^{E}_{\mathbb{C}}=\frac{1}{|\Gamma|}\sum\limits_{\gamma\in\Gamma}\overline{\chi_{\mathbb{C}}(\gamma)}\rho_{E}(\gamma).$

As $\rho_{\mathbb{C}}$ is the trivial representation of $\Gamma$, for all
$\gamma\in\Gamma$ we have
$\chi_{\mathbb{C}}(\gamma)=\mathrm{Tr}\left(\mathrm{Id_{\mathbb{C}}}\right)$
and then

$p^{E}_{\mathbb{C}}=\frac{1}{|\Gamma|}\sum\limits_{\gamma\in\Gamma}\rho_{E}(\gamma).$

Taking the trace of this expression, with
$f=\chi_{E}:\Gamma\rightarrow\mathbb{C}$ and multiplying it by $|\Gamma|$ we
obtain the following:

$f(1)=|\Gamma|\mathrm{Tr}\left(p_{\mathbb{C}}^{E}\right)-\sum\limits_{\underset{\gamma\neq
1}{\gamma\in\Gamma}}f(\gamma).$

As
$\mathrm{Tr}\left(p_{\mathbb{C}}^{E}\right)=\dim\mathrm{Im}\left(p_{\mathbb{C}}^{E}\right)$
is a positive integer, we can look at this expression modulo $|\Gamma|$.
Similarly, if $f:\Gamma\rightarrow\mathbb{C}$ is as in the assumption, we
obtain the thesis.

∎

Let $X_{10}^{g}$ be the subspace of $X_{10}$ where $g\in M_{20}$ acts as the
identity and observe that we have only isolated fixed points on it. Then,
thanks again to the Lefschetz formula, for $g\neq 1$ we have
$\mathcal{L}(g)=\chi\left(X_{10}\right)=\left|X_{10}^{g}\right|$. We compute
this last term with MAGMA and by Lemma 2.1 we obtain

$\mathcal{L}(1)\equiv 24\mod 960\,.$

The surface $X_{10}$ has Euler characteristic equal to $24$, hence it is a K3
surface. Finally we get the equations for the quadrics which define the K3
surface, see Appendix B.

##  3\. Other cases

In order to repeat the construction for other values of $L^{2}$, we first
observe for which $n$ the equation (2) admits solution.

######  Proposition 3.1.

The equation

(3) $4n=(2\lambda-\delta)^{2}+(2\mu-\delta)^{2}+10\delta^{2}\,.$

admits a solution $(\lambda,\mu,\delta)$ if and only if $n$ can not be
expressed as $4^{i}(16j+6)$ for some non-negative integers $i,j$.

###### Proof.

By Ramanujan ternary quadratic form, an even positive integer can be expressed
as $x^{2}+y^{2}+10z^{2}$ for some $x,y,z\in\mathbb{Z}$ if and only if it is
not of the form $4^{i}(16j+6)$ for some non-negative $i,j$, see [15]. Then if
$n\neq 4^{i}(16j+6)$, there exist $x,y,z\in\mathbb{Z}$ such that

(4) $4n=x^{2}+y^{2}+10z^{2}.$

Thus we can compute $(\lambda,\mu,\delta)$ satisfying (3) from the triple
$(x,y,z)$:

$\lambda=\frac{\pm x\pm z}{2},\ \mu=\frac{\pm y\pm z}{2},\ \delta=\pm z.$

Observe that $\lambda,\mu\in\mathbb{Z}$. In fact, if $z$ is even, then by (4)
the sum $x^{2}+y^{2}$ has to be congruent to 0 modulo 4. Since 3 is not a
square modulo 4, then necessarily

$x^{2}\equiv 0\mod 4,\quad y^{2}\equiv 0\mod 4.$

This implies that $x^{2}$ and $y^{2}$ are even, so $x$ and $y$ are even too.
Thus $\lambda,\mu\in\mathbb{Z}$. Similarly, if $z$ is odd, then $z^{2}\equiv
1\mod 4$. Thus by (4)

$x^{2}\equiv 1\mod 4,\quad y^{2}\equiv 1\mod 4,$

$x$ and $y$ are odd too and $\lambda,\mu\in\mathbb{Z}$. ∎

Computations similar to the ones of Section 2 allow to study the existence of
the K3 surface with $L^{2}=4n$ for any value of $n$ (except for cases of
Proposition 3.1). We show the results for $3\leq n\leq 10$, in order to give
examples in several cases. We will see that some values admit just an
embedding, whereas for other values there are more embeddings and we will show
why they are not in the same orbit.

In Table 1, we compute the values of $\lambda$, $\mu$ and $\delta$. To get all
the possibilities, one can act on the given $(\lambda,\mu,\delta)$ with the
elements of the isometry group $\mathcal{O}(\mathbb{L}_{20})$, i.e exchange
$\lambda$ and $\mu$ and multiply the vector by $-1$.

$L^{2}$ | $16$ | $20$ | $24$ | $28$ | $32$ | $36$ | $40$
---|---|---|---|---|---|---|---
$\lambda$ | 2 | 1 | $-1,2$ | - | 2 | 2 | 3 | $-2,3$ | $3$ | $1,-1$
$\mu$ | 0 | 2 | $0,1$ | - | -1 | 2 | 0 | $0,1$ | 1 | $1,-1$
$\delta$ | 0 | 0 | 1 | - | 1 | 0 | 0 | 1 | 0 | 2
Table 1. Cases up to $L^{2}=40$

######  Remark 3.2.

Two vectors $(\lambda,\mu,\delta)$ and
$(\lambda^{\prime},\mu^{\prime},\delta^{\prime})$ with
$\delta\neq\pm\delta^{\prime}$ are not in the same orbit by the action of
$\mathcal{O}(\mathbb{L}_{20})$. This follows the fact that
$\rho_{1}(\lambda,\mu,\delta)=(\mu,\lambda,\delta)$ and both $\rho_{2}$ and
$-id_{\mathbb{L}_{20}}$ change the sign of $\delta$.

Observe that the converse is not true, i.e. there exists vectors
$(\lambda,\mu,\delta),(\lambda^{\prime},\mu^{\prime},\delta^{\prime})$ such
that $\delta=\delta^{\prime}$ but not in the same orbit by the action of
$\mathcal{O}(\mathbb{L}_{20})$. For example for $L^{2}=300$, the two vectors
$(\lambda,\mu,\delta)=(3,6,5)$ and
$(\lambda^{\prime},\mu^{\prime},\delta^{\prime})=(5,0,5)$ are not in the same
orbit, but $\delta=\delta^{\prime}$.

#### $L^{2}=16$

There are $4$ possibilities for $(\lambda,\mu,\delta)$ and they are in the
same orbit modulo isometry. Hence there exists a unique embedding $\langle
16\rangle\hookrightarrow\mathbb{L}_{20}$ which maps $L$ to $2e$. The lattice
$\mathrm{T}_{X}$ obtained in this case is the same as the case $L^{2}=4$ and
we will see in Section 5 the relation between these two examples.

#### $L^{2}=20$

Up to isometry, we obtained two embedding $\langle
20\rangle\hookrightarrow\mathbb{L}_{20}$:

$L\longmapsto e+2f\quad\mathrm{et}\quad L\longmapsto f-h\,.$

These embeddings are not in the same orbit, due to Remark 3.2.

Let us consider the first embedding $L\longmapsto e+2f$; we can choose
$u_{1}=f-2e$ and $u_{2}=e+f+2h$ as generators of the transcendental lattice
and get

$\mathrm{T}_{X}=\left(\begin{array}[]{cc}20&0\\\ 0&40\end{array}\right).$

In the second case with $L\longmapsto f-h$, we take $u_{1}=2f+e+h$ and
$u_{2}=f-3e$ and get

$\mathrm{T}_{X}=\left(\begin{array}[]{cc}20&0\\\ 0&40\end{array}\right).$

######  Remark 3.3.

We observe that for $L^{2}=20$ there are two non isometric vectors
$(\lambda,\mu,\delta)$ but the matrix for the lattice $T_{X}$ is the same,
therefore the same K3 surface admits two different actions of $M_{20}$.

#### $L^{2}=24$

In this case there is no embedding by Proposition 3.1.

#### $L^{2}=28$

In this case $\delta=\pm 1$. If $\delta=1$, then $\lambda,\mu\in\\{2,-1\\}$
and if $\delta=-1$, then $\lambda,\mu\in\\{-2,1\\}$. This gives 8
possibilities for $(\lambda,\mu,\delta)$, all in the same orbit. Thus we can
choose $L\mapsto e+f-h$. Taking $u_{1}=f-e$ and $u_{2}=4e+4f+3h$ we compute

$\mathrm{T}_{X}=\left(\begin{array}[]{cc}8&0\\\ 0&140\end{array}\right)$

and the conditions ($\star$ ‣ 1) hold for this matrix.

#### $L^{2}=32$

In this case one obtains the same $\mathrm{T}_{X}$ as the case $L^{2}=8$. This
case is explained in Section 5.

#### $L^{2}=36$

In this case, there are two possibilities:

$L\mapsto 3e\quad\mathrm{and}\quad L\mapsto 3e+h\,.$

The first embedding which maps $L$ on $3e$ gives the same transcendental
lattice as $L^{2}=4$ because $36=3^{2}\cdot 4$ (see Section 5).

For the second embedding, one can consider $u_{1}=3f+h$ and $u_{2}=2f+e-h$.
The matrix then is

$\mathrm{T}_{X}=\left(\begin{array}[]{cc}36&12\\\ 12&44\end{array}\right).$

#### $L^{2}=40$

There are two possibilities, up to isometry. One is $L\mapsto e+f+2h$ and this
one is studied in [12]. The other one is $L\mapsto 3f+e$. Taking $u_{1}=f-e+h$
and $u_{2}=2e+h$ one gets $u_{1}^{2}=u_{2}^{2}=20,u_{1}\cdot u_{2}=0$, i.e.

