# The Cycle Class of the Supersingular Locus
of principally polarized abelian varieties

Gerard van der Geer  Korteweg-de Vries Instituut, Universiteit van Amsterdam,
Postbus 94248, 1090 GE Amsterdam, The Netherlands<EMAIL_ADDRESS>and  Shushi Harashita  Graduate School of Environment and Information
Sciences, Yokohama National University<EMAIL_ADDRESS>
###### Abstract.

We prove a formula for the cycle class of the supersingular locus in the Chow
ring with rational coefficients of the moduli space of principally polarized
abelian varieties in characteristic $p$. This formula determines this class as
a monomial in the Chern classes of the Hodge bundle up to a factor that is a
polynomial in $p$. This factor is known for $g\leq 3$. We determine the factor
for $g=4$.

###### 1991 Mathematics Subject Classification:

14G,14K,11G

##  1\. Introduction

An abelian variety over a field $k$ of characteristic $p>0$ is called
supersingular if it is isogenous to a product of supersingular elliptic curves
over the algebraic closure of $k$. Equivalently, by [21, Thm. 4.2], if its
formal isogeny type has a Newton polygon with all slopes equal to $1/2$.
Recall that the Newton polygon of an abelian variety starts at $(0,0)$, ends
at $(2g,g)$ and is lower convex and satisfies a symmetry condition. The two
extreme cases are the polygon with slopes $0$ and $1$ and break point $(g,0)$
(the ordinary case) and the case with slope $1/2$ (the supersingular case).

Let ${\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$ be the moduli space of
principally polarized abelian varieties of dimension $g$ in characteristic
$p>0$. The supersingular locus $S_{g}$ is defined as the closed subset of
principally polarized abelian varieties of
${\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$ that are supersingular. This locus
can be considered as the most degenerate stratum in the Newton polygon
stratification on ${\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$. Its dimension is
known by Li and Oort to be $[g^{2}/4]$ and also the number of irreducible
components is known, see below. Besides the Newton polygon stratification
there is another stratification on $\mathcal{A}_{g}\otimes{\mathbb{F}}_{p}$,
the Ekedahl-Oort stratification. While the cycle classes of the Ekedahl-Oort
stratification on ${\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$ are known, the
cycle classes of the Newton polgon strata in general are not. For $g=1$ and
$g=2$ the supersingular locus is a stratum of the Ekedahl-Oort stratification
and thus the class is known. For $g=3$ the supersingular locus is not a
stratum of the Ekedahl-Oort stratification, but its cycle class was determined
in joint work of the first author with Ekedahl, and the result was presented
in [9].

In this paper we will prove a formula for the cycle class of the supersingular
locus in the Chow ring with rational coefficients of a Faltings-Chai
compactification $\tilde{\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$. This
formula determines this class as a monomial in the Chern classes of the Hodge
bundle up to a factor that is a polynomial in $p$. Furthermore, we will
determine the factor for the cycle class of the supersingular locus for $g=4$.
This latter determination builds upon the method used for the case of $g=3$
and calculates the degrees of the top Chern classes of the Hodge bundle on a
component of the supersingular locus. For this we construct an explicit smooth
model of each irreducible component of $S_{4}$. We also give the proof for the
class for $g=3$ that was not published in [9]. Including the well-known
results for $g=1$ and $g=2$ we arrive at the following theorem.

######  Theorem 1.1.

The cycle class of the supersingular locus $S_{g}$ in the Chow ring with
rational coefficients of a Faltings-Chai compactification
$\tilde{\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$ of the moduli space
$\mathcal{A}_{g}\otimes{\mathbb{F}}_{p}$ lies in the tautological ring. More
precisely, its class is of the form

$[S_{g}]=f_{g}(p)\begin{cases}\lambda_{g}\lambda_{g-2}\cdots\lambda_{2}&\text{$g$
even},\\\ \lambda_{g}\lambda_{g-2}\cdots\lambda_{1}&\text{$g$ odd}\,,\\\
\end{cases}$

where $f_{g}(p)$ is a polynomial in $p$ with rational coefficients and
$\lambda_{i}$ is the $i$th Chern class of the Hodge bundle on
$\tilde{\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$. For $g\leq 4$ the cycle
class is given by

$[S_{g}]=\begin{cases}(p-1)\,\lambda_{1}&$g=1$\\\
(p-1)(p^{2}-1)\,\lambda_{2}&$g=2$\\\
(p-1)^{2}(p^{3}-1)(p^{4}-1)\,\lambda_{3}\lambda_{1}&$g=3$\\\
(p-1)^{3}(p^{3}-1)(p^{4}-1)(p^{6}-1)\,\lambda_{4}\lambda_{2}&$g=4$.\\\
\end{cases}$

We also discuss for $g=3$ and $g=4$ the loci in the supersingular locus where
the $a$-number is at least $2$.

###### Contents

  1. 1 Introduction
  2. 2 The moduli space ${\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$
  3. 3 Irreducible components of the supersingular locus
  4. 4 Flag type quotients
  5. 5 Dieudonné modules and displays
  6. 6 The cycle class of the supersingular locus
  7. 7 Moduli of flag type quotients for $g=3$
  8. 8 The cycle class of $S_{3}$
  9. 9 Loci for $g=3$ defined by conditions on the $a$-number
  10. 10 Moduli of flag type quotients for $g=4$
  11. 11 Interpretation of the morphism $\mathcal{F}_{1}\to\mathcal{F}_{2}$
  12. 12 The Hodge bundle on the supersingular locus
  13. 13 Loci with $a$-number $\geq 2$ for $g=4$
    1. 13.1 Loci of the first type.
    2. 13.2 Loci of the second type
  14. 14 The fibres over $\mathcal{F}_{2}({\mathbb{F}}_{p^{2}})$
  15. 15 Superspecial points of $S_{4}$
  16. 16 The cycle class of $S_{4}$ and intersection numbers
  17. 17 Determination of intersection numbers

##  2\. The moduli space ${\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$

By ${\mathcal{A}}_{g}$ we denote the moduli stack of principally polarized
abelian varieties of dimension $g$ and by
$\pi:\mathcal{X}_{g}\to{\mathcal{A}}_{g}$ the universal abelian variety over
${\mathcal{A}}_{g}$. It is a Deligne-Mumford stack defined over
${\mathbb{Z}}$. The moduli space ${\mathcal{A}}_{g}$ carries a natural vector
bundle ${\mathbb{E}}$ of rank $g$, the Hodge bundle, defined as
$\pi_{*}\Omega^{1}_{\mathcal{X}_{g}/{\mathcal{A}}_{g}}$. We denote by
$\tilde{\mathcal{A}}_{g}$ a Faltings-Chai compactification of
${\mathcal{A}}_{g}$ as defined and treated in [6]. The Hodge bundle extends to
$\tilde{A}_{g}$ and will again be denoted by ${\mathbb{E}}$.

In the rest of this paper we consider the moduli stack
$\mathcal{A}_{g}\otimes{\mathbb{F}}_{p}$ in characteristic $p>0$. We set
$\lambda_{i}=c_{i}({\mathbb{E}})$ for the $i$th Chern class of ${\mathbb{E}}$
for $i=1,\ldots,g$. These classes satisfy the relation

$(1+\lambda_{1}+\cdots+\lambda_{g})(1-\lambda_{1}+\cdots+(-1)^{g}\lambda_{g})=1$

and these classes generate a subring $R_{g}$ of the Chow ring ${\rm
CH}_{\mathbb{Q}}^{*}(\tilde{\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p})$ called
the tautological ring, see [9, 5]. For $0\leq n\leq g(g+1)/2$ the graded part
of $R_{g}$ of degree $n$ has a basis
$\lambda_{1}^{e_{1}}\cdots\lambda_{g}^{e_{g}}$ with $0\leq e_{i}\leq 1$ and
$\sum_{i}e_{i}i=n$. The ring $R_{g}$ is a Gorenstein ring with socle generated
by $\lambda_{1}\lambda_{2}\cdots\lambda_{g}$. We will denote the degree of
this $0$-cycle by

$v(g)=\deg\lambda_{1}\lambda_{2}\cdots\lambda_{g}\,,$

the Hirzebruch proportionality constant, and we have

$v(g)=(-1)^{g(g+1)/2}2^{-g}\zeta(-1)\zeta(-3)\cdots\zeta(1-2g),$

where $\zeta(s)$ is the Riemann zeta function. We give a little table with
relevant values:

$g$ $0$ $1$ $2$ $3$ $4$ $v(g)$ $1$ $1/24$ $1/5760$ $1/2903040$ $1/1393459200$

The tautological ring of $\mathcal{A}_{g}\otimes{\mathbb{F}}_{p}$ is the
quotient $R_{g}/(\lambda_{g})\cong R_{g-1}$.

The moduli space ${\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$ carries two
important stratifications, the Ekedahl-Oort stratification and the Newton
polygon stratification, see [22] and [23]. The strata of the Ekedahl-Oort
stratification $\mathcal{V}_{\mu}$ are indexed by Young diagrams or tuples
$\mu=[\mu_{1},\ldots,\mu_{r}]$ of integers with $0\leq r\leq g$ and
$\mu_{i}>\mu_{i+1}$, according to the usage of [9, 4]. The largest open
stratum $\mathcal{V}_{[\emptyset]}$ is the locus of ordinary abelian
varieties. The codimension of $\mathcal{V}_{\mu}$ is $\sum_{i}\mu_{i}$. The
stratification can be extended to $\tilde{\mathcal{A}}_{g}$.

By [9, 4] we can calculate the cycle classes of the closed Ekedahl-Oort strata
in $\mathcal{A}_{g}\otimes{\mathbb{F}}_{p}$ and
$\tilde{\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$. For example the cycle class
of the locus of abelian varieties with $p$-rank $\leq f$ (corresponding to
$\mu=[g-f]$) is

$None$
$[\overline{\mathcal{V}}_{[g-f]}]=(p-1)(p^{2}-1)\cdots(p^{g-f}-1)\lambda_{g-f}$

and the cycle class of the smallest stratum, the locus of superspecial abelian
varieties (corresponding to $\mu=[g,g-1,\ldots,1]$) is

$[\mathcal{V}_{g,g-1,\ldots,1]}]=(p-1)(p^{2}+1)\cdots(p^{g}+(-1)^{g})\lambda_{1}\lambda_{2}\cdots\lambda_{g}\,.$

This formula implies as a special case a result of Ekedahl [3], namely that

$None$ $\sum\frac{1}{\\#{\rm
Aut}(X)}=(p-1)(p^{2}+1)\cdots(p^{g}+(-1)^{g})\,v(g)\,,$

where the sum is over the isomorphism classes of principally polarized
superspecial abelian varieties over $k$ and $v(g)$ the proportionality
constant defined above. A formula for the actual number of isomorphism classes
of superspecial abelian varieties with a level $n\geq 3$ structure is obtained
by multiplying the formula for the degree of $\mathcal{V}_{[g,g-1,\ldots,1]}$
by the degree of the natural map ${\mathcal{A}}_{g}[n]\to{\mathcal{A}}_{g}$
(as stacks) with ${\mathcal{A}}_{g}[n]$ the moduli space of principally
polarized abelian varieties with a level $n$ structure.