$\mathrm{T}_{X}=\left(\begin{array}[]{cc}20&0\\\ 0&20\end{array}\right).$

######  Remark 3.4.

Observe that in [12, Lemma 3.1, Prop. 3.3], the author proves that the surface
$X=Km(E_{i}\times E_{i})$ is the unique surface admitting the action of a
symplectic group $G$ such that $G_{0}\simeq M_{20}$ and
$G/G_{0}\simeq\mathbb{Z}/4\mathbb{Z}$. In this case
$T_{X}=\left(\begin{array}[]{cc}4&0\\\ 0&4\end{array}\right)$. The fact that
we find here two different surfaces (with two different trascendental
lattices) is not contradicting the unicity proven in [12], since here we are
not assuming the action of $G$ such that $G_{0}\simeq M_{20}$ and
$G/G_{0}\simeq\mathbb{Z}/4\mathbb{Z}$.

Let $X_{\mathrm{Ko}}$ be the surface with
$T_{X_{\mathrm{Ko}}}=\left(\begin{array}[]{cc}4&0\\\ 0&4\end{array}\right)$
according to the notation of [3, Section 3], and let $X^{\prime}$ be the K3
surface with transcendental lattice
$T_{X^{\prime}}=\left(\begin{array}[]{cc}20&0\\\ 0&20\end{array}\right)$. We
can observe that $T_{X^{\prime}}=5T_{X_{\mathrm{Ko}}}$ and this relates to the
fact that by [20, Section 5], $X^{\prime}$ is the Kummer surface
$\mathrm{Km}\left(E_{5i}\times E_{5i}\right)$, where $E_{\tau}$ is the usual
notation for the elliptic curve $\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$.

##  4\. About the projective models

In order to describe the projective models of K3 surfaces with an action of
$M_{20}$, we want to understand when the linear system $|L|$ is hyperelliptic,
i.e. when there exists a surface $S\subset\mathbb{P}^{n}$ such that the map
$X\rightarrow S$ given by $|L|$ is $2:1$, and when $L$ defines an embedding.
With the previous notations, we will show the following:

######  Theorem 4.1.

Given the K3 surface $X$ with the embedding given by $L$ and $L^{2}=4n$, we
have the following results:

  1. 1.

$L$ is an ample class;

  2. 2.

the linear system $|L|$ is not hyperelliptic;

  3. 3.

the projective model is only defined by quadrics.

For the following, we assume that $L$ is $M_{20}$-invariant and $L^{2}=4n$,
with $n\in\mathbb{Z}_{\geq 1}$. The proof of Theorem 4.1 will be done in the
following four Sections.

###  4.1. $L$ is ample

Recall that $\mathrm{NS}(X)\supset\mathbb{Z}L\oplus\mathbb{L}_{20}^{\perp}$
and consider $C\in\mathrm{NS}(X)$ a $(-2)$-curve. Then, it exists
$v\in\mathbb{L}_{20}^{\perp}$ and two integers $\alpha$ and $\eta\neq 0$ such
that

$C=\frac{\alpha L+v}{\eta}.$

Observe that we can choose these two integers to be positive. In fact, by
Riemman-Roch Theorem [10, Section V, Theorem 1.6], we have for any divisor $D$
on a K3 surface:

$h^{0}(D)+h^{0}(-D)=2+\frac{D^{2}}{2}+h^{1}(D)$

where $h^{i}(D)=\dim\mathrm{H}^{i}(D)$, for $i\geq 0$. In our case, $D=C$ is a
$(-2)$-curve, so the sum $h^{0}(C)+h^{0}(-C)$ is strictly positive. By
definition, $h^{0}(C)$ is strictly positive if and only if $C$ is an effective
divisor. Hence $C$ or $-C$ is effective and so we can choose the integers
$\alpha$ and $\eta$ to be positive.

Now, taking the intersection with $L$, we have $L\cdot
C=\frac{\alpha}{\eta}L^{2}$. Remark that if $\alpha=0$, then $C$ lives in
$\mathbb{L}_{20}^{\perp}$, which does not contain any $(-2)$-class by [14,
Lemma 4.2 b)]. Hence $L\cdot C>0$, which implies by [10, Chapter V, Theorem
1.10] that $L$ is an ample class.

###  4.2. $|L|$ has no fixed part

By [16, Section 3.8], either $|L|$ has no fixed part or $L=aE+\Gamma$, with
$a$ a positive integer, $E$ an elliptic curve and $\Gamma$ an irreducible
$(-2)$-curve such that $E\cdot\Gamma=1$. Suppose that we are in the second
case. Consider $v,w\in\mathbb{L}_{20}^{\perp}$ and the integers
$\alpha,\beta\geq 0$ and $\eta,\eta^{\prime}>0$ such that

$E=\frac{\alpha L+v}{\eta}\quad\mathrm{and}\quad\Gamma=\frac{\beta
L+w}{\eta^{\prime}}\,.$

As before, we can chose all the integers $\alpha,\beta,\eta,\eta^{\prime}$ to
be positive due to Riemann-Roch Theorem. By simple computations we have
$E\cdot L=E\cdot\Gamma=1$ which implies $\eta=4n\alpha$. Now, we compute the
index $I$ of $\mathbb{Z}L\oplus\mathbb{L}_{20}^{\perp}$ in $\mathrm{NS}(X)$.
Using the notations introduced in Section 1,
$\left|\det\mathrm{NS}(X)\right|=\left|\det\mathrm{T}_{X}\right|=4\left(4ac-b^{2}\right)$
and
$\left|\det\left(\mathbb{Z}L\oplus\mathbb{L}_{20}^{\perp}\right)\right|=640n$.
Then by [2, Chapter I, Lemma 2.1] we have

$I^{2}=\left[\mathrm{NS}(X):\mathbb{Z}L\oplus\mathbb{L}_{20}^{\perp}\right]^{2}=\frac{160n}{4ac-b^{2}}\,.$

Remember that $E=\frac{\alpha L+v}{4n\alpha}$ is an element of
$\mathrm{Pic}(X)$ and remark that $4n\alpha$ is the smaller integer such that
$4n\alpha E\in\mathbb{Z}L\oplus\mathbb{L}_{20}$. Then $(4n\alpha)^{2}$ must
divide
$\left[\mathrm{NS}(X):\mathbb{Z}L\oplus\mathbb{L}_{20}^{\perp}\right]^{2}$,
that is

(5) $\exists m\in\mathbb{N},\,10=n\alpha^{2}\left(4ac-b^{2}\right)m\,.$

So the integer $n$ should divide $10$. Hence $n\in\\{1,2,5,10\\}$, that is
$L^{2}\in\\{4,8,20,40\\}$. We already know that for $n=4,8$, the linear system
$|L|$ has no fixed part by [3]. It remains to study the two last cases.

  * •

Case $n=5$: We know by the table in Appendix A that $4ac-b^{2}=200$, so the
equation (5) becomes $100m\alpha^{2}=1$, which is impossible to solve for
$m\in\mathbb{N},\alpha\in\mathbb{Z}_{\geq 0}$.

  * •

Case $n=10$: In this case we have two possibilities for the transcendental
lattice $T_{X}$. The one which corresponds to the Kondo’s case was already
studied in [3] and the linear system has no fixed part. For the other one, the
equation (5) is impossible to solve as in the previous case.

Thus for all $n\in\mathbb{Z}_{\geq 1}$, the equation (5) has no solution.
Hence the linear system $|L|$ has no fixed part.

###  4.3. $|L|$ is not hyperelliptic

We first prove the following lemma:

######  Lemma 4.2.

If the linear system $|L|$ is hyperelliptic, then $n$ divides $40$.

###### Proof.

Let us assume that $|L|$ is hyperelliptic. By [17, Theorem 5.2] $L$ is
hyperelliptic only in the two following cases:

  * •

There exists an irreducible curve $E$ of genus 1 such that $E\cdot L=2$.

  * •

There exists an irreducible curve $B$ of genus 2 such that $L=2B$.

First, assume that there exists an irreducible curve $B$ of genus $2$ such
that $L=2B$. Remark that $B^{2}=2$, so $L^{2}=4B^{2}=8$. However, we already
know that in this case the model is not hyperelliptic by [3].

Now suppose that there exists an irreducible curve $E\in\mathrm{NS}(X)$ of
genus $1$ such that $E\cdot L=2$. We can consider two integers $\alpha$ and
$\eta$ such that $E=\frac{\alpha L+v}{\eta}$, and as before we can chose these
integers to be positive. By taking the product with $L$ we have
$\frac{4n\alpha}{\eta}=2$ which implies $\eta=2n\alpha$ and so $E=\frac{\alpha
L+v}{2n\alpha}$. Recall that we have

$\left[\mathrm{NS}(X):\mathbb{Z}L\oplus\mathbb{L}_{20}^{\perp}\right]^{2}=\frac{160n}{4ac-b^{2}}\,.$

As $E=\frac{\alpha L+v}{2n\alpha}$ is in $\mathrm{NS}(X)$, it follows that
$4n^{2}\alpha^{2}$ divides $\frac{160n}{4ac-b^{2}}$, that is

(6) $\exists m\in\mathbb{N},\,40=n\alpha^{2}\left(4ac-b^{2}\right)m\,.$

Hence $n$ divides $40$, so the lemma. ∎

To complete the proof that $|L|$ is not hyperelliptic, it remains to study the
divisors of 40, i.e. $n\in\\{1,2,4,5,8,10,20,40\\}$ which correspond to
$L^{2}\in\\{4,8,16,20,32,40,80,160\\}$. We already know that in the cases
$L^{2}=4,8$, the projective model is not hyperelliptic.

  * •

Case $L^{2}\in\\{16,32,80,160\\}$. In each of these four cases, we know that
there exists $M$ such that $L=2M$ and $M^{2}=4m$, with $m>0$ an integer.
Assume that there exists an irreducible curve $E$ of genus $1$ such that
$EL=2$. This implies that $EM=1$, which is impossible because $E\cdot C>1$,
see [17, Scholie 3.9.6].

  * •

Case $L^{2}=20$. In this case the equation (6) becomes $25\alpha^{2}m=1$ which
is impossible to solve.

  * •

Case $L^{2}=40$. In this last case we have two embeddings. In the case studied
by Kondo, we already know that the model is not hyperelliptic by [3]. In the
other case, the equation (6) is the same as the case $L^{2}=20$, so it is
impossible to solve.

Hence up to now we have proven that for all integer $n\geq 1$, the linear
system $|L|$ has no fixed part and it is never hyperelliptic. In other words,
it always defines an embedding $\langle
4n\rangle\hookrightarrow\mathbb{L}_{20}$.

###  4.4. Quadrics on the surface

We now prove the last part of Theorem 4.1, i.e. that the surface is defined by
an intersection of quadrics.

By [17, Theorem 7.2] this is true except in the two following cases:

  * •

there exists an irreducible curve $E$ of genus 1 with $E\cdot L=3$, or

  * •

$L=2B+F$ where $B$ is an irreducible curve of genus 2 and $F$ is an
irreducible rational curve such that $B\cdot F=1$.

For the first case, assume that there exists an irreducible curve $E$ of genus
$1$ such that $E\cdot L=3$. As before, we have $L^{2}=4n$ with $n>3$ and we
can write $E=\frac{\alpha L+v}{\eta}$. By computing the product $E\cdot L$, we
obtain $\eta=\frac{4n\alpha}{3}$. We observe that we have to assume $n\in
3\mathbb{Z}$ or $\alpha\in 3\mathbb{Z}$. Using again the index of
$\mathbb{Z}L\oplus\mathbb{L}_{20}^{\perp}$ in $\mathrm{NS}(X)$ we have

$\exists m\in\mathbb{Z},\,90=n\alpha^{2}\left(4ac-b^{2}\right)m\,.$

Hence $n$ divides $90$ that is $L^{2}\in\\{20,24,36,40,60,72,120,180,360\\}$.
Looking at the Appendix A for all these cases, we have no solutions for the
previous equation. The second point is not possible either, since
$\left(2B+\Gamma\right)^{2}=10\neq 4n$.

This concludes the proof of Theorem 4.1. We can be more precise and compute
the exact number $Q_{L^{2}}$ of quadrics which define the K3 surface
associated to $L$.

######  Proposition 4.3.

Let $X\subset\mathbb{P}^{N}$ with the embedding $L$ such that $L^{2}=4n$. The
number of quadrics defining $X$ in $\mathbb{P}^{N}$ is

$Q_{4n}=2n^{2}-3n+1.$

###### Proof.

Suppose that we have $X\subset\mathbb{P}^{N}$ with the embedding determined by
$L$ such that $L^{2}=4n$. We know that $N=2n+1$ by Proposition 1.5. Following
[17, Section 6.5.3], one can compute the number of quadrics defining the
surface as

$\dim H^{0}(2L)-\dim S^{2}H^{0}(L),$

where $\dim S^{2}H^{0}(L)$ is the total number of quadrics in the projective
space $\mathbb{P}^{2n+1}$, that is $\binom{2n+3}{2}$. By Riemann Roch theorem
[10, Section V, Theorem 1.6] we can compute $\dim H^{0}(2L)=2+8n$. Finally, we
conclude that the number of quadrics defining $X$ is

$Q_{4n}:=\binom{2n+3}{2}-(2+8n)=2n^{2}-3n+1.$

∎

##  5\. Polarizations which are not primitives

With the computations of $T_{X}$ (see Appendix A), one could remark that there
are similarities between some cases. For example, we get the same lattice
$T_{X}$ when $L^{2}=4$ and $L^{2}=16$, as well as in cases $L^{2}=8$ and
$L^{2}=32$. Also, there is a relation between these embeddings. Actually, we
have the following result:

######  Proposition 5.1.

Given the embedding $\langle 4n\rangle\hookrightarrow\mathbb{L}_{20}$, then,
for all integers $r\geq 1$, there exists an embedding $\langle
r^{2}4n\rangle\hookrightarrow\mathbb{L}_{20}$ and the lattice $\mathrm{T}_{X}$
obtained for $L$ is the same as the one for $rL$.

###### Proof.

For the existence of the embedding $\langle
r^{2}4n\rangle\hookrightarrow\mathbb{L}_{20}$, we can assume that there exists
$\lambda_{0},\mu_{0},\delta_{0}\in\mathbb{Z}$ such that
$L\mapsto\lambda_{0}e+\mu_{0}f+\delta_{0}h$. Then we have

$L^{2}=(2\lambda_{0}-\delta_{0})^{2}+(2\mu_{0}-\delta_{0})^{2}+10\delta_{0}^{2}$

which implies

$\begin{array}[]{cl}(rL)^{2}=r^{2}L^{2}&=r^{2}\left((2\lambda_{0}-\delta_{0})^{2}+(2\mu_{0}-\delta_{0})^{2}+10\delta_{0}^{2}\right)\\\
&=(2r\lambda_{0}-r\delta_{0})^{2}+(2r\mu_{0}-r\delta_{0})^{2}+10\left(r\delta_{0}\right)^{2}\\\
\end{array}$

Hence
$rL\mapsto\left(r\lambda_{0}\right)e+\left(r\mu_{0}\right)f+\left(r\delta_{0}\right)h$
is the desired embedding.

Now we want to study the transcendental lattice $\mathrm{T}_{X}$. For all
integers $\lambda$, $\mu$ and $\delta$, we have

$\langle\lambda e+\mu f+\delta h,rL\rangle=0\Longleftrightarrow\langle\lambda
e+\mu f+\delta h,L\rangle=0.$

Hence we can choose the same $\lambda$, $\mu$ and $\delta$ for each multiple
of $L$ and obtain every time the same lattice $\mathrm{T}_{X}$. ∎

The previous proposition allows to describe explicitly some non-primitive
cases, i.e. cases where the polarization is multiple of some other
polarization $L$.

Throughout this section, $X_{4n}$ will denote the K3 surface admitting the
polarization $L$ with $L^{2}=4n$. Thanks to Proposition 4.4, we are able to
compute the number of quadrics defining $X_{4n}$. In cases which are not
primitive, we can be more precise and give explicitly the equations of the
quadrics defining the surface. To do this, we first recall the Veronese
embedding of degree $d$, see [9]:

$\begin{array}[]{cccc}\nu_{d}^{n}:&\mathbb{P}^{n}&\longrightarrow&\mathbb{P}^{m}\\\
&\left(x_{0}:\dots:x_{n}\right)&\longmapsto&\left(x_{0}^{d}:x_{1}^{d}:\dots:x_{n}^{d}:x_{0}^{d-1}x_{1}:x_{0}^{d-1}x_{2}:\dots\right)\\\
\end{array}\,.$

Since we use monomials of degree $d$ in $n$ variables, $m=\binom{n+d}{d}-1$.

###  5.1. Cases $L^{2}=4\cdot(4n)$

We will first show explicitly some examples and then state the general
property.

######  Example 5.2 (Case $L^{2}=16$).

To study the case $L^{2}=16$, we first consider the surface $X_{\mathrm{Mu}}$
with the polarisation given by $M$, with $M^{2}=4$. By Proposition 1.6,
$X_{\mathrm{Mu}}$ is the zero locus of a quartic in $\mathbb{P}^{3}$ whose
equation is given in (1).

Now let $L=2M$ and we consider the Veronese embedding of degree 2:

$\nu_{2}^{3}:\mathbb{P}^{3}\rightarrow\mathbb{P}^{9}.$

Let $(y_{0}:\ldots:y_{9})$ be the coordinates of $\mathbb{P}^{9}$. The image
of the equation defining $X_{Mu}$ in $\mathbb{P}^{9}$ via $\nu_{2}^{3}$ is
given by

(7)
$y_{0}^{2}+y_{1}^{2}+y_{2}^{2}+y_{3}^{2}-6(y_{4}^{2}+y_{5}^{2}+y_{6}^{2}+y_{7}^{2}+y_{8}^{2}+y_{9}^{2})$

The image $\nu_{2}^{3}(\mathbb{P}^{3})$ is defined by the quadrics given by
the zero loci of the $2\times 2$ minors of the matrix

$\mathcal{A}:=\left(\begin{array}[]{cccc}y_{0}&y_{4}&y_{5}&y_{6}\\\
y_{4}&y_{1}&y_{7}&y_{8}\\\ y_{5}&y_{7}&y_{2}&y_{9}\\\
y_{6}&y_{8}&y_{9}&y_{3}\\\ \end{array}\right)\,.$

This gives 20 equations of quadrics. Let us consider the K3 surface $X_{16}$
in $\mathbb{P}^{9}$ given by the 20 quadrics obtained as minors of the matrix
$\mathcal{A}$ plus the quadric defined by (7). The number of quadrics is
$Q_{16}=21$, as desired. As observed in Proposition 5.1, the surface $X_{16}$
admits a polarization $L$ such that $L^{2}=4M^{2}=16$ and it inherits the
action of $M_{20}$, as well as the action of $\mu_{2}$ described in [3,
Section 4].

######  Example 5.3 (Case $L^{2}=64$).

We have previously shown a model of $X_{16}$ in $\mathbb{P}^{9}$ defined by
$Q_{16}=21$ quadrics. In order to exhibit a projective model of a K3 surface
$X_{64}$ with a polarization $L$ with $L^{2}=64$, we consider the Veronese
embedding $\nu_{2}^{9}:\mathbb{P}^{9}\rightarrow\mathbb{P}^{54}$. The 21
quadrics defining $X_{16}\subseteq\mathbb{P}^{9}$ give 21 hyperplanes in
$\mathbb{P}^{54}$, thus their intersection determines a 33-dimensional space.
The desired surface $X_{64}\subset\mathbb{P}^{33}$ is given by the restriction
of the quadrics defining $\nu_{2}^{9}(\mathbb{P}^{9})$ to this 33-dimensional
space.