##  3\. Irreducible components of the supersingular locus

The number of irreducible components of the supersingular locus $S_{g}$ in
$\mathcal{A}_{g}\otimes{\mathbb{F}}_{p}$ was determined by Deuring for $g=1$,
by Katsura and Oort for $g=2$ ([15, 16]) and in general by Li and Oort for
$g\geq 3$, [18, 4.9]. The actual number of irreducible components in
${\mathcal{A}}_{g}\otimes\overline{\mathbb{F}}_{p}$ is given by a class number
$h_{p}(g)$ for $g$ odd and a similar class number $h_{p}^{\prime}(g)$ for $g$
even. Here $h_{p}(g)$ (resp. $h_{p}^{\prime}(g)$) is the class number of the
principal (resp. non-principal) genus in the hermitian space $B^{g}$, with $B$
the definite quaternion algebra ramified at $p$ and $\infty$. These class
numbers are difficult to deal with, see for example [13, p. 147], and one gets
better and more useful formulas by counting in a stacky way, that is, taking
into account weights equal to the inverse of the order of the automorphism
groups of the objects that one counts. For example, for $g=1$ the class number
of the quaternion algebra $B={\mathbb{Q}}_{p,\infty}$ over ${\mathbb{Q}}$
split outside $p$ and $\infty$, is given by

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<EMAIL_ADDRESS>to8.33334pt{\kern-1.0pt\cleaders\hbox{\kern 0.5pt\vrule
height=0.2pt,depth=0.2pt,width=1.0pt\kern
<EMAIL_ADDRESS>to9.33333pt{\kern-0.5pt\cleaders\hbox{\kern 0.5pt\vrule
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<EMAIL_ADDRESS>to6.66666pt{\kern 0.0pt\cleaders\hbox{\kern 0.5pt\vrule
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0.0pt}}\kern-1.0pt\right)\mathclose{}}\right)\frac{1}{4}\,.$

But a stacky interpretation of this number leads to the much more elegant
formula

$\sum\frac{1}{\\#{\rm Aut}(E)}=\frac{p-1}{24}$

with the summation over all isomorphism classes of supersingular elliptic
curves defined over $\overline{\mathbb{F}}_{p}$.

We will denote by $N_{g}$ the number of irreducible components of the
supersingular locus, where each irreducible components is counted with a
weight $w$ with $1/w=\\#{\rm Aut}(X_{\eta})$, where $X_{\eta}$ denotes the
principally polarized abelian variety corresponding to the generic point of
the irreducible component.

This number $N_{g}$ has the property that the number $N_{g}[n]$ of irreducible
components of the supersingular locus on the moduli space
${\mathcal{A}}_{g}[n]$ with a level $n\geq 3$ structure equals

$N_{g}[n]=N_{g}\cdot\deg({\mathcal{A}}_{g}[n]\to{\mathcal{A}}_{g})\,.$

######  Proposition 3.1.

The number $N_{g}$ of irreducible components of the supersingular locus in
$\mathcal{A}_{g}\otimes\overline{\mathbb{F}}_{p}$ is

$\begin{cases}(p-1)(p^{2}+1)(p^{3}-1)\cdots(p^{g}-1)\,v(g)&\text{for $g$
odd},\\\ (p^{2}-1)(p^{6}-1)\cdots(p^{2g-2}-1)\,v(g)&\text{for $g$ even}.\\\
\end{cases}$

The stacky interpretation that we use reduces to the mass of the principal
(resp. non-principal genus) and can be deduced from [3] or [12]. One finds
this mass formula also in [8, p. 123].

For odd $g$ the irreducible components of $S_{g}$ are in bijective
correspondence with the isomorphism classes of superspecial principally
polarized abelian varieties of dimension $g$, hence the formula for $N_{g}$
follows immediately from Ekedahl’s result (2). For even $g$ one has a
correction factor

$\frac{(p+1)(p^{3}+1)\cdots(p^{g-1}+1)}{(p^{2}+1)(p^{4}+1)\cdots(p^{g}+1)}\,.$

Here for $g$ even the numerator can be interpreted as the number of totally
isotropic subspaces of dimension $g/2$ in a $g$-dimensional hermitian space
over ${\mathbb{F}}_{p^{2}}$ with conjugation given by Frobenius, while the
denominator equals the number of totally isotropic subspaces of dimension
$g/2$ in a symplectic space of dimension $g$ over ${\mathbb{F}}_{p^{2}}$.

##  4\. Flag type quotients

Work of Oda and Oort ([20]) makes it possible to parametrize the irreducible
components of the supersingular locus $S_{g}$ by so-called flag type
quotients. For an abelian variety $X$ over a field of characteristic $p$ we
will denote the subgroup scheme $\ker(F)\cap\ker(V)$ by $A(X)$ with $F$ and
$V$ Frobenius and Verschiebung on $X$. It is a subgroup scheme of order
$p^{a(X)}$ with $a(X)$ the $a$-number of $X$. A supersingular abelian has
$1\leq a(X)\leq g$ and if $a(X)=g$ and $g\geq 2$ then $X$ is isomorphic to the
base change to $k$ of a product $E^{g}$ with $E$ a supersingular elliptic
curve defined over ${\mathbb{F}}_{p}$.

For a supersingular abelian variety $X$ of dimension $g$ one has
$a(X/A(X))\geq\min(g,a(X)+1)$, see [17]. By starting with $X=X_{0}$ and
putting $X_{i+1}=X_{i}/A(X_{i})$ one arrives after $g-1$ steps at a
superspecial abelian variety $X_{g-1}$, that is, an abelian variety with with
$a(X_{g-1})=g$. Then the kernel of the dual map is contained in
$\ker(F^{g-1})$, hence one finds a homomorphism $Y\to X$ with
$Y=X_{g-1}^{(p^{g-1})}$. This implies the fact that for a supersingular
abelian variety $X$ there exists a minimal isogeny $\rho:E^{g}\to X$ with $E$
a supersingular elliptic curve with the property that any other homomorphism
$h:Z\to X$ of a superspecial abelian variety $Z$ factors uniquely through
$\rho$. If $a(X)=1$ this minimal isogeny is obtained in $g-1$ steps

$Y_{g-1}\to Y_{g-2}\to\cdots\to Y_{0}=X$

where $Y_{g-1}=E^{g}\otimes{\rm Spec}(k)$ and $Y_{i}=Y_{g-1}/G_{i}$ for
$i=1,\ldots,g-1$ with $G_{i}=\ker(\rho)\cap Y_{g-1}[F^{g-1-i}]$. If $a(X)>1$
this sequence need not be unique. Taking into account also the polarizations
leads to the definition of a (polarized) flag type quotient.

######  Definition 4.1.

A polarized flag type quotient of dimension $g$ is a diagram of abelian
varieties and homomorphisms

$\textstyle{Y_{g-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{g-1}}$$\scriptstyle{\eta_{g-1}}$$\textstyle{Y_{g-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{g-2}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{1}}$$\textstyle{Y_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{0}}$$\textstyle{Y^{t}_{g-1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Y^{t}_{g-2}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
Y_{0}^{t}}$

with $Y_{g-1}$ superspecial and $\eta_{g-1}$ a polarization with kernel
$Y_{g-1}[F^{g-1}]$ satisfying

  1. (1)

$\ker(\rho_{i})\subset A(Y_{i})$ is of order $p^{i}$;

  2. (2)

$\ker(\eta_{i})\subseteq\ker(F^{i-j}\circ V^{j})$ for $0\leq j\leq i/2$ .

This flag type quotient is called rigid if $G_{i}=G_{0}\cap G[F^{g-1-i}]$ with
$G_{0}=\ker(Y_{g-1}\to Y_{0})\cap Y_{g-1}[F^{g-1}]$. The term ‘rigid’ refers
to the fact that in this case the corresponding flag type is unique.

The main references for flag type quotients are [20] and [18, 9.6–9.7].

##  5\. Dieudonné modules and displays

The theory of Dieudonné modules makes it possible to describe flag type
quotients in terms of Dieudonné modules.

Here $k$ will denote an algebraically closed field of characteristic $p$ and
$W=W(k)$ the ring of Witt vectors of $k$. We define a ring

$A=W[F,V]/(FV-p,VF-p,Fa-a^{\sigma}F,aV-Va^{\sigma},\,\forall a\in W)$

and set $A_{1,1}:=A/(F-V)$.

A polarized flag type quotient as described in Definition 4.1 corresponds to a
flag of contravariant Dieudonné modules

$M_{0}\subset M_{1}\subset M_{2}\subset M_{3}$

satisfying

  1. (1)

$M_{g-1}=A_{1,1}^{g}$ provided with a quasi-polarization

$\langle\,,\,\rangle:M_{g-1}\otimes_{W}M_{g-1}^{t}\to Q(W)$

that induces an identification $M_{g-1}^{t}=F^{g-1}M_{g-1}$;

  2. (2)

$(F,V)M_{i}\subset M_{i-1}$ and $\dim(M_{i}/M_{i-1})=i$ for $i=0,\ldots,g-1$;

  3. (3)

$(F,V)^{i}M_{i}\subset M_{i}^{t}$ for $i=0,\ldots,g-1$.

We call such a flag a polarized Dieudonné flag of length $g$. It is called
rigid if $M_{i}=M_{0}+F^{g-1-i}M_{g-1}$ for $i=0,\ldots,g-1$. We observe that
rigidity implies

$M_{i}=M_{m}+F^{g-1-i}M_{g-1}\quad\text{\rm for $m<i\leq(g-1)$}\,.$

We can translate rigid polarized flag type quotients in terms of displays,
replacing Dieudonné modules by displays. We recall the definition of displays
(cf. [24, Section 1]). Let $R$ be a commutative unitary ring. Let $W(R)$ be
the ring of Witt vectors and $Q(W)$ its field of fractions. Let
$\mathfrak{f}:W(R)\to W(R)$ be Frobenius and $\mathfrak{v}:W(R)\to W(R)$
Verschiebung. Set $I_{R}=\mathfrak{v}(W(R))$. A display over $R$ is a
quadruple $(P,Q,F,V^{-1})$ consisting of a finitely generated projective
$W(R)$-module $P$, a $W(R)$-submodule $Q$ of $P$ and homomorphisms
$F:P^{(p)}\to P$ and $V^{-1}:Q^{(p)}\to P$, where
$M^{(p)}:=W(R)\otimes_{\mathfrak{f},W(R)}M$, with the properties:

  1. (i)

$I_{R}P\subset Q\subset P$ and there exists a decomposition of $P$ into a
direct sum of $W(R)$-modules $P=L\oplus T$, such that $Q=L\oplus I_{R}T$;

  2. (ii)

$V^{-1}$ is epimorphism;

  3. (iii)

For $x\in P$ and $w\in W(R)$ we have $V^{-1}(1\otimes\mathfrak{v}(w)x)=wFx$.

By [24, Lemma 9], we have an isomorphism

$None$ $V^{-1}\oplus F:(L\oplus T)^{(p)}\to P.$

The matrix (with respect to a basis of $P$) associated to this isomorphism is
a generalization of the classical display ([19]).

######  Remark 5.1.

If $R$ is a perfect field, then $P$ is the usual Dieudonné module,
$I_{R}=p\,W(R)$ and $Q$ is the $V$-image $VP$, so $Q$ is determined by the
Dieudonné module $P$. But if $R$ is not a perfect field, then $Q$ is not
determined by $P$ with $F,V$; conversely $P$ is determined by the
$V^{-1}$-image of $Q^{(p)}$.

By a result of Li-Oort [18, 3.7] the moduli space of polarized Dieudonné flags
of length $g$ exists and is projective. Moreover, by [18, 3.7] the moduli of
rigid polarized Dieudonné flags of length $g$ exists and is quasi-projective,
and by [18, 7.6] it is non-singular.

##  6\. The cycle class of the supersingular locus

In this section we will show that the cycle class of the supersingular locus
$S_{g}$ in $\mathcal{A}_{g}\otimes{\mathbb{F}}_{p}$ lies in the tautological
ring $R_{g}$ generated by the Chern classes $\lambda_{i}$ ($i=1,\ldots,g$) of
the Hodge bundle ${\mathbb{E}}$ on a Faltings-Chai compactification of
$\mathcal{A}_{g}\otimes{\mathbb{F}}_{p}$ and give a formula for it that fixes
the class up to a multiplicative constant.

Here the cycle class is taken in the Chow ring with rational coefficients of a
Faltings-Chai compactification
$\tilde{\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$ of
$\mathcal{A}_{g}\otimes{\mathbb{F}}_{p}$.