######  Example 5.4 (Case $L^{2}=32$).

By [3, Section 5] the K3 surface $X_{BH}$ admits a polarization $M$ such that
$M^{2}=8$. Taking $L=2M$, we obtain a polarization with $L^{2}=32$ on the same
surface. We would like to describe its projective model which is in
$\mathbb{P}^{17}$ by Proposition 1.5. By Proposition 1.6, case 3., $X_{BH}$ is
the intersection of 3 quadrics in $\mathbb{P}^{5}$.

In order to show the projective model of $X_{BH}$ in $\mathbb{P}^{17}$ we
consider the Veronese embedding
$\nu_{2}^{5}:\mathbb{P}^{5}\rightarrow\mathbb{P}^{20}$. The image of
$\mathbb{P}^{5}$ via $\nu_{2}^{5}$ is the zero locus of quadrics and their
number is

$\frac{1}{2}\left(\binom{6}{2}^{2}+\binom{6}{2}\right)-\binom{6}{4}=105\ .$

The last term $\binom{6}{4}$ considers the Plücker relations. The images of
$q_{1}$, $q_{2}$ and $q_{3}$ via $\nu_{2}^{5}$ give three linear equations,
thus $q_{1},q_{2},q_{3}$ define 3 hyperplanes in $\mathbb{P}^{20}$. Their
intersection is a $17$-dimensional space and in this projective space of
dimension 17 the equation of $X$ is given by the restrictions of the $105$
equations defining $\nu_{2}^{5}(\mathbb{P}^{5})$.

In general let $M^{2}=4n$. Recall that the number of quadrics defining
$X_{4n}$ by Proposition 4.3 is

$Q_{4n}=2n^{2}-3n+1.$

If we consider $L=2M$, therefore $L^{2}=4M^{2}=16n$ and we expect to find a
projective model of $X_{16n}$ in $\mathbb{P}^{8n+1}$ by Proposition 1.5. We
consider the Veronese embedding of degree 2

$\nu_{2}^{2n+1}:\mathbb{P}^{2n+1}\hookrightarrow\mathbb{P}^{2n^{2}+5n+2}\,.$

The $Q_{4n}$ quadrics defining $X_{4n}$ give via $\nu_{2}$ a set of
independent hyperplanes in $\mathbb{P}^{2n^{2}+5n+2}$, whose number is
$Q_{4n}=2n^{2}-3n+1$. Thus we obtain a subspace of dimension

$(2n^{2}+5n+2)-(2n^{2}-3n+1)=8n+1$

as expected by Proposition 1.5. The surface $X_{16n}$ is given by the
restriction of the quadrics defining $\nu_{2}(\mathbb{P}^{2n+1})$ to this
space of dimension $8n+1$.

Observe that this argument works for any value of $M^{2}=4n$. We showed
explicitley what happens when $M^{2}=4$, $M^{2}=16$ and $M^{2}=8$ in the
previous examples.

###  5.2. Cases $L^{2}=4r^{2}$

Another interesting case is when we take $L=rM$ with $M^{2}=4$. Let $X_{Mu}$
be the K3 surfaces defined in Proposition 1.6, with $M^{2}=4$, whose
projective model is given by the zero locus of a polynomial $f_{4}$ of degree
4 in $\mathbb{P}^{3}$. We show how to obtain the projective model of the
surface with embedding $L=rM$, thus having $L^{2}=4r^{2}$.

######  Example 5.5 (Case $L^{2}=36$, $r=3$).

We expect a model of $X_{36}$ in $\mathbb{P}^{19}$ by Propositon 1.5.
According to [9, Example 2.4], one can prove $X_{Mu}=\\{f_{4}=0\\}$ can also
be defined as the intersection of the zero loci of

$q_{2}(x_{0},\ldots,x_{3})f_{4}(x_{0},\ldots,x_{3})=0$

where $q_{2}(x_{0},\ldots,x_{3})$ is one of the 10 elements of the basis of
polynomials of degree 2 in $(x_{0},\ldots,x_{3})$, i.e.:

$x_{0}^{2},\ x_{1}^{2},\ x_{2}^{2},\ x_{3}^{2},\ldots x_{2}x_{3}.$

Thus $X_{Mu}$ is defined as the intersection of the zero loci of 10
polynomials $g_{1},\ldots,g_{10}$ of degree 6. Let
$\nu_{3}^{3}:\mathbb{P}^{3}\rightarrow\mathbb{P}^{19}$ be the Veronese
embedding of degree 3. The image of each $g_{i}$ is a polynomial of degree 2
in $\mathbb{P}^{19}$. We thus obtain $X_{36}$ as the intersection of the 10
quadrics in $\mathbb{P}^{19}$ given by the $g_{i}$’s.

######  Example 5.6 (Case $L^{2}=100$, $r=5$).

If $r=5$ and we consider $L=5M$ with $M^{2}=4$, then $L^{2}=100$. Let $f_{4}$
be the polynomial of degree 4 defining the surface $X_{Mu}$. The zero locus
$\\{f_{4}=0\\}$ can also be described by the intersection of the zero loci of

(8) $x_{0}f_{4}=x_{1}f_{4}=x_{2}f_{4}=x_{3}f_{4}=0.$

Via the Veronese embedding
$\nu^{3}_{5}:\mathbb{P}^{3}\rightarrow\mathbb{P}^{\binom{8}{5}-1}=\mathbb{P}^{55}$
the surface $X_{Mu}$ has as image a K3 surface admitting a polarisation
$L=5M$. The equations (8) define 4 hyperplanes in $\mathbb{P}^{55}$, thus the
model of $X_{100}$ is contained in $\mathbb{P}^{51}$.

More generally, for any $r>3$, via the Veronese embedding
$\nu^{3}_{r}:\mathbb{P}^{3}\rightarrow\mathbb{P}^{\binom{r+3}{3}-1}$ one
obtains a K3 surface admitting a polarization $L=rM$ and thus $L^{2}=4r^{2}$.
This surface is obtained as the image of $X_{Mu}$ via $\nu^{3}_{r}$. As
observed, the surface $X_{Mu}$ can be described by the intersection of the
zero loci of

(9) $q_{1}f_{4}=\ldots=q_{s}f_{4}=0$

with $s=\binom{r-1}{3}$ and the $q_{i}$’s are monomials of degree $r-4$.
Equations (9) define $\binom{r-1}{3}$ hyperplanes in
$\mathbb{P}^{\binom{r+3}{3}-1}$ thus one obtains a model for the K3 surface
$X_{4r^{2}}$ in a space of dimension

$\binom{r+3}{3}-1-\binom{r-1}{3}=2r^{2}+1$

as expected.

##  6\. Existence of infinitely many K3 surfaces with an action of $M_{20}$

It is interesting to ask whether there exists an infinite number of K3
surfaces with an action of $M_{20}$. According to previous Sections, one can
consider the polarization $L^{2}=4p$ with $p$ a prime. If equation (2) admits
a solution for $L^{2}=4p$, one can repeat the arguments used before and by
Theorem 1.3 one obtains a K3 with the action of $M_{20}$. Moreover, since $p$
is a prime, this surface is a new one, meaning that it does not admit a
polarization $L^{2}=4m$ with $m<p$.

######  Theorem 6.1.

There exist infinitely many K3 surfaces admitting the action of $M_{20}$.

###### Proof.

It suffices to show that there are infinite number of primes $p$ such that the
polarization $L^{2}=4p$ defines an embedding and thus a K3 surface. In order
to answer this, we need to check that there is an infinite number of primes
$p$ such that the equation

(10) $4p=(2\lambda-\delta)^{2}+(2\mu-\delta)^{2}+10\delta^{2}.$

admits a solution $(\lambda,\mu,\delta)$.

We observe that if $p\equiv 1\ (4)$, one can take $\delta=0$. The equation
becomes

$p=\lambda^{2}+\mu^{2}$

and Fermat’s theorem of two squares [7, Chapter 1.1] ensures the existence of
two integers $\lambda,\mu$. We now use Dirichlet’s Theorem in order to ensure
the existence of infinitely many primes $p\equiv 1\ (4)$.

######  Theorem (Dirichlet).

Given two positive coprime integers $a,d$, there exist infinitely many primes
of the form $am+d$.

Taking $a=4,d=1$, the result guarantees that there exists infinitely many
primes of the form $4m+1$ and thus infinitely many primes such that the
equation (10) admits a solution $(\bar{\lambda},\bar{\mu},0)$. It follows that
there is an infinite number of K3 surfaces with an action of $M_{20}$.∎

##  Appendix A Table of $T_{X}$

We recall the notations of Table LABEL:tab_complete. Let $X$ be a K3 surface
admitting a faithful and symplectic action of the Mathieu group $M_{20}$ and
let $T_{X}$ be its transcendental lattice, with
$T_{X}=\left(\begin{array}[]{cc}4a&2b\\\ 2b&4c\end{array}\right)$ and $a,b,c$
as in Section 1, satisfying ($\star$ ‣ 1). We denote by $L\in\mathrm{Pic}(X)$
a $M_{20}$-invariant ample class such that $L^{2}=4n$, with
$n\in\mathbb{Z}_{\geq 1}$ and by $Q_{L^{2}}$ the number of quadrics that
describe the projective model, according to Proposition 4.3. Finally,
$I=\sqrt{\frac{160n}{4ac-b^{2}}}$ is the index of
$\mathbb{Z}L\oplus\mathbb{L}_{20}^{\perp}$ in $\mathrm{NS}(X)$. When
$n=1,2,10$, the surfaces are described in Proposition 1.6.