######  Theorem 6.1.

The cycle class of the supersingular locus on
$\mathcal{A}_{g}\otimes{\mathbb{F}}_{p}$ in ${\rm
CH}_{\mathbb{Q}}(\tilde{\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p})$ is a non-
zero multiple of $\lambda_{g}\lambda_{g-2}\cdots\lambda_{1}$ if $g$ is odd and
of $\lambda_{g}\lambda_{g-2}\cdots\lambda_{2}$ if $g$ is even. The multiple is
a polynomial in $p$.

###### Proof.

For the proof we will use the presentation of Frobenius on the covariant
Dieudonné module $M$ with $p$-rank $0$ and $a$-number $1$ as given by Oort in
[23]. His description of the display of such a module $M$ is as follows. With
$W$ the Witt ring of $k$, an algebraically closed field of characteristic
$p>0$, there exists a $W$-basis $e_{1},\ldots,e_{g},e_{g+1},\ldots,e_{2g}$
such that Frobenius is given by the formulas

$\displaystyle Fe_{j}=\sum_{i=1}^{2g}\gamma_{ij}e_{i},\quad(1\leq j\leq g)\,,$
$\displaystyle e_{j}=V(\sum_{i=1}^{2g}\gamma_{ij}e_{i})\quad(g+1\leq j\leq
2g)\,,$

where $\gamma=(\gamma_{ij})$ is a $W$-valued $2g\times 2g$ matrix which is
symplectic in the sense that

$\gamma\left(\begin{matrix}0&1_{g}\\\ -1_{g}&0\\\
\end{matrix}\right)\gamma^{t}=\left(\begin{matrix}0&1_{g}\\\ -1_{g}&0\\\
\end{matrix}\right)\,.$

We write $\gamma$ as a matrix of $g\times g$ blocks

$\gamma=(\gamma_{ij})=\left(\begin{matrix}a&b\\\ c&d\\\
\end{matrix}\right)\,.$

We denote the Frobenius endomorphism of the Witt ring $W$ by $\sigma$. Note
that the $\sigma$-linear map $F$ is given by the matrix

$\left(\begin{matrix}a&pb\\\ c&pd\\\ \end{matrix}\right)\,.$

Oort shows ([23, p. 191] that if $M$ has $p$-rank $0$ and $a$-number $1$ we
may choose the basis such that the matrix $\gamma$ is of the form

$a_{ij}=d_{ij}=\begin{cases}1&i=j+1\\\ 0&i\neq j+1\\\ \end{cases},\quad
c_{ij}=\begin{cases}1&(i,j)=(1,g)\\\ 0&else\\\ \end{cases},\\\ $

and $b_{ig}=0$ for $i\neq 1$. Given such a basis in normal form, we have
according to [23, Lemma 2.6] that there exists a $P\in A$ such that

$None$ $F^{2g}e_{1}=Pe_{1}\quad\text{\rm with}\quad
P=\sum_{i=1}^{g}\sum_{j=g+1}^{2g}p^{j-g}\gamma_{ij}^{{\sigma}^{2g-j}}F^{2g+i-j-1}\,,$

with $Fx=x^{\sigma}F$ for $x\in W[F,V]$ and repeated application of $F$ is in
the $\sigma$-linear sense (cf. [23, p. 195]).

######  Lemma 6.2.

We have $\gamma_{1,j}=0$ for $j=g+1,\ldots,2g-1$. Moreover, the square matrix

$\tilde{\gamma}=\left(\begin{matrix}\gamma_{2,g+1}&\ldots&\gamma_{2,2g-1}\\\
\vdots&&\vdots\\\ \gamma_{g,g+1}&\cdots&\gamma_{g,2g-1}\\\
\end{matrix}\right)$

is symmetric.

###### Proof.

We have $ab^{t}=ba^{t}$ and $b^{t}d=d^{t}b$. In view of the shape of the
matrices $a$ and $d$ the result follows as $\gamma$ is symplectic. ∎

######  Remark 6.3.

We know that the Ekedahl-Oort stratum $\mathcal{V}_{\mu}$ with $\mu=[g,1]$
corresponding to $p$-rank $0$ and $a$-number $2$ has codimension $1$ in the
$p$-rank $0$ locus $V_{0}$, hence the generic point of every irreducible
component of $V_{0}$ has $a=1$. Moreover, by the results of Li-Oort [18] we
know that each irreducible component of the supersingular locus $S_{g}$ has an
open dense subset where the $a$-number equals $1$.

One can read off supersingularity from the matrix $\tilde{\gamma}$ using
Oort’s result on the action of $F$ on $e_{1}$ given in (4), see [23, Prop.
2.7].

######  Corollary 6.4.

The module $M$ is supersingular if $\gamma_{ij}\equiv 0\,(\bmod\,p)$ for
$2\leq i\leq g-1$, $g+1\leq j\leq 2g-2$ with $i+j\leq 2g$.

Note that because of the symmetry this gives a priori

$\sum_{j=1}^{\lfloor g/2\rfloor}(g-2j)=\frac{g(g-1)}{2}-[\frac{g^{2}}{4}]=\dim
V_{0}-\dim S_{g}$

conditions for supersingularity, where $V_{0}$ is the $p$-rank zero locus.

The strategy is now to impose consecutively conditions that together imply
supersingularity by Corollary 6.4; we begin by requiring the vanishing modulo
$p$ of the column of entries that is the transpose of

$(\gamma_{2,g+1},\ldots,\gamma_{g-1,g+1})\,,$

and continue by requiring the vanishing modulo $p$ of the column of entries
whose transpose is

$(\gamma_{3,g+2},\ldots,\gamma_{g-2,g+2})\,,$

and so on, till finally the column with transpose
$(\gamma_{g/2,3g/2-1},\gamma_{g/2+1,3g/2-1})$ of length $2$ for $g$ even or
the vanishing of the single entry $\gamma_{(g+1)/2,(3g-1)/2}$ for $g$ odd.

For example, for $g=5$ we require the vanishing modulo $p$ of the red entries
in the symmetric matrix

$\tilde{\gamma}=\left(\begin{matrix}\color[rgb]{1,0,0}\gamma_{26}&\color[rgb]{1,0,0}\gamma_{27}&\color[rgb]{1,0,0}\gamma_{28}&\gamma_{29}\\\
\color[rgb]{1,0,0}\gamma_{36}&\color[rgb]{1,0,0}\gamma_{37}&\gamma_{38}&\gamma_{39}\\\
\color[rgb]{1,0,0}\gamma_{46}&\gamma_{47}&\gamma_{48}&\gamma_{49}\\\
\gamma_{56}&\gamma_{57}&\gamma_{58}&\gamma_{59}\\\ \end{matrix}\right)$

giving $4$ conditions.

In terms of displays, we have an $\mathfrak{f}$-linear map $V^{-1}\oplus
F:M=L\oplus T\overset{}{\to}M$, see (3). We write $F/p$ for the composition

$VM/pM\to M/pM\xrightarrow{V^{-1}\oplus F}M/pM\to M/VM\,.$

This map is given by the square matrix $(\gamma_{ij})_{1\leq i\leq g,g+1\leq
j\leq 2g}$. Then by the vanishing indicated in Lemma 6.2 we may restrict to
submodules of rank $g-1$ generated by $g-1$ consecutive generators in $VM/pM$
and $M/VM$:

$G=\langle e_{g+1},e_{g+2},\ldots,e_{2g-1}\rangle\longrightarrow H=\langle
e_{2},e_{3},\ldots,e_{g}\rangle\,.$

We have increasing filtrations for $i=1,\ldots,g$ given by

$G_{i}=\langle e_{g+1},e_{g+2},\ldots,e_{g+i}\rangle\quad\text{\rm and}\quad
H_{i}=\langle e_{2},e_{g-1},\ldots,e_{g+1-i}\rangle\,.$

That the $p$-rank is zero means that the image of $G_{g-1}$ is in $H_{g-1}$.
If we identify ${\rm Lie}(X)$ with $VM/pM$ for the abelian variety $X$
corresponding to the dual of $M$ (cf. [1, 4.3.12] and [18, 5.4, 7.4]), we can
view the induced map $F/p:G_{g-1}\to H_{g-1}$ as a symmetric morphism between
vector bundles of rank $g-1$ made from the Hodge bundle and its dual by
Frobenius twists. Since we wish to have the filtrations we will have to work
on a cover of the $p$-rank zero locus $V_{0}$.

We now consider $G\langle 1\rangle$, the module generated by $e_{g+1}$. We
require that it maps to zero modulo $p$ under $F/p:G\langle 1\rangle\to
H\langle 1\rangle$ with the module $H\langle i\rangle$ generated by
$e_{i+1},\ldots,e_{g-i}$. We can view the semi-linear map $G\langle
1\rangle\to H\langle 1\rangle$ defined by $F/p$ modulo $p$ as a morphism of a
line bundle to a vector bundle of rank $g-2$, where these bundles are made
from the Hodge bundle by truncations and Frobenius twists. We consider the
locus where this morphism vanishes. The vanishing of this morphism corresponds
to the vanishing modulo $p$ of the vector
$(\gamma_{2,g+1},\ldots,\gamma_{g-1,g+1})$.

If this morphism vanishes then by the symmetry $\gamma_{2,g+2}$ vanishes
modulo $p$ and we can consider a morphism $G\langle 2\rangle\to H\langle
2\rangle$ induced by $F/p$ with $G\langle j\rangle=G_{j}/G_{j-1}$ generated by
$e_{g+j}$ and require its vanishing modulo $p$. By induction, assuming the
vanishing modulo $p$ of the semi-linear morphism

$None$ $G\langle j\rangle\longrightarrow H\langle j\rangle$

for $j=1,\ldots,s$, we get a next morphism $G\langle s+1\rangle\to H\langle
s+1\rangle$. We require inductively that these morphisms vanish for
$j=1,\ldots[(g-1)/2]$ on an appropriate covering space of $V_{0}$ where we
have the filtrations. Supersingularity follows if the conditions that the
induced map $G\langle j\rangle\to H\langle j\rangle$ is zero are satisfied
successively for $j=1,\ldots[(g-1)/2]$.

The locus where the morphism (5) for $j=1$ is zero has cycle class the
$(g-2)$th Chern class of the dual of $G\langle 1\rangle\otimes(H\langle
1\rangle)^{\vee}$.

We now work on the space of flags $\mathfrak{F}$ on the cohomology $H^{1}_{\rm
dR}$ of the universal principally polarized abelian variety as introduced in
[4, Section 3]; more precisely, we work on the closure of the final stratum
$\mathfrak{F}_{w}$ corresponding to $p$-rank zero. Here the Hodge bundle
${\mathbb{E}}$ has a filtration ${\mathbb{E}}(i)$ for $i=1,\ldots,g$ with
${\rm rank}({\mathbb{E}}(i))=i$.

We can view the induced map $F/p:G_{g-1}\to H_{g-1}$ as a symmetric morphism
between modules of rank $g-1$ that induces a morphism of vector bundles
$G\langle 1\rangle\to H\langle 1\rangle$ on $\mathfrak{F}_{w}$. The vector
bundles induced by $G$ and $H$ have filtrations and the Chern classes of their
graded quotients are of the form $\pm p^{r_{i}}\ell_{i}$ where
$\ell_{i}=c_{1}({\mathbb{E}}(i)/{\mathbb{E}}(i-1))$ ($i=1,\ldots,g$) are the
Chern classes of the graded quotients of the Hodge bundle on the final stratum
and $r_{i}\in{\mathbb{Z}}$.

The conditions on the vanishing modulo $p$ of rows of entries can now be
viewed as a degeneracy condition for a symmetric morphism between vector
bundles on $\mathfrak{F}_{w}$.

We shall calculate the cycle class of the Zariski closure of the degeneracy
locus of this map over the open part of $V_{0}$ where $a=1$. This Zariski
closure is contained in the supersingular locus as the Newton polygon can only
go up under specialization. Moreover, for $g\geq 2$ each irreducible component
of $S_{g}$ has an open dense set with $a=1$, hence intersects the degeneracy
locus over $V_{0}$.