Table 2. Table with all cases  $n$ | $L^{2}$ | $Q_{L^{2}}$ | $T_{X}$ | $a$ | $b$ | $c$ | $\langle 4n\rangle\hookrightarrow\mathbb{L}_{20}$ | $I=\sqrt{\frac{160n}{4ac-b^{2}}}$
---|---|---|---|---|---|---|---|---
1 | 4 | 1 | $\left(\begin{array}[]{cc}4&0\\\ 0&40\end{array}\right)$ | 1 | 0 | 10 | $L\mapsto e$ | 2
2 | 8 | 3 | $\left(\begin{array}[]{cc}8&4\\\ 4&12\end{array}\right)$ | 2 | 2 | 3 | $L\mapsto e+f$ | 4
3 | 12 | 10 | $\left(\begin{array}[]{cc}8&0\\\ 0&60\end{array}\right)$ | 2 | 0 | 15 | $L\mapsto h$ | 2
4 | 16 | 21 | $\left(\begin{array}[]{cc}4&0\\\ 0&40\end{array}\right)$ | 1 | 0 | 10 | $L\mapsto 2e$ | 4
5 | 20 | 36 | $\left(\begin{array}[]{cc}20&0\\\ 0&40\end{array}\right)$ | 5 | 0 | 10 | $L\mapsto e+2f$ | 2
$L\mapsto f-h$
6 | 24 | - | - | - | - | - | - | -
7 | 28 | 80 | $\left(\begin{array}[]{cc}8&0\\\ 0&140\end{array}\right)$ | 2 | 0 | 35 | $L\mapsto e+f-h$ | 4
8 | 32 | 105 | $\left(\begin{array}[]{cc}8&4\\\ 4&12\end{array}\right)$ | 2 | 2 | 3 | $L\mapsto 2e+2f$ | 8
9 | 36 | 136 | $\left(\begin{array}[]{cc}4&0\\\ 0&40\end{array}\right)$ | 1 | 0 | 10 | $L\mapsto 3e$ | 6
$\left(\begin{array}[]{cc}36&12\\\ 12&44\end{array}\right)$ | 9 | 6 | 11 | $L\mapsto 3e+h$ | 2
10 | 40 | 171 | $\left(\begin{array}[]{cc}4&0\\\ 0&4\end{array}\right)$ | 1 | 0 | 1 | $L\mapsto e+f+2h$ | 20
$\left(\begin{array}[]{cc}20&0\\\ 0&20\end{array}\right)$ | 5 | 0 | 5 | $L\mapsto e+3f$ | 4
15 | 60 | 406 | $\left(\begin{array}[]{cc}8&0\\\ 0&12\end{array}\right)$ | 2 | 0 | 3 | $L\mapsto 2e+2f-h$ | 5
$\left(\begin{array}[]{cc}20&0\\\ 0&120\end{array}\right)$ | 5 | 0 | 25 | $L\mapsto e-2h$ | 1
18 | 72 | 595 | $\left(\begin{array}[]{cc}8&4\\\ 4&12\end{array}\right)$ | 2 | 2 | 3 | $L\mapsto 3e+3f$ | 12
$\left(\begin{array}[]{cc}8&4\\\ 4&92\end{array}\right)$ | 2 | 2 | 23 | $L\mapsto 3e+3f+2h$ | 4
30 | 120 | 1711 | $\left(\begin{array}[]{cc}20&10\\\ 10&20\end{array}\right)$ | 5 | 5 | 5 | $L\mapsto 3e+f-2h$ | 8
45 | 180 | 3916 | $\left(\begin{array}[]{cc}20&0\\\ 0&40\end{array}\right)$ | 5 | 0 | 10 | $L\mapsto 3f-3h$ | 6
$L\mapsto 3e+6f$
$\left(\begin{array}[]{cc}20&0\\\ 0&360\end{array}\right)$ | 5 | 0 | 90 | $L\mapsto 4e+6f+h$ | 2
$L\mapsto 3e+4f+4h$
90 | 360 | 15931 | $\left(\begin{array}[]{cc}20&0\\\ 0&20\end{array}\right)$ | 5 | 0 | 5 | $L\mapsto 3e+9f$ | 12
$\left(\begin{array}[]{cc}4&0\\\ 0&4\end{array}\right)$ | 1 | 0 | 1 | $L\mapsto 3e+3f+6h$ | 60
$\left(\begin{array}[]{cc}20&0\\\ 0&180\end{array}\right)$ | 5 | 0 | 45 | $L\mapsto 3e+7f-2h$ | 4
$\left(\begin{array}[]{cc}8&4\\\ 4&20\end{array}\right)$ | 2 | 2 | 5 | $L\mapsto 3e+3f-4h$ | 20

##  Appendix B Quadrics for $X\subset\mathbb{P}^{7}$

The surface $X$ of Section 2 is described as the intersection of 10 quadrics
in $\mathbb{P}^{7}$. We detail now the equations of the quadrics
$F_{i}(x_{1},\ldots,x_{8})$, with $i\in\\{1,\ldots,10\\}$, obtained by
computations by MAGMA using a program by C. Bonnafé. In what follows, $a$ is a
primitive root of unity of order 20.

$\begin{split}F_{1}:=\textstyle\frac{1}{15}&(-736a^{7}+528a^{6}-352a^{5}-528a^{4}-736a^{3}-304)x_{1}x_{7}+\textstyle\frac{1}{15}(736a^{7}+272a^{6}+192a^{5}-272a^{4}+736a^{3}-576)x_{1}x_{8}+\\\
\textstyle\frac{1}{15}&(64a^{7}+208a^{6}+368a^{5}-208a^{4}+64a^{3}-64)x_{2}x_{7}+\textstyle\frac{1}{15}(-16a^{7}-192a^{6}-112a^{5}+192a^{4}-16a^{3}+416)x_{2}x_{8}+\\\
\textstyle\frac{1}{15}&(-176a^{7}-32a^{6}-112a^{5}+32a^{4}-176a^{3}+176)x_{3}x_{5}+\textstyle\frac{1}{5}(-32a^{7}+16a^{6}-64a^{5}-16a^{4}-32a^{3}-128)x_{3}x_{6}+\\\
\textstyle\frac{1}{5}&(48a^{7}+96a^{6}+96a^{5}-96a^{4}+48a^{3}-128)x_{4}x_{5}+\textstyle\frac{1}{5}(48a^{7}-64a^{6}+16a^{5}+64a^{4}+48a^{3}+352)x_{4}x_{6},\end{split}$

$\begin{split}F_{2}:=&(-224a^{7}-144a^{6}-160a^{5}+144a^{4}-224a^{3}+256)x_{1}x_{7}+(32a^{7}+240a^{6}+16a^{5}-240a^{4}+32a^{3}-448)x_{1}x_{8}+\\\
&(16a^{7}+96a^{6}+16a^{5}-96a^{4}+16a^{3}-128)x_{2}x_{7}+(64a^{7}-80a^{6}+32a^{5}+80a^{4}+64a^{3}+112)x_{2}x_{8}+\\\
&(-16a^{7}-64a^{6}-16a^{5}+64a^{4}-16a^{3}+128)x_{3}x_{5}+(-96a^{7}-16a^{6}-48a^{5}+16a^{4}-96a^{3})x_{3}x_{6}+\\\
&(-32a^{7}+112a^{6}-112a^{4}-32a^{3}-192)x_{4}x_{5}+(192a^{7}-48a^{6}+96a^{5}+48a^{4}+192a^{3}+144)x_{4}x_{6},\end{split}$

$\begin{split}F_{3}:=&(-96a^{7}-16a^{6}-48a^{5}+16a^{4}-96a^{3}+32)x_{1}x_{5}+\textstyle\frac{1}{3}(-32a^{7}+144a^{6}-80a^{5}-144a^{4}-32a^{3}-176)x_{1}x_{6}+\\\
&\textstyle\frac{1}{3}(32a^{7}+112a^{6}-112a^{4}+32a^{3}-240)x_{2}x_{5}+\textstyle\frac{1}{3}(80a^{7}+32a^{6}+64a^{5}-32a^{4}+80a^{3}+16)x_{2}x_{6}+\\\
&(-16a^{7}-16a^{3})x_{3}x_{7}+\textstyle\frac{1}{3}(32a^{7}+48a^{6}+80a^{5}-48a^{4}+32a^{3}-16)x_{3}x_{8}+(-16a^{7}-16a^{5}-16a^{3}+16)x_{4}x_{7}+\\\
&(16a^{7}+32a^{6}-32a^{4}+16a^{3}+16)x_{4}x_{8},\end{split}$

$\begin{split}F_{4}:=&\textstyle\dfrac{1}{3}(-112a^{7}-256a^{6}-32a^{5}+256a^{4}-112a^{3}+400)x_{1}x_{5}+\textstyle\frac{1}{3}(-144a^{7}+64a^{6}-32a^{5}-64a^{4}-144a^{3}-32)x_{1}x_{6}+\\\
&\textstyle\frac{1}{3}(-96a^{7}+112a^{6}-80a^{5}-112a^{4}-96a^{3}-176)x_{2}x_{5}+\textstyle\frac{1}{3}(48a^{7}+32a^{6}+80a^{5}-32a^{4}+48a^{3}-112)x_{2}x_{6}+\\\
&\textstyle\frac{1}{3}(-16a^{6}-16a^{5}+16a^{4}+80)x_{3}x_{7}+\textstyle\frac{1}{3}(16a^{7}+32a^{6}-32a^{4}+16a^{3}-96)x_{3}x_{8}+(-16a^{7}-32a^{6}+32a^{4}-16a^{3}+80)x_{4}x_{7}+\\\
&(-16a^{7}+32a^{6}+16a^{5}-32a^{4}-16a^{3}-48)x_{4}x_{8},\end{split}$

$\begin{split}F_{5}:=&(32a^{7}-160a^{6}+160a^{4}+32a^{3}+192)x_{1}x_{4}+32x_{2}x_{3}+(-32a^{7}+32a^{6}-64a^{5}-32a^{4}-32a^{3}-32)x_{2}x_{4}+\\\
&(-64a^{6}-16a^{5}+64a^{4}+128)x_{5}^{2}+(-96a^{7}+32a^{6}-64a^{5}-32a^{4}-96a^{3}-64)x_{5}x_{6}+(64a^{6}+16a^{5}-64a^{4}-32)x_{6}^{2}+\\\
&(32a^{7}-32a^{6}+16a^{5}+32a^{4}+32a^{3}+32)x_{7}^{2}+(-96a^{7}-32a^{6}-64a^{5}+32a^{4}-96a^{3})x_{7}x_{8}+\\\
&(32a^{7}+32a^{6}+48a^{5}-32a^{4}+32a^{3}-64)x_{8}^{2},\end{split}$