We know that the codimension of the degeneracy locus equals the number of
conditions imposed by Corollary 6.4 in the supersingular case, hence also for
the intermediate cases defined by the vanishing of $G\langle j\rangle\to
H\langle j\rangle$. The theory of degeneracy loci [7] tells that the cycle
classes of these degeneracy loci on $\mathfrak{F}_{w}$ are polynomials in the
classes $\ell_{i}$.

To calculate these, we begin by remarking that the cycle class of the $p$-rank
zero locus $V_{0}$ in $\tilde{\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$ is a
multiple of $\lambda_{g}$ by [4]. We carry out induction and assume that the
image under the Gysin map from $\mathfrak{F}_{w}$ to
$\mathcal{A}_{g}\otimes{\mathbb{F}}_{p}$ of the class of the locus over
$V_{0}$ where $F/p$ maps $G\langle s\rangle$ to zero in $H\langle s\rangle$
for $s=1,\ldots,j-1$ is a multiple of
$\lambda_{g}\lambda_{g-2}\cdots\lambda_{g+2-2j}$.

The locus where the morphism $G\langle j\rangle\to H\langle j\rangle$ is zero
has as cycle class the $(g-2j)$th Chern class of the dual of $G\langle
j\rangle\otimes(H\langle j\rangle)^{\vee}$. With $r=g-2j={\rm rank}(G\langle
j\rangle)$ this Chern class is

$(-1)^{r}(c_{r}(H\langle j\rangle)-c_{r-1}(H\langle j\rangle)c_{1}(G\langle
j\rangle))\,.$

In order to calculate the class of the corresponding locus on
$\tilde{\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$ we have to apply a Gysin map
from the Chow group of $\mathfrak{F}_{w}$ to the Chow group of
$\tilde{\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$ and calculate the image of
the class of the degeneracy locus.

We first look at the case $j=1$.

######  Lemma 6.5.

The pushdown to $\tilde{\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$ of the
classes $c_{g-2}(H\langle 1\rangle)$ and $c_{g-3}(H\langle
1\rangle)c_{1}(G\langle 1\rangle)$ on $\mathfrak{F}_{w}$ are multiples of
$\lambda_{g-2}$.

###### Proof.

The filtration on ${\mathbb{E}}$ is extended to the de Rham bundle by
${\mathbb{E}}_{g+i}=({\mathbb{E}}_{g-i})^{\bot}$ as in [4, Section 3]. This
symplectic pairing is different from the one used the description of the
display used in [23]. Since we use covariant Dieudonné modules we have to take
duals and Frobenius twists to relate the Chern roots of $G\langle j\rangle$
and $H\langle j\rangle$ to those of the Hodge bundle. The Chern roots of
$G\langle j\rangle$ and $H\langle j\rangle$ are determined by the filtrations
$G_{i}$ and $H_{i}$. We write $l_{i}$ for these roots, while writing
$\ell_{i}$ for the roots of ${\mathbb{E}}$. Then the Chern roots of $H\langle
1\rangle$ given by this filtration are $l_{2},...,l_{g-1}$ and that of
$G\langle 1\rangle$ is $-l_{1}$. The Chern class $c_{g-2}(H\langle 1\rangle)$
is then the $(g-2)$th elementary symmetric function in $l_{2},\ldots,l_{g-1}$.
The $(g-2)$th symmetric function in $l_{2},\ldots,l_{g-1}$ is a Frobenius
twist of the $(g-2)$th symmetric function in $l_{1},\ldots,l_{g-2}$ (cf. the
proof of [4, Lemma 12.3]) and is a multiple of
$\lambda_{g-2}(g-2)=c_{g-2}({\mathbb{E}}(g-2))$. Now by [4, Lemma 12.3] the
pushdown of $\lambda_{g-2}(g-2)$ equals a non-zero multiple of
$\lambda_{g-2}$. The morphism from $\mathfrak{F}$ to $\mathcal{A}_{g}$ is
fibered by generically finite morphisms $\pi_{i}$ defined by forgetting a step
of the flag
${\mathbb{E}}(i)\subsetneq\mathbb{E}(i+1)\subsetneq\cdots\subsetneq{\mathbb{E}}(g)$.
We have for the Chern classes $\lambda_{r}(i)=c_{r}({\mathbb{E}}(i))$ of the
partial flag the formula
$(\pi_{i})^{*}(\lambda_{r}(i+1))=\ell_{i+1}\lambda_{r-1}(i)+\lambda_{r}(i)$.
For the Chern roots $l_{i}$ that we use here a similar formula holds.
Therefore, again by [4, Lemma 12.3], the pushdown of $c_{g-3}(H\langle
1\rangle)c_{1}(G\langle 1\rangle)$ is also a multiple of $\lambda_{g-2}$. ∎

We conclude that the class of the locus where $G\langle 1\rangle\to H\langle
1\rangle$ vanishes on $V_{0}$ is a multiple of the class $\lambda_{g-2}$ on
$V_{0}$. Since this is a Chern class of a vector bundle on
$\tilde{\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$ and the class of $V_{0}$ is a
multiple of $\lambda_{g}$ we find that the class of the vanishing locus in
$\tilde{\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$ of this bundle morphism on
$V_{0}$ is a multiple of $\lambda_{g}\lambda_{g-2}$.

We now carry out induction. We restrict to the locus $Z$ where the consecutive
morphisms $G\langle s\rangle\to H\langle s\rangle$ for $s=1,\ldots,j-1$
vanish. Then the class of the locus of vanishing of (5) equals up to a sign
the $(g-2j)$th Chern class of $G\langle j\rangle\otimes(H\langle
j\rangle)^{\vee}$ and this is $c_{g-2j}(H\langle j\rangle)-c_{g-2j-1}(H\langle
j\rangle)c_{1}(G\langle j\rangle)$. By the argument given in Lemma 6.5 the
class $c_{g-2j}(H\langle j\rangle)$ is a non-zero multiple of the $(g-2j)$th
elementary symmetric function in $g-2j$ consecutive classes $\ell_{i}$. We can
view this as obtained by applying a Frobenius power to
$\ell_{1},\ldots,\ell_{g-2j}$, or use [4, Lemma 12.3], hence this elementary
symmetric function represents a multiple of $\lambda_{g-2j}(g-2j)$. The
pushdown of this is a multiple of $\lambda_{g-2j}$. The argument for
$c_{g-2j-1}(H\langle j\rangle)c_{1}(G\langle j\rangle)$ is similar, as in
Lemma 6.5. Therefore the cycle class of the vanishing locus is a multiple of
$\lambda_{g-2j}$ on the locus $Z$ and $Z$ has as class a multiple of
$\lambda_{g}\lambda_{g-2}\cdots\lambda_{2g+2-2j}$. As $\lambda_{g-2j}$ is the
Chern class of a vector bundle on
$\tilde{\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$ we find as cycle class on
$\tilde{\mathcal{A}}_{g}\otimes{\mathbb{F}}_{p}$ a multiple of
$\lambda_{g}\lambda_{g-2}\cdots\lambda_{2g-2j}$.

∎

######  Remark 6.6.

i) By analyzing more precisely the characteristic classes of the degeneracy
loci in the proof, it should be possible to determine the multiple $f(p)$ as a
polynomial in $p$, but this involves many subtleties. ii) By interpreting
Newton polygon strata contained in the $p$-rank zero locus as degeneracy loci
as done in the proof of Theorem 6.1 we see that the cycle classes of these
loci lie in the tautological ring. This suggests that all Newton polygon
classes are tautological.

##  7\. Moduli of flag type quotients for $g=3$

In this section and the next we calculate the cycle class of the supersingular
locus $S_{3}$. We consider an irreducible component of the space of polarized
flags of Dieudonné modules for $g=3$, defined by the choice of a quasi-
polarization on $A_{1,1}^{3}$. This space is the Zariski closure of the moduli
of rigid polarized Dieudonné flags. A description was given in [18, p. 58].
Thus we look at polarized flags
$(E^{3},\eta)=(Y_{2},\eta)\buildrel{\rho_{2}}\over{\longrightarrow}(Y_{1},\eta_{1})\buildrel{\rho_{1}}\over{\longrightarrow}Y_{0},\eta_{0})$
corresponding to a polarized flag of Dieudonné modules

$M_{0}\subset M_{1}\subset M_{2}=A_{1,1}^{3}=A\langle x,y,z\rangle$

with the quasi-polarization given by

$\langle x,Fx\rangle=\langle y,Fy\rangle=\langle z,Fz\rangle=1/p\,.$

Since $FM_{2}\subset M_{1}$ with $\dim(M_{1}/FM_{2})=1$ the module $M_{1}$ is
determined by a $1$-dimensional subspace of $M_{2}/FM_{2}$, say generated by a
vector $v=ax+by+cz$. The condition $(F,V)M_{1}\subset M_{1}^{t}$ requires
$\langle v,Fv\rangle\in W$, that is, if we view the coefficients $a,b,c$ as
elements of $k$, the condition $(F,V)M_{1}\subset M_{1}^{t}$ is satisfied if
and only if

$a^{p+1}+b^{p+1}+c^{p+1}=0\,.$

Thus the moduli space $\mathcal{F}_{1}$ of truncated flags $M_{1}\subset
M_{2}$ can be identified with a Fermat curve
$\mathcal{X}_{p+1}\subset{\mathbb{P}}^{2}={\rm Gr}(2,3)$. The module $M_{0}$
is determined by a $2$-dimensional subspace $M_{0}/FM_{1}\subset
M_{1}/FM_{1}$. Assuming rigidity, we see that it is spanned by two vectors

$w_{1}=v_{0},\qquad w_{2}=\alpha Fx+\beta Fy+\gamma Fz\,,$

and the condition $M_{0}\subseteq M_{0}^{t}$ gives $a\alpha+b\beta+c\gamma=0$.
This implies that $M_{1}/M_{0}$ defines a sheaf isomorphic to
$\mathcal{O}_{\mathcal{F}_{1}}(1)$. Moreover, the degree $p^{2}$ homomorphism

$\eta_{1}:Y_{1}\to Y_{0}\xrightarrow{\sim}Y_{0}^{t}\to Y_{1}^{t}$

shows that $M_{1}/M_{1}^{t}$ is self dual, and it defines a locally free sheaf
isomorphic to
$\mathcal{O}_{\mathcal{F}_{1}}(1)\oplus\mathcal{O}_{\mathcal{F}_{1}}(-1)$.

This implies that the moduli space of rigid polarized Dieudonné flags with
given quasi-polarization $\eta$ admits a structure

$\mathcal{F}_{0}^{0}\to\mathcal{F}_{1}\to\mathcal{F}_{2}=\text{\rm point}$

with $\mathcal{F}_{0}^{0}$ the open dense part of the
${\mathbb{P}}^{1}$-bundle
$\mathcal{F}_{0}={\mathbb{P}}(\mathcal{O}_{\mathcal{F}_{1}}(1)\oplus\mathcal{O}_{\mathcal{F}_{1}}(-1))$
that is the complement of the unique section with negative self-intersection
number. The Zariski closure is obtained by taking the full
${\mathbb{P}}^{1}$-bundle $\mathcal{F}_{0}$.

The morphism $\mathcal{F}_{0}\to
S_{3}\subset\mathcal{A}_{3}\otimes{\mathbb{F}}_{p}$ is of degree $1$ onto its
image, and the image forms an irreducible component of $S_{3}$. The natural
morphism to $\mathcal{A}_{3}\otimes{\mathbb{F}}_{p}$ contracts the section.

##  8\. The cycle class of $S_{3}$

Here we give the proof of the formula for the cycle class of $S_{3}$ stated in
[9, Thm. 11.3]. The first author learned from Ekedahl at that time how to
calculate the Hodge bundle for flag type quotients. Ekedahl employed this in
[2, Cor. 3.4]. This idea was used in [9] to calculate the cycle class of
$S_{3}$. As done at the time of [9], here we will not use the results of
Section 6.

The Chow rings with rational coefficients of ${\mathcal{A}}_{3}$ and
$\tilde{\mathcal{A}}_{3}$ are known by [10]. The ring ${\rm
CH}_{\mathbb{Q}}^{*}(\tilde{\mathcal{A}}_{3})$ is generated by the Chern
classes of the Hodge bundle and boundary classes $\sigma_{1}$ and
$\sigma_{2}$. A priori the class of $S_{3}$ is a linear combination of the
generators of ${\rm CH}_{\mathbb{Q}}^{4}({\tilde{\mathcal{A}}_{3}})$, viz.
$\lambda_{1}^{4}$, $\lambda_{1}^{3}\sigma_{1}$,
$\lambda_{1}^{2}\sigma_{1}^{2}$ and $\lambda_{1}\sigma_{1}\sigma_{2}$, see
[10]. But since $S_{3}\cdot\sigma_{1}^{2}=0=S_{3}\cdot\sigma_{2}$ we see from
the multiplication table 3f in [10, p. 765] that the class of $S_{3}$ is a
multiple of $\lambda_{1}^{4}=8\,\lambda_{1}\lambda_{3}$. Alternatively, this
follows from the fact that $S_{3}$ is contained in $V_{0}$, the $p$-rank $0$
locus, whose class is a multiple of $\lambda_{3}$.

######  Theorem 8.1.

The class of the supersingular locus for genus $3$ in the Chow ring with
rational coefficients of a Faltings-Chai compactification of
$\mathcal{A}_{3}\otimes{\mathbb{F}}_{p}$ is given by

$[S_{3}]=(p-1)^{2}(p^{3}-1)(p^{4}-1)\,\lambda_{1}\lambda_{3}\,.$

###### Proof.

The class $[S_{3}]$ is a multiple of $\lambda_{1}\lambda_{3}$ and the multiple
can be determined by calculating the intersection number with $\lambda_{2}$.
Using the flag type quotients we see above that an irreducible component of
the supersingular locus $S_{3}$ in $\mathcal{A}_{3}\otimes{\mathbb{F}}_{p}$ is
the image of a surface $\mathcal{F}_{0}$ under a map
$\mathcal{F}_{0}\to\mathcal{A}_{3}\otimes{\mathbb{F}}_{p}$ of degree $1$ and
$\mathcal{F}_{0}$ is of the form

$\mathcal{F}_{0}\buildrel{\pi_{0}}\over{\longrightarrow}\mathcal{F}_{1}\buildrel{\pi_{1}}\over{\longrightarrow}\mathcal{F}_{2}={\rm
point}\,,$

where $\mathcal{F}_{i}$ parametrizes partial flag type quotients
$Y_{2}\to\cdots\to Y_{i}$. More precisely, a component of $S_{3}$ is the image
under a morphism of a ${\mathbb{P}}^{1}$-bundle $\mathcal{B}=\mathcal{F}_{0}$
over the Fermat curve $\mathcal{F}_{1}=\mathcal{X}_{p+1}$ of degree $p+1$ in
${\mathbb{P}}^{2}$ that blows down the unique section $S$ with negative self-
intersection number of the ${\mathbb{P}}^{1}$-bundle
${\mathbb{P}}(\mathcal{O}(1)\oplus\mathcal{O}(-1))$ over $\mathcal{X}_{p+1}$.
A point of $\mathcal{F}_{1}$ corresponds to the choice of a subgroup scheme
$\alpha_{p}^{2}$ in $E^{3}[F]$.

If we use contravariant Dieudonné modules over a geometric point of
${\mathcal{F}}_{i}$ we have for $i=0$ and $i=1$ an exact squence

$0\to pM_{i+1}/pM_{i}\to VM_{i}/pM_{i}\to VM_{i+1}/pM_{i+1}\to
VM_{i+1}/VM_{i}\to 0.$

Over ${\mathcal{F}}_{i}$, we can identify ${\rm Lie}(Y_{i})^{\vee}$ with
$VM_{i}/pM_{i}$ (cf. [1, 4.3.12] and [18, 5.4, 7.4]), more precisely with
${\rm Q}_{i}/I_{\mathcal{O}_{\mathcal{F}_{i}}}{\rm P}_{i}$, where $({\rm
P}_{i},{\rm Q}_{i},F,V^{-1})$ is the display associated to $Y_{i}$. (Note that
${\rm Q}_{i}$ and $I_{\mathcal{O}_{\mathcal{F}_{i}}}{\rm P}_{i}$ become
$VM_{i}$ and $pM_{i}$ respectively if we pull back them to the spectrum of a
perfect field.) By the exact sequence we have in the Grothendieck group
$K_{0}(\mathcal{F}_{i})$ the relation

${\rm Lie}(Y_{i})^{\vee}={\rm Lie}(Y_{i+1})^{\vee}-Q_{i}+Q_{i}^{(p)}$

with $Q_{i}$ the locally free $\mathcal{O}_{\mathcal{F}_{1}}$-module defined
by $VM_{i+1}/VM_{i}$. Here ${\rm Lie}(Y_{i+1})$ denotes the pull back under
$\pi_{i}$. We pull back the relation

${\rm Lie}(Y_{1})^{\vee}={\rm Lie}(Y_{2})^{\vee}-Q_{1}+Q_{1}^{(p)}$

under $\pi_{0}$ to $K_{0}(\mathcal{F}_{0})$ and then find in
$K_{0}(\mathcal{F}_{0})$ suppressing the $\pi_{0}^{*}$

${\rm Lie}(Y_{0})^{\vee}=[3]-Q_{1}+Q_{1}^{(p)}-Q_{0}+Q_{0}^{(p)}\,,$

where the $[3]$ stands for the class of the trivial rank $3$ bundle
$\pi_{0}^{*}\pi_{1}^{*}({\rm Lie}(Y_{2}))^{\vee}$. From the short exact
sequence

$0\to VM_{1}/pM_{1}\to VM_{2}/pM_{2}\to VM_{2}/VM_{1}\to 0$

we get the exact sequence of vector bundles

$0\to U_{1}\to\pi_{1}^{*}({\rm Lie}(Y_{2})^{\vee})\to Q_{1}\to 0$

with ${\rm rank}(Q_{1})=2$ that comes from the universal tautological exact
sequence of bundles on the Grassmannian. Here $U_{1}$ has rank $1$ and
$\pi_{1}^{*}{\rm Lie}(Y_{2})$ is trivial. This implies that $[Q_{1}]=-[U_{1}]$
in the Grothendieck group of vector bundles and so the total Chern class of
${\rm Lie}(Y_{0})^{\vee}$ is given by

$(1-\ell_{1})(1-p\ell_{1})^{-1}(1+\ell_{0})^{-1}(1+p\ell_{0})\,,$

where $\ell_{i}=c_{1}(Q_{i})$. Now $\ell_{1}$ lives on the curve
$\mathcal{F}_{1}=\mathcal{X}_{p+1}$, so $\ell_{1}^{2}=0$. This gives for the
classes $\lambda_{1}$ and $\lambda_{2}$ the relations

$\lambda_{1}=(p-1)(\ell_{0}+\ell_{1}),\quad\lambda_{2}=(p-1)^{2}\ell_{0}\ell_{1}-(p-1)\ell_{0}^{2}\,.$

The identity $\lambda_{1}^{2}=2\lambda_{2}$ that holds in the tautological
ring $R_{3}$ implies that $(p-1)^{2}(\ell_{0}^{2}-\ell_{1}^{2})=0$, hence
$\ell_{0}^{2}=0$. Since $\deg(\ell_{1})=p+1$ on $\mathcal{F}_{1}$ and
$\ell_{0}$ represents $\mathcal{O}(1)$ on the fibres of
$\mathcal{F}_{0}\to\mathcal{F}_{1}$ we find $\deg(\ell_{0}\ell_{1})=p+1$. We
thus find that $\deg(\lambda_{2})=(p+1)(p-1)^{2}$ on each irreducible
component of $S_{4}$. We get

$\displaystyle\deg(\lambda_{2}[S_{3}])$ $\displaystyle=(p+1)(p-1)^{2}\,N_{3}$
$\displaystyle=(p+1)(p-1)^{2}(p-1)(p^{2}+1)(p^{3}-1)\,v(3)\,.$

On the other hand, $\deg(\lambda_{1}\lambda_{2}\lambda_{3})=v(3)$ and this
implies the result. ∎

The morphism $\pi_{0}:\mathcal{F}_{0}\to\mathcal{F}_{1}$ is a
${\mathbb{P}}^{1}$-bundle over a Fermat curve of degree $p+1$ with a section
with image $S$. The Picard group of $\mathcal{F}_{0}$ is generated by the
pullback under $\pi_{0}$ of the Picard group of $\mathcal{F}_{1}$ and by the
class of the section $S$.

######  Proposition 8.2.

We have $[S]=\ell_{0}-\ell_{1}$ and $S^{2}=-2(p+1)$.

###### Proof.

Let $X$ be a fibre of $\pi_{0}$. We have $XS=1$ and $(S-\ell_{0})X=0$, hence
$S-\ell_{0}=\pi_{0}^{*}(D)$ with $D$ a divisor class on $\mathcal{F}_{1}$.
This gives $(S-\ell_{0})^{2}=0$. The identity $\lambda_{1}^{2}=2\,\lambda_{2}$
implies $\ell_{0}^{2}=0$ and thus $S^{2}-2\ell_{0}S=0$. Now we use the fact
that $S$ is contracted under the map of $\mathcal{F}_{0}$ to
$\mathcal{A}_{3}\otimes{\mathbb{F}}_{p}$. This implies that $\lambda_{1}$
restricted to $S$ vanishes, hence $(\ell_{0}+\ell_{1})S=0$. We thus get
$S^{2}=2\ell_{0}S=-2\ell_{1}S$ and on the other hand
$S^{2}=\ell_{0}S+\pi_{0}^{*}(D)S=-\ell_{1}S+\pi_{0}^{*}(D)S$, hence
$\pi_{0}^{*}(D)=-\ell_{1}$ and $S=\ell_{0}-\ell_{1}$. The fact that
$\ell_{0}\ell_{1}=p+1$ and $\ell_{0}^{2}=\ell_{1}^{2}=0$ imply
$S^{2}=-2(p+1)$. ∎

##  9\. Loci for $g=3$ defined by conditions on the $a$-number

We now discuss the subloci of $S_{3}$ defined by the inequality $a\geq 2$.
Here $a$ indicates the $a$-number of an abelian variety. Let $J$ with
$\\#J=N_{3}$ be the set of irreducible components of $S_{3}$ (where we count
in the stacky way). Each irreducible component of $S_{3}$ is the image under a
morphism of a ${\mathbb{P}}^{1}$-bundle $\mathcal{F}_{0}\to\mathcal{F}_{1}$
that blows down a section. The curve $\mathcal{F}_{1}$ has $p^{3}+1$ points
rational over ${\mathbb{F}}_{p^{2}}$ and
$\\#\mathcal{F}_{0}({\mathbb{F}}_{p^{2}})=(p^{3}+1)(p^{2}+1)$ and each point
of $\mathcal{F}_{0}({\mathbb{F}}_{p^{2}})$ defines a superspecial abelian
variety. Let $\sqcup_{j\in J}\mathcal{F}_{0}^{j}$ be the disjoint union of the
smooth models of the irreducible components of $S_{3}$. Under the natural
morphism

$m:\sqcup_{j\in J}\mathcal{F}_{0}^{j}\longrightarrow
S_{3}\subset\mathcal{A}_{3}\otimes{\mathbb{F}}_{p}\,.$

the $N_{3}(p^{3}+1)(p^{2}+1)$ superspecial points of $\sqcup_{j\in
J}\mathcal{F}_{0}^{j}$ map to $N_{3}$ superspecial points of $S_{3}$. Thus
each superspecial point of $S_{3}$ is the image of $(p^{3}+1)(p^{2}+1)$ points
and this multiplicity can be explained as follows. On each surface
$\mathcal{F}_{0}^{j}$ a section is contracted giving a factor $p^{3}+1$, while
the image of an ${\mathbb{F}}_{p^{2}}$-rational fibre of
$\mathcal{F}_{0}^{j}\to\mathcal{F}_{1}^{j}$ lies on the image of $p^{2}+1$
surfaces $\mathcal{F}_{0}^{j}$. This can be checked by using Ekedahl-Oort
strata and their classes as follows.