$\begin{split}F_{6}:=&32x_{1}x_{3}+(32a^{7}-32a^{6}+64a^{5}+32a^{4}+32a^{3}+32)x_{1}x_{4}+(-32a^{7}-32a^{6}+32a^{4}-32a^{3})x_{2}x_{4}+\\\
&(16a^{7}-48a^{6}+48a^{4}+16a^{3}+64)x_{5}^{2}+(-64a^{7}-32a^{5}-64a^{3})x_{5}x_{6}+(16a^{7}+16a^{6}-32a^{5}-16a^{4}+16a^{3}-32)x_{6}^{2}+\\\
&(16a^{7}+16a^{6}-16a^{4}+16a^{3})x_{7}^{2}+(-32a^{5}+64)x_{7}x_{8}+(16a^{7}+16a^{6}-32a^{5}-16a^{4}+16a^{3}-32)x_{8}^{2},\end{split}$

$\begin{split}F_{7}:=&\textstyle\frac{1}{5}(-304a^{7}+352a^{6}-128a^{5}-352a^{4}-304a^{3}-496)x_{1}^{2}+(64a^{7}+32a^{5}+64a^{3}+32)x_{1}x_{2}+(-16a^{7}-16a^{3}+16)x_{2}^{2}+\\\
&\textstyle\frac{1}{5}(-48a^{7}-96a^{6}-96a^{5}+96a^{4}-48a^{3}+48)x_{3}^{2}+\textstyle\frac{1}{5}(-656a^{7}+128a^{6}+128a^{5}-128a^{4}-656a^{3}+16)x_{4}^{2}+\\\
&\textstyle\frac{1}{5}(608a^{7}-64a^{6}+256a^{5}+64a^{4}+608a^{3}+672)x_{5}x_{7}+\textstyle\frac{1}{5}(-384a^{7}-288a^{6}-448a^{5}+288a^{4}-384a^{3}+384)x_{5}x_{8}+\\\
&\textstyle\frac{1}{5}(-192a^{7}-544a^{6}-64a^{5}+544a^{4}-192a^{3}+512)x_{6}x_{7}+\textstyle\frac{1}{5}(-224a^{7}+192a^{6}+192a^{5}-192a^{4}-224a^{3}-736)x_{6}x_{8},\end{split}$

$\begin{split}F_{8}:=&\textstyle\frac{1}{5}(-256a^{7}+368a^{6}-32a^{5}-368a^{4}-256a^{3}-704)x_{1}^{2}+\textstyle\frac{1}{5}(288a^{7}-224a^{6}+96a^{5}+224a^{4}+288a^{3}+192)x_{1}x_{2}+\\\
&\textstyle\frac{1}{5}(-32a^{7}+16a^{6}-64a^{5}-16a^{4}-32a^{3}+32)x_{2}^{2}+\textstyle\frac{1}{5}(-128a^{7}-16a^{6}+64a^{5}+16a^{4}-128a^{3}-32)x_{3}^{2}+\\\
&\textstyle\frac{1}{5}(32a^{7}-96a^{6}-256a^{5}+96a^{4}+32a^{3}-32)x_{3}x_{4}+\textstyle\frac{1}{5}(-224a^{7}+272a^{6}-448a^{5}-272a^{4}-224a^{3}-416)x_{4}^{2}+\\\
&\textstyle\frac{1}{5}(352a^{7}-576a^{6}+224a^{5}+576a^{4}+352a^{3}+768)x_{5}x_{7}+\textstyle\frac{1}{5}(-608a^{7}+64a^{6}-256a^{5}-64a^{4}-608a^{3}+128)x_{5}x_{8}+\\\
&\textstyle\frac{1}{5}(-448a^{7}-96a^{6}-96a^{5}+96a^{4}-448a^{3}+448)x_{6}x_{7}+\textstyle\frac{1}{5}(192a^{7}+544a^{6}+64a^{5}-544a^{4}+192a^{3}-512)x_{6}x_{8},\end{split}$

$\begin{split}F_{9}:=&\textstyle\frac{1}{9}(-160a^{7}-352a^{5}-160a^{3}-16)x_{1}^{2}+(32a^{7}+32a^{6}-32a^{4}+32a^{3}-32)x_{1}x_{2}+16x_{2}^{2}+\textstyle\frac{1}{9}(-32a^{6}-32a^{5}+32a^{4}+16)x_{3}^{2}+80x_{4}^{2}+\\\
&\textstyle\frac{1}{9}(128a^{7}-64a^{6}-128a^{5}+64a^{4}+128a^{3}+160)x_{5}x_{7}+\textstyle\frac{1}{9}(-32a^{7}+96a^{6}-32a^{5}-96a^{4}-32a^{3}-224)x_{5}x_{8}+\\\
&\textstyle\frac{1}{9}(-32a^{7}+32a^{6}-96a^{5}-32a^{4}-32a^{3}+96)x_{6}x_{7}+\textstyle\frac{1}{9}(128a^{7}-64a^{5}+128a^{3}-160)x_{6}x_{8},\end{split}$

$\begin{split}F_{10}:=&(-96a^{7}+96a^{6}-64a^{5}-96a^{4}-96a^{3}-144)x_{1}^{2}+(96a^{7}-32a^{6}+64a^{5}+32a^{4}+96a^{3}-32)x_{1}x_{2}+\\\
&(-32a^{6}-32a^{5}+32a^{4}+16)x_{2}^{2}+80x_{3}^{2}+(-32a^{6}+32a^{5}+32a^{4}+16)x_{4}^{2}-32x_{5}x_{7}+(32a^{7}-32a^{6}-32a^{5}+32a^{4}+32a^{3}+32)x_{5}x_{8}+\\\
&(32a^{7}+32a^{6}+32a^{5}-32a^{4}+32a^{3}-32)x_{6}x_{7}+(-64a^{6}-64a^{5}+64a^{4}+32)x_{6}x_{8}\end{split}$

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# Condition monitoring of wind turbine blades via learning-based methods

Giovanni Zaniboni<EMAIL_ADDRESS>Alessio Dallabona
<EMAIL_ADDRESS>Johnny Nielsen<EMAIL_ADDRESS>Dimitrios Papageorgiou
<EMAIL_ADDRESS>
###### Abstract

This paper addresses the topic of condition monitoring of wind turbine blades
and presents a learning-based approach to fault detection. The proposed scheme
utilises Principal Components Analysis and Autoencoders to derive data-driven
models from root-bending moment and other measurements. The models are trained
with real data obtained from a fault-free wind turbine, and then validated on
data corresponding to unknown health condition. Online test statistics,
employing static thresholds and Generalized Likelihood Ratio tests, are used
on residual signals generated by discrepancies between the actual and
reconstructed measurements to detect deviations from nominal operation. The
efficacy and effectiveness of the proposed methods are demonstrated using
real-life data collected from wind turbines experiencing blade faults.

###### keywords:

Wind Turbines , Fault Detection , Condition Monitoring , Principal components
analysis , Autoencoders

††journal: Renewable Energy

[1]organization=Department of Electronics, Information and Bioengineering
(DEIB), addressline=Via Ponzio 34/5, city=Milano, postcode=20133,
country=Italy [2]organization=Department of Electrical and Photonics
Engineering, Technical University of Denmark, addressline=Elektrovej 326,
city=Lyngby, postcode=2800, country=Denmark [3]organization=Vestas Wind System
A/S, addressline=Hedeager 42, city=Aarhus, postcode=8200, country=Denmark

##  1 Introduction

The Wind Turbine (WT) market has experienced significant growth in recent
years and is projected to continue expanding over the next few decades.
Governments, such as the European Commission, are committing substantial
resources to transition towards renewable energy sources, aiming for complete
carbon neutrality by 2050 [1].

As the number of wind turbines continues to rise, there is a growing need to
identify new suitable locations for new wind farms. The limited availability
of optimal sites is reflected in the trend of constructing taller turbines
with bigger blades for making the investment economically viable in less
optimal areas. The design of bigger blades requires different solutions, with
the consequence of dealing with different fault scenarios that have higher
economical impact. Even with the current blade designs state, about $14.9\%$
of the faults on WT are blade failures [2], caused by both endogenous and
exogenous factors [3, 4, 5]. Furthermore, blades are the structural element of
the turbine most affected by damage, due to their high complexity and
prolonged exposition to strong dynamical loads [3]. As a consequence, interest
arose in the investigation of condition monitoring and fault detection
strategies to be able to detect faults preemptively before they become more
severe and thus considerably more costly.

Model-based Fault Detection (FD) strategies are the most traditional schemes
used in wind turbines. Starting form a linearised model of the system,
residuals are commonly generated starting from Kalman filters and the
collected data [6]. In [7] and [8] a closed-loop scheme that describes the WT
model is provided, which was then used for residual generation using Kalman
Filters. In the latter, wind speed was taken as a reliable measurement for the
design of the Kalman filter. However, it is generally difficult to obtain
accurate local wind speed measurements [9]. This issue was addressed in [10],
where the authors estimated the wind speed via estimating the aerodynamic
torque.

Such limitations of conventional model-based methods can oftentimes be
overcome by leveraging data-driven methods that only rely on sensor readings
as opposed to estimating unmeasured signals [11]. Principal Component Analysis
(PCA) was implemented in [12] and [13] to detect sensors and actuator faults
in wind turbines. In [14], a fault detection algorithm was developed that
monitors the data collected online from a plant, whose model was built using
Principal Component Analysis. The authors in [15] presented a method for
sensor and actuator fault detection and proposed a way to address the
linearity of the PCA by performing multiple model training stages. During each
stage, data were collected from a different Operating Region of the WT. A
different data-driven fault detection approach was proposed in [16], where an
alarm signal was computed as a safety threshold on the root bending moments.