Each ${\mathbb{F}}_{p^{2}}$-rational point of $\mathcal{F}_{1}^{j}$ determines
a fibre in the ${\mathbb{P}}^{1}$-bundle
$\mathcal{F}_{0}^{j}\to\mathcal{F}_{1}^{j}$ and the image under $m$ of such a
fibre provides a component of the Ekedahl-Oort locus $\mathcal{V}_{[3,2]}$.
This locus $\mathcal{V}_{[3,2]}$ consists of a finite union of
${\mathbb{P}}^{1}$s. By [4] we know the class of this locus:

$[\overline{\mathcal{V}}_{[3,2]}]=(p-1)^{2}(p^{6}-1)\,\lambda_{2}\lambda_{3}\,.$

Since the degree of the determinant $\lambda_{1}$ of the Hodge bundle
restricted to such a ${\mathbb{P}}^{1}$ is $p-1$, we find that
$\overline{\mathcal{V}}_{[3,2]}$ has

$m_{3,2}=\frac{\deg([\overline{\mathcal{V}}_{[3,2]}]\,\lambda_{1})}{p-1}=(p-1)(p^{6}-1)\,v(3)$

irreducible components, each a copy of ${\mathbb{P}}^{1}$. Here we count in
the stacky sense. Each such component contributes $p^{2}+1$ superspecial
points and we see from

$m_{3,2}\,(p^{2}+1)=\deg(\overline{\mathcal{V}}_{[3,2,1]})\,(p^{3}+1)$

that this fits with the fact that through a superspecial point there pass
$p^{3}+1$ components of $\overline{\mathcal{V}}_{[3,2]}$. In fact, under the
map $\mathcal{F}_{0}^{j}\to\mathcal{A}_{3}\otimes{\mathbb{F}}_{p}$ a section
is blown down and this section intersects the $p^{3}+1$ fibres of
$\mathcal{F}_{0}^{j}\to\mathcal{F}_{1}^{j}$ over
$\mathcal{F}_{1}^{j}({\mathbb{F}}_{p^{2}})$.

We can also check that each such fibre lies on $p^{2}+1$ irreducible
components of $S_{3}$; hence we find for the number of superspecial points

$N_{3}\,(p^{3}+1)(p^{2}+1)=\deg({\overline{\mathcal{V}}_{[3,2,1]}})(p^{2}+1)(p^{3}+1)\,$

in agreement with the fact that $\mathcal{V}_{[3,2,1]}$ is the superspecial
locus and that $N_{g}$ equals the degree of the superspecial locus for odd
$g$.

##  10\. Moduli of flag type quotients for $g=4$

In this section we construct a smooth model for each irreducible component of
the supersingular locus $S_{4}$. The model is obtained by taking the Zariski
closure of the moduli of rigid flag type quotients for $g=4$ and by showing
that this moduli space is smooth.

We consider the space $\mathcal{M}=\mathcal{M}_{\eta}$ of polarized flags of
covariant Dieudonné modules

$M_{0}\subset M_{1}\subset M_{2}\subset M_{3}$

satisfying

  1. (1)

$M_{3}=A_{1,1}^{4}$ provided with $\eta$, a fixed quasi-polarization
$\langle\,,\,\rangle$ that induces an identification $M_{3}^{t}=F^{3}M_{3}$;

  2. (2)

$(F,V)M_{i}\subset M_{i-1}$ and $\dim(M_{i}/M_{i-1})=i$;

  3. (3)

$(F,V)^{i}M_{i}\subset M_{i}^{t}$.

We say that it is rigid if $M_{i}=M_{0}+F^{3-i}M_{3}$ for $i=0,\ldots,3$.

######  Theorem 10.1.

The Zariski closure $\mathcal{F}_{0}$ of the moduli space of rigid polarized
Dieudonné flags of length $4$ with given quasi-polarization on $M_{3}$ inside
$\mathcal{M}$ is non-singular.

###### Proof.

We assume we have a skeleton of $M_{3}$, that is, generating elements
$x_{1},x_{2},x_{3},x_{4}$ with $(F-V)x_{i}=0$ and such that the pairing
defined by $\eta$ satisfies

$\langle x_{1},F^{4}x_{4}\rangle=\langle x_{2},F^{4}x_{3}\rangle=1\,.$

For a rigid polarized Dieudonné flag $M$ the module $M_{2}$ is generated by
$FM_{3}$ and a vector

$v_{0}=\sum_{i=1}^{4}a_{i}x_{i}\,\in M_{3}/FM_{3}$

with the condition $\langle v_{0},F^{2}v_{0}\rangle\in W$. Viewing the
coefficients $a_{i}$ as lying in $k$, this amounts to the equation

$f:=a_{1}a_{4}^{p^{2}}-a_{1}^{p^{2}}a_{4}+a_{2}a_{3}^{p^{2}}-a_{2}^{p^{2}}a_{3}=0\,.$

This defines a smooth surface $\mathcal{F}_{2}$ in ${\mathbb{P}}^{3}$. This
surface was studied in detail by Katsura [14]. Locally on this surface we may
assume without loss of generality that $a_{1}\neq 0$ and that $a_{1}=1$. Now
$M_{1}$ is generated by $FM_{2}$ and a $2$-dimensional subspace $M_{1}/FM_{2}$
in $M_{2}/FM_{2}$. Since $a_{1}=1$ we can assume that this $2$-dimensional
subspace is generated by non-zero elements $v$ and $w$ with

$None$ $v=a_{5}v_{0}+a_{6}Fx_{2}+a_{7}Fx_{3}+a_{8}Fx_{4},\quad
w=a_{9}Fx_{2}+a_{10}Fx_{3}+a_{11}Fx_{4}\,.$

We then have the conditions

$None$ $\langle v,Fv\rangle\in W,\quad\langle v,Fw\rangle\in W,\quad\langle
Fv,w\rangle\in W\,.$

Viewing the coefficients as elements of $k$ we find three equations all
divisible by $a_{5}$. But $a_{5}=0$ yields a flag that is not rigid; indeed,

$M_{1}+FM_{3}=(F,V)M_{2}+Aw+FM_{3}=(F,V)(Av+FM_{3})+Aw+FM_{3}\subset FM_{3}$

but $M_{2}\not\subset FM_{3}$, hence $M_{2}\neq M_{1}+FM_{3}$, contradicting
rigidity. Removing the factor $a_{5}$ from the equations (7) by considering
$\langle v,Fv\rangle/a_{5},\langle v,Fw\rangle/a_{5}$ and $\langle
Fv,w\rangle/a_{5}^{p}$, we get the equations

$None$ $\displaystyle g_{1}:=$ $\displaystyle
a_{1}a_{8}^{p}-a_{1}^{p}a_{5}^{p-1}a_{8}+a_{2}a_{7}^{p}-a_{2}^{p}a_{5}^{p-1}a_{7}+a_{3}^{p}a_{5}^{p-1}a_{6}-a_{3}a_{6}^{p}=0,$
$\displaystyle g_{2}:=$ $\displaystyle
a_{1}a_{11}^{p}+a_{2}a_{10}^{p}-a_{3}a_{9}^{p}=0\,,$ $\displaystyle g_{3}:=$
$\displaystyle a_{1}^{p}a_{11}+a_{2}^{p}a_{10}-a_{3}^{p}a_{9}=0\,.$

######  Remark 10.2.

The reader may verify that if the point
$(1:a_{2}:a_{3}:a_{4})\in\mathcal{F}_{2}(k)$ is not rational over
${\mathbb{F}}_{p^{2}}$ then we may choose as $w$ the element

$(F-V)\,v_{0}\,.$

Indeed, it satisfies $g_{2}=0$ and $g_{3}=0$ for any non-zero choice of $v$;
namely with $a_{1}=1$ we have
$a_{1}(a_{4}^{p^{2}}-a_{4})+a_{2}(a_{3}^{p^{2}}-a_{3})-a_{3}(a_{2}^{p^{2}}-a_{2})=0$
and similarly for $g_{3}$.

Now we first look at a point with $a_{5}\neq 0$. If both $a_{9}$ and $a_{10}$
vanish we have by $g_{3}=0$ that $w=0$. So we may assume that, say, $a_{9}\neq
0$ and then have $a_{1}=a_{5}=a_{9}=1$ and by changing $v$ to $v-a_{6}w$ we
may assume $a_{6}=0$. The Jacobian matrix of the equations
$f,g_{1},g_{2},g_{3}$ with respect to the variables $a_{j}$ for
$j=2,3,4,7,8,10,11$ is

$\left(\begin{matrix}a_{3}^{p^{2}}&-a_{2}^{p^{2}}&-a_{1}^{p^{2}}&0&0&0&0\\\
a_{7}^{p}&-a_{6}^{p}&0&-a_{2}^{p}a_{5}^{p-1}&-a_{1}^{p}a_{5}^{p-1}&0&0\\\
a_{10}^{p}&-a_{9}^{p}&0&0&0&0&0\\\ 0&0&0&0&0&a_{2}^{p}&a_{1}^{p}\\\
\end{matrix}\right)$

and this is of rank $4$.

Next we look at the case where $a_{5}=0$. The vanishing of $a_{9}$ and
$a_{10}$ implies by $g_{3}$ that $a_{11}=0$, so we may assume that $a_{9}\neq
0$ or $a_{10}\neq 0$. Again without loss of generality we may assume
$a_{9}\neq 0$. Changing $v$ by a multiple of $w$ we may assume $a_{6}=0$. If
now $a_{7}=0$ then $g_{1}$ forces $a_{8}=0$, hence $v=0$. So we may assume
that $a_{7}\neq 0$. Then it suffices to treat the case of
$a_{1}=a_{7}=a_{9}=1$ and $a_{6}=0$. Then the Jacobian matrix of the equations
$f,g_{1},g_{2},g_{3}$ with respect to the variables $a_{j}$ for
$j=2,3,4,5,8,10,11$ is

$\left(\begin{matrix}a_{3}^{p^{2}}&-a_{2}^{p^{2}}&-a_{1}^{p^{2}}&0&0&0&0\\\
a_{7}^{p}&0&0&(a_{1}^{p}a_{8}+a_{2}^{p}a_{7})a_{5}^{p-2}&-a_{1}^{p}a_{5}^{p-1}&0&0\\\
a_{10}^{p}&-a_{9}^{p}&0&0&0&0&0\\\ 0&0&0&0&0&a_{2}^{p}&a_{1}^{p}\\\
\end{matrix}\right)$

which is of rank $4$ as required. ∎

By writing $\mathcal{F}_{i}$ for the Zariski closure in $\mathcal{M}$ of the
moduli space of rigid polarized Dieudonné flags $M_{i}\subset\cdots\subset
M_{3}$ we get a sequence

$\mathcal{F}_{0}\buildrel{\pi_{0}}\over{\longrightarrow}\mathcal{F}_{1}\buildrel{\pi_{1}}\over{\longrightarrow}\mathcal{F}_{2}\buildrel{\pi_{2}}\over{\longrightarrow}\mathcal{F}_{3}={\rm
point}$

with $\dim\mathcal{F}_{i}=4-i$ for $i=0,1,2$.

We now describe the fibres of the morphism
$\pi_{1}:\mathcal{F}_{1}\to\mathcal{F}_{2}$. We start by remarking that by
using the symmetry of $\mathcal{F}_{2}$ there is no loss of generality if we
look at the fibre of a point $(a_{1}:a_{2}:a_{3}:a_{4})$ of $\mathcal{F}_{2}$
with $a_{1}=1$. If one of $a_{2},a_{3},a_{4}$ lies in ${\mathbb{F}}_{p^{2}}$
then the point lies on one of the lines of $\mathcal{F}_{2}$. Indeed, if
$a_{4}\in{\mathbb{F}}_{p^{2}}$ then such a line is parametrically
$(1:t:t:a_{4})$, while if, say, $a_{2}\in{\mathbb{F}}_{p^{2}}$ then such a
line is $(t:a_{2}:1:t)$.

For describing the fibre over a point $(1:a_{2}:a_{3}:a_{4})$ we consider the
equations

$None$
$a_{8}^{p}+a_{2}a_{7}^{p}-a_{3}a_{6}^{p}-a_{5}^{p-1}(a_{8}+a_{2}^{p}a_{7}-a_{3}^{p}a_{6})=0\,,$

and

$None$ $a_{11}^{p}+a_{2}a_{10}^{p}-a_{3}a_{9}^{p}=0,\quad
a_{11}+a_{2}^{p}a_{10}-a_{3}^{p}a_{9}=0\,.$

By the two equations $g_{2}$, $g_{3}$ of (10) we eliminate $a_{11}$ and get

$None$
$\frac{a_{10}^{p}}{a_{9}^{p}}=\frac{a_{3}^{p^{2}}-a_{3}}{a_{2}^{p^{2}}-a_{2}}\,.$

In the neighborhood of an ${\mathbb{F}}_{p^{2}}$-valued point of
$\mathcal{F}_{2}$, say $(1:a_{2}:a_{3}:a_{4})$, the expressions
$a_{2}-a_{2}^{p^{2}}$ and $a_{3}-a_{3}^{p^{2}}$ are local coordinates. This
shows that the function field of $\mathcal{F}_{1}$ can be generated over the
function field of $\mathcal{F}_{2}$ by adjoining the $p$th root of
$(a_{2}-a_{2}^{p^{2}})/(a_{3}-a_{3}^{p^{2}})$ as determined by (11) and then
adjoining a further element via an Artin-Schreier equation (9). Hence the
degree of inseparability of $\mathcal{F}_{1}$ over $\mathcal{F}_{2}$ is $p$.

Over an open neighborhood $U$ of a ${\mathbb{F}}_{p^{2}}$-rational point with
local coordinates $a_{2}-a_{2}^{p^{2}}$ and $a_{3}-a_{3}^{p^{2}}$, the
equation (11) describes an inseparable cover of the blow-up of $U$ (in
$U\times{\mathbb{P}}^{1}$ with coordinates $(u:v)$ on ${\mathbb{P}}^{1}$)
given by

$u(a_{2}-a_{2}^{p^{2}})-v(a_{3}-a_{3}^{p^{2}})=0,\quad
u/v=(a_{10}/a_{9})^{p}\,.$

Thus we see that the morphism $\pi_{1}:\mathcal{F}_{1}\to\mathcal{F}_{2}$
factors via an inseparable cover of the blow-up $\tilde{\mathcal{F}}_{2}$ of
$\mathcal{F}_{2}$ in the ${\mathbb{F}}_{p^{2}}$-rational points.

If we have a point not on a line we may assume $a_{9}=1$ and then that
$a_{6}=0$. The reduced fibre is a curve in ${\mathbb{P}}^{2}$ with coordinates
$(a_{5},a_{7},a_{8})$ given by

$a_{8}^{p}+a_{2}a_{7}^{p}-a_{5}^{p-1}(a_{8}+a_{2}^{p}a_{7})=0\,.$

This is a curve with one singularity of order $p$, a cusp located at $a_{5}=0$
and $a_{8}+a_{2}^{1/p}a_{7}=0$.

Next we consider the case of a point on a line. Since the automorphism group
of $\mathcal{F}_{2}$ acts transitively on the set of lines defined over
${\mathbb{F}}_{p^{2}}$ (by Witt’s theorem, see [14]) we may assume that the
line is given as $(1:t:0:0)$. The last two equations give
$(t^{p^{2}}-t)a_{10}^{p}=0$ and the first equation yields
$a_{8}^{p}+ta_{7}^{p}-a_{5}^{p-1}(a_{8}+t^{p}a_{7})=0$, again a curve with a
cusp. So if the point is not a ${\mathbb{F}}_{p^{2}}$-valued point of
$\mathcal{F}_{2}$ we get $a_{10}^{p}=0$ and as reduced fibre again a curve
with a single singularity, a cusp. If $t\in{\mathbb{F}}_{p^{2}}$, then the
first equation splits as the union of $p$ lines passing through one point.

We summarize.

######  Proposition 10.3.

Let $\tilde{\mathcal{F}}_{2}$ be the blow-up of $\mathcal{F}_{2}$ in all
${\mathbb{F}}_{p^{2}}$-rational points. The morphism
$\pi_{1}:\mathcal{F}_{1}\to\mathcal{F}_{2}$ factors through
$\mathcal{F}_{1}\to\tilde{\mathcal{F}}_{2}\to\mathcal{F}_{2}$. The morphism
$\pi_{1}^{\prime}:\mathcal{F}_{1}\to\tilde{\mathcal{F}}_{2}$ has
inseparability degree $p$. The reduced fibre over a
non-${\mathbb{F}}_{p^{2}}$-rational point is an irreducible curve with one
singularity, a cusp singularity of order $p$, while the fibre over a point on
an exceptional curve is a union of $p$ lines meeting in one point.

##  11\. Interpretation of the morphism $\mathcal{F}_{1}\to\mathcal{F}_{2}$

The morphism $\pi_{1}:\mathcal{F}_{1}\to\mathcal{F}_{2}$ is inseparable and
factors through the blown-up surface $\tilde{\mathcal{F}}_{2}$. We give an
interpretation of this factorization by describing the blow-up
$\tilde{\mathcal{F}}_{2}$ in terms of Dieudonné modules and by showing that
$\mathcal{F}_{1}$ is realized in a natural ${\mathbb{P}}^{2}$-bundle over
$\tilde{\mathcal{F}}_{2}$.

We begin with a moduli interpretation of the fibers of
$\tilde{\mathcal{F}}_{2}\to\mathcal{F}_{2}$.

######  Proposition 11.1.

The fiber of $\tilde{\mathcal{F}}_{2}\to\mathcal{F}_{2}$ over a point
$(M_{2}\subset M_{3})\in{\mathcal{F}}_{2}(k)$ is given by a set of lines in a
$2$-dimensional vector space

$\\{L\subset V^{-1}M_{2}^{t}/FM_{2}\mid\dim L=1,\ \text{ $L$ contains
$(F,V)M_{2}\bmod FM_{2}$}\\}.$

###### Proof.

We begin by observing two facts:

  1. (i)

$(FM_{2}\subset)\ (F,V)M_{2}\subset V^{-1}M_{2}^{t}$

  2. (ii)

$V^{-1}M_{2}^{t}/FM_{2}$ is a $k$-vector space of dimension two.

Indeed, (i) follows from $V(F,V)M_{2}\subset(F,V)^{2}M_{2}\subset M_{2}^{t}$.
To prove (ii), consider the dual of $FM_{3}\subset M_{2}\subset M_{3}$:

$M_{3}^{t}\subset M_{2}^{t}\subset V^{-1}M_{3}^{t}.$

By $V^{-1}M_{3}^{t}=F^{-1}M_{3}^{t}=F^{2}M_{3}\subset FM_{2}$, we have
$M_{2}^{t}\subset FM_{2}$. This means that $V$ (and therefore $p$) kills
$V^{-1}M_{2}^{t}/FM_{2}$, whence $V^{-1}M_{2}^{t}/FM_{2}$ is a $k$-vector
space. Looking at the inclusions $M_{2}^{t}\subset FM_{2}\subset
V^{-1}M_{2}^{t}$, we have

$\displaystyle\dim V^{-1}M_{2}^{t}/FM_{2}$ $\displaystyle=\dim
V^{-1}M_{2}^{t}/M_{2}^{t}-\dim FM_{2}/M_{2}^{t}$ $\displaystyle=4-\dim
FM_{2}/F^{2}M_{3}-\dim V^{-1}M_{3}^{t}/M_{2}^{t}$ $\displaystyle=4-1-1=2$

and this proves ii). If $(M_{2}\subset M_{3})$ represents a point of
$\mathcal{F}_{2}$ that is not rational over ${\mathbb{F}}_{p^{2}}$ then
$FM_{2}\neq VM_{2}$ and $L$ is unique. If $(M_{2}\subset M_{3})$ represents a
${\mathbb{F}}_{p^{2}}$-rational point, then $FM_{2}=VM_{2}$ and the fibre is a
${\mathbb{P}}^{1}$. ∎

######  Remark 11.2.

We point out that the Dieudonné module $V^{-1}M_{2}^{t}/FM_{2}$ is self-dual.

We now describe the morphism $\mathcal{F}_{1}\to\tilde{\mathcal{F}}_{2}$. On
$\tilde{\mathcal{F}}_{2}$ we have by Proposition 11.1 the subspace $L\subset
V^{-1}M_{2}^{t}/FM_{2}$. It determines a $W$-module $\tilde{L}$ with

$(F,V)M_{2}\subset\tilde{L}\subset V^{-1}M_{2}^{t}\,,$

the inverse image of $L$ under the projection $V^{-1}M_{2}^{t}\to
V^{-1}M_{2}^{t}/FM_{2}$. It has the property that outside
$\tilde{\pi}_{2}^{-1}(\mathcal{F}_{2}({\mathbb{F}}_{p^{2}}))$ we have
$\tilde{L}=(F,V)M_{2}$. We can now consider over a point of
$\tilde{\mathcal{F}}_{2}$ the $3$-dimensional vector space $M_{2}/\tilde{L}$.
This should define a rank $3$ vector bundle $B$, but as the equations show we
can realize $B$ only after an inseparable base change.

######  Lemma 11.3.

The threefold $\mathcal{F}_{1}$ is a divisor in a ${\mathbb{P}}^{2}$-bundle
${\mathbb{P}}(B)$ with $B$ the rank $3$ vector bundle defined by
$M_{2}/\tilde{L}$ over a surface $\tilde{\mathcal{F}}_{2}^{\prime}$ obtained
by an inseparable base change
$\tilde{\mathcal{F}}_{2}^{\prime}\to\tilde{\mathcal{F}}_{2}$ of degree $p$.

###### Proof.

Recall that in order to define $M_{1}\subset M_{2}$, we chose a basis

$v=a_{5}v_{0}+a_{6}Fx_{2}+a_{7}Fx_{3}+a_{8}Fx_{4},\quad
w=a_{9}Fx_{2}+a_{10}Fx_{3}+a_{11}Fx_{4}\,$

as in (6) with $\langle v,Fv\rangle$, $\langle v,Fw\rangle$, $\langle
Fv,w\rangle$ all in $W$. The equations ($g_{2}$) and ($g_{3}$) correspond to
the inseparable base change
$\tilde{\mathcal{F}}_{2}^{\prime}\to\tilde{\mathcal{F}}_{2}$ given on the
locus with $a_{1}\neq 0$ by (11)

$(a_{9}/a_{10})^{p}=(a_{2}-a_{2}^{p^{2}})/(a_{3}-a_{3}^{p^{2}})\,.$

Then on $\tilde{\mathcal{F}}_{2}^{\prime}$ we have the bundle
${\mathbb{P}}(B)$. If $a_{5}\neq 0$ the morphism
${\mathcal{F}}_{1}\to\tilde{\mathcal{F}}_{2}$ is defined by sending
$(M_{1}\subset M_{2}\subset M_{3})$ to the point defined by $L:=M_{1}\cap
V^{-1}M_{2}^{t}\bmod FM_{2}$. Indeed, by $(F,V)M_{2}\subset M_{1}$, the
subspace $L$ contains $(F,V)M_{2}\bmod FM_{2}$, and $L$ is the one-dimensional
space generated by $w$ of (6), since one can check $\langle
Vw,M_{2}\rangle=\\{0\\}$ and if $a_{5}\neq 0$, then $\langle
Vv,M_{2}\rangle\neq\\{0\\}$.