Neural network have also been explored because of their capability of dealing
with underlying nonlinear functions. Autoencoders were applied in [17], to
detect faults occurring on the gearbox and generator. Other works have
continued on this path, training the autoencoders to recognise different
faults happening on the WT, such as [18], and [19]. An AE model trained with
Long Short-Term Memory layers was proposed in [20] to recognise when a blade
cracks.

Although data-driven methods for fault detection in wind turbines has been
studied in the recent literature, dedicated studies on condition monitoring of
the blades has been sparsely reported. This paper explicitly focuses on the
design and application of learning-based fault diagnosis in WT. The main
contributions of this paper are

  * 1.

Derivation of data-driven models for reconstruction of root-bending moments
real measurements.

  * 2.

Design of a condition-monitoring framework utilising two versions of data-
driven models, namely on based on PCA and one on autoencoders.

  * 3.

Integration of statistical change detection into the learning-based framework
for robust diagnosis of anomalies.

  * 4.

Experimental validation of models and comparative assessment of the condition
monitoring methods using data from real-life wind turbines.

The rest of this paper is structured as follows: Section 2 describes the
characteristics of the wind turbine from which the data was collected and
presents the fault scenarios that will be studied. A brief introduction to the
methods is presented in Section 3. In Section 4 the details about the design
of the different models are presented, and the experimental results obtained
by applying the developed algorithms are summarised in Section 5. Finally,
conclusions are drawn in Section 6.

##  2 System description and problem statement

All the data that were used for the scope of this paper were provided by
Vestas Wind System A/S, and were collected from a 2.2 MW variable-speed,
variable-pitch horizontal axis wind turbine. The signals that are available
from the measurements, and that will be used to train the models are:

  1. 1.

Flap bending moments acting on each blade (3 measurements)

  2. 2.

Edge bending moments acting on each blade (3 measurements)

  3. 3.

The Rotor speed

  4. 4.

The Wind speed

  5. 5.

The Grid power

  6. 6.

Pitch angles of each blade (3 measurements)

According to the wind speed, the power output and the rotor speed, 5 different
operating regions are defined, where the turbine is operated with different
control strategies. Such signals, together with the consequent blade pitch
angle are normalized and presented in Figure 1, highlighting the different
operating regions at which the turbine is operating at.

Figure 1: Normalized data for determining the current operating region.

A long recording with the turbine not subject to fault is utilized for fitting
the learning-based models. Such data are normalized and split into training,
validation and testing for designing and verifying the design under nominal
conditions. Some faulty data are available for the same turbine subject to
different blade faults. These data were not accounted for in the design stage
and are used to test the models under real-life faulty scenarios.

The final goal is to create a condition monitoring framework for detecting
blade faults, which starting from the listed measurements is able to detect
anomalies by finding discrepancies in their reconstructions.

###  2.1 Failure Modes

The fault scenario that were considered in the investigation can be divided in
three main categories:

  1. 1.

Two common phenomenon that might affect the blade are icing, caused by the
formation of ice because of low temperatures, and the formation of cracks
along the surface of the blades. Such events could entail a rather sudden
change in the moment of inertia of the blade or its aerodynamic properties,
hence showing a sudden jump in the measurement of the root bending moments, as
stated in [21]. Hence, they were simulated on the available data from the
healthy turbine by adding an offset to the original bending moment
measurements. A bias in the measurement could also be caused by a fault
occurring in the sensor itself.

  2. 2.

Another common fault is represented by the case of a sensor measurements being
stuck because of its incorrect setup on the blade, or by defects in the
sensors itself. This kind of fault was simulated by changing the measured
bending moment signal to become constant after the fault occured.

  3. 3.

Finally, it was decided to use real-life data collected from a blade
undergoing a failure. For this reason, Vestas Wind Systems A/S provided some
data collected with the turbine experiencing a fault.

##  3 Methods and theoretical overview

Both the implemented Fault Detection methods rely on the measurements provided
by the different wind turbine sensors, to develop a data-driven model of the
healthy system by finding the main features between the collected data. This
first stage is the training stage and is performed offline.

Once the model is trained, the testing Stage is performed online: new data are
collected from the wind turbine, into an unknown state, and then reconstructed
by the trained model. The fault detection algorithm is obtained by computing a
suitable statistic to compare the data reconstructed by the trained model and
the real data measured by the sensors (usually by means of Square Prediction
Error or Mean Average Error), and by comparing the result with the fault-free
case obtained in the training stage by means of a GLR test. If the two
statistics are similar, then the GLRT assumes that the new data are from a
healthy plant, while if they are showing strong differences, it detects a
fault. Two approaches are proposed (PCA and Autoencoders), differing both in
the model structure and in the training method.

###  3.1 Introduction to PCA

Principal Component Analysis (PCA) is an unsupervised, linear method for
dimensional reduction in large multivariate datasets. It transforms a
normalized data matrix $\boldsymbol{X}\in\mathbb{R}^{n\times m}$ (with $n$
samples and $m$ features) into a lower-dimensional space by applying singular
value decomposition (SVD) to its covariance matrix:

$\boldsymbol{S}=\frac{1}{n-1}\boldsymbol{X}^{T}\boldsymbol{X}=\boldsymbol{P\Lambda
P}^{T}.$ (1)

Here, $\boldsymbol{P}$ contains the eigenvectors, and $\boldsymbol{\Lambda}$
is a diagonal matrix with eigenvalues in descending order. Traditional PCA,
however, is limited by its assumption of linear and time-invariant
relationships between features. To address nonlinearities and capture temporal
dependencies in datasets, Dynamic PCA (dPCA) was introduced. dPCA extends the
conventional PCA framework by incorporating a time window $\omega$, resulting
in an expanded data matrix
$\boldsymbol{X_{d}}\in\mathbb{R}^{n-(\omega-1)\times\omega m}$:

$\boldsymbol{X_{d}}(k)=\begin{bmatrix}\boldsymbol{X}(k-\omega+1)&\ldots&\boldsymbol{X}(k)\end{bmatrix},$
(2)

where $k$ ranges from $1$ to $n-(\omega-1)$.

Upon applying PCA to $\boldsymbol{X_{d}}$, the significant principal
components are retained based on their eigenvalues. Each row
$\boldsymbol{X_{d}}(k)$ can be decomposed into its principal subspace
projection $\boldsymbol{\hat{x}}(k)$ and the residual $\boldsymbol{e}(k)$:

$\boldsymbol{X_{d}}(k)=\boldsymbol{\hat{x}}(k)+\boldsymbol{e}(k).$ (3)

The model’s effectiveness is quantified using the Squared Prediction Error
(SPE) statistic for each data point, providing a measure of deviation from the
modelled behaviour. The SPE statistic measures, for every row of
$\boldsymbol{X_{d}}$,

$SPE(k)=\sum_{i=1}^{m\omega}\left(\hat{X}_{d}^{(i)}(k)-X_{d}^{(i)}(k)\right)^{2}=\boldsymbol{X_{d}}(k)(\boldsymbol{I}-\boldsymbol{\hat{P}\hat{P}}^{T})(\boldsymbol{I}-\boldsymbol{\hat{P}\hat{P}}^{T})^{T}\boldsymbol{X_{d}}(k)^{T}$
(4)

which is the sum of all the squared components of $\boldsymbol{e}(k)$. This
dynamic approach is particularly beneficial for systems like wind turbines,
where operational dynamics introduce significant nonlinearity and time-
dependent behavior in sensor data.

###  3.2 Introduction to Autoencoders

Autoencoders (AEs) are a type of neural network designed for unsupervised
learning, primarily aimed at data reconstruction. They consist of two main
components: an encoder and a decoder. The encoder compresses the input data
$\boldsymbol{X}(k)\in\mathbb{R}^{m}$ into a lower-dimensional latent space
$\boldsymbol{Z}(k)\in\mathbb{R}^{q}$, through a series of transformations:

$\boldsymbol{E}_{i}(k)=\begin{cases}\varphi_{1}^{e}(\boldsymbol{W}_{1}^{e}\boldsymbol{X}(k)+\boldsymbol{b}_{1}^{e}),&\text{if
}i=1\\\
\varphi_{i}^{e}(\boldsymbol{W}_{i}^{e}\boldsymbol{E}_{i-1}(k)+\boldsymbol{b}_{i}^{e}),&\text{otherwise}\end{cases}$
(5)

and

$\boldsymbol{Z}(k)=\varphi^{z}(\boldsymbol{W}^{z}\boldsymbol{E}_{\zeta}(k)+\boldsymbol{b}^{z}),$
(6)

where $\zeta$ denotes the number of encoder layers. The decoder, in a similar
fashion, is then leveraged to reconstruct the original input from these latent
variables, culminating in the output $\boldsymbol{\hat{X}}(k)$.

The performance of an AE is commonly evaluated by the mean squared error (MSE)
between the original and reconstructed data, minimized during training:

$MSE(\boldsymbol{X},\boldsymbol{\hat{X}})=\frac{1}{n}\sum_{k=1}^{n}\|\boldsymbol{X}(k)-\boldsymbol{\hat{X}}(k)\|^{2}.$
(7)

Given the non-linear and time-variant nature of wind turbine (WT) data,
standard AEs are often insufficient. Therefore, the model is extended with
recurrent structures such as Long Short-Term Memory (LSTM) layers to capture
temporal dependencies, and convolutional layers for spatial feature extraction
in data. Additionally, a dynamic expansion of the input data matrix
$\boldsymbol{X}$ to
$\boldsymbol{X_{d}}\in\mathbb{R}^{(n-(\omega-1))\times(m\omega)}$ allows the
AE to adapt to varying operational conditions by expanding the input space
temporally. These adaptations are essential for modeling the complex dynamics
of WT systems under different environmental conditions.

###  3.3 General Likelihood Ratio Test

The General Likelihood Ratio Test (GLRT) is employed to determine whether a
signal adheres to a specified probabilistic model under null hypothesis
$\mathcal{H}_{0}$, or if it instead follows a different distribution under
alternative hypothesis $\mathcal{H}_{1}$ characterized by a different mean
value. Specifically, for signals that normally distribute, the GLRT is
computed as:

$g(k)=\frac{1}{2\sigma^{2}M}\max_{k-M+1\leq j\leq
k}\left(\sum_{i=k-M+1}^{k}(z(i)-\mu_{0})^{2}\right).$ (8)

where $z(i)$ represents the realization of the signal, and $\mu_{0}$ and
$\sigma$ are the mean and standard deviation of the signal under
$\mathcal{H}_{0}$. The calculation within a running window $M$ is leveraged to
enhance computational efficiency. The statistic $g(k)$ is compared against a
threshold $h$ to decide between $\mathcal{H}_{0}$ (non-faulty behavior) and
$\mathcal{H}_{1}$ (fault indication). $M$ and $h$ are design parameters that
can be calculated starting from the statistical property of the signal under
$\mathcal{H}_{0}$ and the expected variation under $\mathcal{H}_{1}$ [22].

##  4 Design of condition monitoring framework

The condition monitoring framework consist of an offline training stage and an
online implementation one. The former involves a pre-processing of the data
presented in Section 2 for training the models, models that are then utilized
in the former for reconstructing new real-time data that are pre-processed in
the same fashion. The comparison between the original and reconstructed data
is processed for assessing the presence of faults. A summary of the entire
framework is presented in Figure 2.

Figure 2: Block diagram for the condition monitoring framework based on the
wind turbine measurements and the reconstructed data.

###  4.1 PCA

Because of the limitation of standard PCA introduced in Section 3.1, a dPCA
model is developed. All the signals defined in Section 2 were used for
training the model. Thus, every datapoint in the training dataset
$\boldsymbol{X}(k)$ measured at time instant $k$ had $12$ components. The
collected data matrix for training the model is
$\boldsymbol{X_{0}}\in\mathbb{R}^{n\times 12}$, where $n$ would change
according to the size of the training dataset chosen for each operating
region. To address PCA’s constraint of being able to only identify linear
correlations, the data are split for the different operating regions and equal
number of model is trained. The first operating region is not considered,
since it is not of interest developing a model for detecting faults in a non-
operational turbine. The process of identifying distinct training datasets
involved analyzing all recorded data and comparing the behaviour of various
variables (e.g. the rotor speed, grid power and pitch angles) to properly
separate all the operating regions.

As a final parameter needed to build the training datasets for dPCA, the time
window was set as $\omega=100$, which corresponds to a time of $10$ seconds.
The value has been chosen for being able to show enough variation in the
signals, for allowing the model to find time varying correlations between the
data and approximating nonlinearities in the relationships between the various
measured variables.

After performing SVD on the normalized data matrix, the value $l$ for
dimensionality reduction was chosen according to the Cumulative Variance (CV),
defined as

$CV=\frac{\sum_{i=1}^{l}\lambda_{i}}{\sum_{i=1}^{m\omega}\lambda_{i}},$ (9)

where $\lambda_{i},1\leq i\leq m\omega$ is the i-th orthonormal loading found
by PCA. The resulting numbers of principal components for a chosen CV of
$90\%$ applied to all four operating regions are summarised in Table 1.

Zone | II | III | IV | V
---|---|---|---|---
$l$ | 4 | 6 | 6 | 8
Table 1: Table showing the chosen amount of PCs for each of the operating
regions

The resulting Squared Prediction Errors (SPEs) for the reduced models are
presented in Figure 3.

Figure 3: SPE computed over the training datasets of the $4$ operating regions

The value is fairly constant except for the presence of spikes, which are
probably consequent to the presence of data outliers and the difficulty in
having a clear distinction among the several operating regions. To attenuate
the presence of the spikes, that could lead to the detection of false alarms,
a low-pass filter has been designed. The filtered signal can be used for
deriving a threshold with which it will be compared during the online
implementation for detecting the presence of a fault. The threshold was
computed by setting a probability of a false alarm, in this case $P_{F}=1\%$,
and then choosing the threshold as the value such that it is smaller than the
$P_{F}$ percentage of all the computed SPE points of the training dataset.

###  4.2 Autoencoders

Autoencoders were implemented as an improvement of the PCA approach, with
their ability to model the nonlinearities present in the wind turbine
dynamics. The AE approach allows, starting from just one training dataset, to
obtain a model of the healthy wind turbine that covers all five operating
regions, provided that the data chosen for the training stage covers all of
them.

In the data collection, for each data point, $9$ features were considered:

  1. 1.

Flap bending moments on all three blades (3 measurements)

  2. 2.

Edge bending moments on all three blades (3 measurements)

  3. 3.

Rotor speed

  4. 4.

Wind speed

  5. 5.

Grid power.

Figure 4: Outline of the autoencoder architecture with both convolutional and
LSTM layers. The input vector $\boldsymbol{x_{d}}(k)$ is collecting the
measured data over the sliding window $\omega$, and the AE reconstructs it as
$\boldsymbol{\hat{x}_{d}}(k).$

The number is reduced in comparison with the PCA model to decrease the
computational time, which in the case of 12 features would have been
considerably longer. The time window for including dynamical correlation was
set to $10$ seconds, to match the PCA model. In fact, a shorter time window
wouldn’t allow the coverage of enough variation in the data. Convolutional and
LSTM layers have been combined for improving the modeling capabilities, the
activation function was chosen to be the hyperbolic tangent. The architecture
of the chosen AE, together with the main parameters of interest is outlined in
Figure 4.

The model has been trained with the described dataset. 200 epochs were
selected to avoid overfitting, while still being able to model the non-linear
relations among the measurements. The results from the training process are
stated in Figure 5.

Figure 5: AE Model training and validation

The performance of the trained model was evaluated by calculating the MAE,
stated in Figure 6. Such metrics allowed to choose the current architecture
among others, as the best performing one in reconstructing the nominal data.
Similarly to the PCA case, the presence of spikes led to the design of a LPF
for reducing the probability of false detection, in case a static threshold is
applied directly to the current metrics. The threshold can similarly be
calculated as it was done previously for the PCA case.

Figure 6: $MAE_{train}$ computed over the training dataset using the
architecture with both convolutional and LSTM layers

###  4.3 GLR Test

A GLR test is designed for leveraging the statistical properties of the
reconstruction error of each model in order to better assess the presence of a
change. The natural filtering action consequent from evaluating the signal
over a time window and from knowing the standard deviation of the underlying
gaussian distribution, replaces the LPF previously designed and improves the
signal separation between the non-faulty and faulty case for better assessing
the presence of faults.

The test is implemented according to Equation 8, with the $h$ and $M$
parameters being calculated to guarantee a certain probability of detection
$P_{D}=0.99$ and false detection $P_{F}=0.01$ for $\mu_{1}=2\mu_{0}$.

##  5 Experimental results

Both the PCA and the AE models are tested for the faulty scenarios presented
in Section 2.1. While defining the experiment for the third category is
straightforward, thanks to the availability of faulty data, the first two
categories require the corruption of (unused) healthy data to mimic the
previously described faulty scenarios. An overview of the experiment setting
for each faulty scenario is presented in Table 2, where $M_{og}(t)$ represents
the blade root bending moment measured on the original healthy turbine at time
$t$, while $t_{f}$ is the time instant where the fault occurs.

Table 2: Overview of the faulty data for the different fault scenarios. Fault scenario |  Description
---|---
Flap bending moment offset $\Delta_{M}$ |  $M=M_{og}+\Delta_{M},$ $\Delta_{M}=1\times 10^{6}\,\text{Nm}$
Edge bending moment offset $\Delta_{M}$ |  $M=M_{og}+\Delta_{M},$ $\Delta_{M}=1\times 10^{6}\,\text{Nm}$
Flap bending moment stuck to $M_{st}$ |  $M=M_{st}=0.8M_{og}(t_{f})$
Edge bending moment stuck to $M_{st}$ |  $M=M_{st}=0.8M_{og}(t_{f})$
Real-life scenario 1: catastrophic fault |  Sudden jump in the flap bending moment signal
Real-life scenario 2: slower-forming |  Exponential increase in the flap bending moment

Starting from the real-life scenario, both the reconstruction error and the
result of applying the GLR are presented in Figure 7. The reconstruction error
changes in both cases and the filtered version allows to detect the fault by
the direct use of a threshold. The GLR proves its effectiveness in improving
signal separation so that the rate of false alarms is lowered.

Figure 7: Results from applying algorithms on real faulty data. Comparison
between reconstruction error and application of the GLR

The same models are applied to the (new) healthy data corrupted with sensor
faults. Because of the better detection performance obtained with the GLR, the
direct threshold methods results are omitted. The results are presented in
Figure 8. The capability of the AE of learning the nonlinear features of the
model is reflected in a better performance over all the possible faults. In
particular, the PCA is in general not able to detect the fault associated with
the Edge-bending-moment sensor, while it is still able to detect faults in the
flap-bending moment sensor by leveraging the correct model according to the
operating region the wind turbine is working at. Even under such conditions,
the AE is better performing by ensuring strong detectability and no false
alarms in the test that have been performed.

Figure 8: Results from applying algorithms on faulty data obtained by
injecting sensor faults. Comparison of change detection properties between PCA
and AE.

##  6 Conclusions

This paper presented a condition monitoring framework for detecting faults in
wind turbine blades based on learning based-models, trained to reconstruct a
set of measurements under healthy conditions and showing discrepancies
otherwise.

Starting with pre-processed real-life data from a healthy wind turbine, both a
linear model based on PCA and a nonlinear model based on autoencoders were
trained to reconstruct the input data. The trained models were then tested on
faulty data. Although both PCA and autoencoder-based models demonstrated their
capability in detecting faults in wind turbine blades, the autoencoder-based
approach proved to be the most effective thanks to its ability of dealing with
all the tested fault scenarios and by leveraging a single model for all the
operating regions.

Future work involves optimizing the algorithm for real-time implementation and
testing different model architectures to improve detection performance.

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