For $a_{5}\neq 0$ we find from $\langle v,Fv\rangle\in W$ an equation

$a_{1}a_{8}^{p}+a_{2}a_{7}^{p}-a_{3}a_{6}^{p}-a_{5}^{p-1}(a_{1}^{p}a_{8}+a_{2}^{p}a_{7}-a_{3}^{p}a_{6})=0\,.$

This defines a rational curve with a cusp in
${\mathbb{P}}^{2}={\mathbb{P}}(M_{2}/\tilde{L})$. As $\mathcal{F}_{1}$ is
defined as the closure of the space of rigid flags, we obtain that this
equation defines $\mathcal{F}_{1}$ in ${\mathbb{P}}(B)$. Observe that in order
to analyze this we may assume as we did in the preceding section that
$a_{1}=1$ and $a_{2}\neq 0$ and then $a_{6}=0$ and the curve can be written in
coordinates $(a_{5}:a_{7}:a_{8})$ as

$a_{1}a_{8}^{p}+a_{2}a_{7}^{p}-a_{5}^{p-1}(a_{1}^{p}a_{8}+a_{2}^{p}a_{7})=0\,.$

The cusp is determined by $a_{5}=0$ and $a_{8}+a_{2}^{1/p}a_{7}=0$. ∎

In particular we see that after an inseparable base change the bundle $B$
admits a nowhere vanishing section.

##  12\. The Hodge bundle on the supersingular locus

The description of principally polarized supersingular abelian varieties of
dimension $4$ via a flag gives us for each irreducible component $S$ of
$S_{4}$ a morphism $\mathcal{F}_{0}\to S$ and a fibration of $\mathcal{F}_{0}$

$\mathcal{F}_{0}\buildrel{\pi_{0}}\over{\longrightarrow}\mathcal{F}_{1}\buildrel{\pi_{1}}\over{\longrightarrow}\mathcal{F}_{2}\to\mathcal{F}_{3}\,,$

where $\mathcal{F}_{i}$ for $i=0,\ldots,3$ is the closure of the moduli space
of rigid polarized flag type quotients $Y_{3}\to\cdots\to Y_{i}$. Note that
$\mathcal{F}_{3}$ is a point. We have seen above that these spaces
$\mathcal{F}_{i}$ are non-singular.

######  Lemma 12.1.

For each irreducible component $S$ of $S_{4}$ in
$\mathcal{A}_{4}\otimes{\mathbb{F}}_{p}$ the natural morphism
$\mathcal{F}_{0}\to S$ is a morphism of degree $p$.

###### Proof.

Let $x$ be a geometric point of ${\mathcal{F}_{2}}$. Let ${\mathcal{F}}_{1,x}$
be the fiber $\pi_{1}^{-1}(x)$. We claim that
$\pi_{0}^{-1}({\mathcal{F}}_{1,x})\to\mathcal{A}_{g}$ is a $p$-to-$1$ map onto
the image. Indeed if $x$ is represented by $(a_{1},a_{2},a_{3},a_{4})\in
k^{4}$ for an algebraically closed field $k$, then the fiber $\pi_{1}^{-1}(x)$
is described in $a_{5},\ldots,a_{11}$ by

$\displaystyle g_{1}:=$ $\displaystyle
a_{1}a_{8}^{p}-a_{1}^{p}a_{5}^{p-1}a_{8}+a_{2}a_{7}^{p}-a_{2}^{p}a_{5}^{p-1}a_{7}+a_{3}^{p}a_{5}^{p-1}a_{6}-a_{3}a_{6}^{p}=0,$
$\displaystyle g_{2}:=$ $\displaystyle
a_{1}a_{11}^{p}+a_{2}a_{10}^{p}-a_{3}a_{9}^{p}=0\,,$ $\displaystyle g_{3}:=$
$\displaystyle a_{1}^{p}a_{11}+a_{2}^{p}a_{10}-a_{3}^{p}a_{9}=0\,.$

But $g_{2}$ is the $p$-th power of

$g_{2}^{\prime}:=a_{1}^{1/p}a_{11}+a_{2}^{1/p}a_{10}-a_{3}^{1/p}a_{9}\,.$

The space defined by $g_{1},g^{\prime}_{2},g_{3}$, say
${\mathcal{F}}^{\prime}_{1,x}$, coincides on an open part of $\mathcal{F}_{1}$
with the fiber of ${\mathcal{V}}_{11}\to{\mathcal{V}}_{2}$ studied in [18,
9.7], where ${\mathcal{V}}_{2}$ corresponds to our ${\mathcal{F}}_{2}$ and
${\mathcal{V}}_{11}$ is the non-garbage component considered in [18, 9.7].
Note that ${\mathcal{F}}^{\prime}_{1,x}$ is a closed subscheme of
${\mathcal{F}}_{1,x}$. Thanks to the proof by Li and Oort (cf. [18, 7.11]),
the map $(\pi_{0})^{-1}({\mathcal{F}}^{\prime}_{1})\to{\mathcal{A}}_{g}$ is
one-to-one on its image as stacks; indeed, the proof of Li and Oort was done
by fiberwise arguments. The claim follows. ∎

The space $\mathcal{F}_{i}$ carries an abelian variety $\mathcal{Y}_{i}$. Its
cotangent bundle along the zero section may be described by Dieudonné theory.
Using contravariant Dieudonné theory with the Dieudonné module $M_{i}$ of a
fibre $Y_{i}$ of $\mathcal{Y}_{i}$, we have

${\rm Lie}(Y_{i})^{\vee}=VM_{i}/pM_{i}\,.$

The flag type quotient provides an inductive construction. For $i=2,1,0$ we
have the exact sequence

$0\to pM_{i+1}/pM_{i}\to VM_{i}/pM_{i}\to VM_{i+1}/pM_{i+1}\to
VM_{i+1}/VM_{i}\to 0\,.$

In the Grothendieck group of vector bundles on $\mathcal{F}_{i}$ we thus get
the identity

$[{\rm Lie}(\mathcal{Y}_{i})^{\vee}]=[\pi_{i}^{*}({\rm
Lie}(\mathcal{Y}_{i+1})^{\vee})]-[Q_{i}]+[Q_{i}^{(p)}]\,,$

where $Q_{i}$ is the locally free $\mathcal{O}_{\mathcal{F}_{i}}$-module of
rank $i+1$ corresponding to $VM_{i+1}/VM_{i}$. Moreover, the exact sequence
for $i=1$ and $i=2$

$0\to VM_{i}/pM_{i+1}\to VM_{i+1}/pM_{i+1}\to VM_{i+1}/VM_{i}\to 0$

gives us an exact sequence of $\mathcal{O}_{\mathcal{F}_{i}}$-modules

$0\to U_{i}\to\pi_{i}^{*}({\rm Lie}(\mathcal{Y}_{i+1})^{\vee})\to Q_{i}\to 0$

with $U_{i}$ the locally free $\mathcal{O}_{\mathcal{F}_{i}}$-module defined
by $VM_{i}/pM_{i+1}$. For $i=0$ we have

$0\to VM_{0}/VM_{1}^{t}\to VM_{1}/VM_{1}^{t}\to VM_{1}/VM_{0}\to 0$

and this gives a short exact sequence of
$\mathcal{O}_{\mathcal{F}_{0}}$-modules

$0\to U_{0}\to\pi_{0}^{*}(K_{1})\to Q_{0}\to 0$

with $K_{1}$ the locally free sheaf corresponding to the Dieudonné module of
$\ker(\mathcal{Y}_{1}\buildrel{\eta_{1}}\over{\longrightarrow}\mathcal{Y}_{1}^{t})$.

In the following we will abuse the notation $Q_{i}$ also for the pullback of
$Q_{i}$ to $\mathcal{F}_{i-1}$ in order to simplify notation. Since ${\rm
Lie}(\mathcal{Y}_{3})^{\vee}$ is trivial of rank $4$ we get from the above the
class of the Hodge bundle ${\mathbb{E}}={\rm Lie}(\mathcal{Y}_{0})^{\vee}$ in
the Grothendieck group of vector bundle on $\mathcal{F}_{0}$.

######  Proposition 12.2.

The class of Hodge bundle of $\mathcal{Y}_{0}$ in the Grothendieck group of
vector bundles on $\mathcal{F}_{0}$ is given by

$\displaystyle{}[{\mathbb{E}}]$
$\displaystyle=4-[Q_{2}]+[Q_{2}^{(p)}]-[Q_{1}]+[Q_{1}^{(p)}]-[Q_{0}]+[Q_{0}^{(p)}]$
$\displaystyle=4+[U_{2}]-[U_{2}^{(p)}]-[Q_{1}]+[Q_{1}^{(p)}]-[Q_{0}]+[Q_{0}^{(p)}]\,,$

where $U_{2}$ and $Q_{0}$ have rank $1$, while $Q_{1}$ has rank $2$.

Note that here we abuse the notation $Q_{i}$ for the pull back of $Q_{i}$ to
$\mathcal{F}_{0}$.

We now set

$\ell_{i}=c_{1}(Q_{i})\quad\text{\rm for $i=0,1,2$.}$

We may consider $\ell_{i}$ as a class living on $\mathcal{F}_{i}$, but we will
denote the pull back $\pi_{0}^{*}(\ell_{1})$, $\pi_{1}^{*}(\ell_{2})$ and
$\pi_{0}^{*}(\pi_{1}^{*}(\ell_{2}))$ also by $\ell_{1}$, $\ell_{2}$ in order
to simplify notation.

Proposition 12.2 implies the following.

######  Proposition 12.3.

The total Chern class $c({\mathbb{E}})$ of the Hodge bundle on
$\mathcal{F}_{0}$ is given by

$c({\mathbb{E}})=\frac{(1-\ell_{2})(1+p\,\ell_{1}+p^{2}c_{2}(Q_{1}))(1+p\,\ell_{0})}{(1-p\,\ell_{2})(1+\ell_{1}+c_{2}(Q_{1}))(1+\ell_{0})}\,.$

######  Corollary 12.4.

We have $c_{2}(Q_{1})=(\ell_{0}^{2}+\ell_{1}^{2}-\ell_{2}^{2})/2$. Moreover,
the class $\ell_{0}^{2}$ is a pullback from $\mathcal{F}_{1}$.

###### Proof.

We deduce $\lambda_{1}=(p-1)(\ell_{0}+\ell_{1}+\ell_{2})$ and

$2\,\lambda_{2}-\lambda_{1}^{2}=(p^{2}-1)(2\,c_{2}(Q_{1})-\ell_{0}^{2}-\ell_{1}^{2}+\ell_{2}^{2})$

and since $\lambda_{1}^{2}=2\,\lambda_{2}$ on $\mathcal{A}_{4}$ the formula
for $c({\mathbb{E}})$ follows. Since $Q_{1}$ lives on $\mathcal{F}_{1}$ it
implies that the class $\ell_{0}^{2}$ is a pullback from $\mathcal{F}_{1}$. ∎

##  13\. Loci with $a$-number $\geq 2$ for $g=4$

The abelian variety corresponding to the generic point of an irreducible
component $S$ of $S_{4}$ has $a$-number equal to $1$. An irreducible component
of the closed stratum of $S$ where $a\geq 2$ is of one of two types, as shown
in [18, Section 9.9]. See also [11]. The first type maps under
$\mathcal{F}_{0}\to\mathcal{F}_{2}$ to a line on $\mathcal{F}_{2}$, while the
second type maps dominantly to $\mathcal{F}_{0}$ or maps to a point of
$\mathcal{F}_{2}({\mathbb{F}}_{p^{2}})$.

###  13.1. Loci of the first type.

The first type parametrizes flag types $M_{3}\supset M_{2}\supset M_{1}\supset
M_{0}$ such that there exists a totally isotropic subspace $I$ of
$M_{3}/FM_{3}$ such that $M_{1}\subset N$ with $N\subset M_{3}$ the submodule
generated by $I$ and $FM_{3}$. Since the automorphism group of $M_{3}$ acts
transitively on totally isotropic subspaces defined over
${\mathbb{F}}_{p^{2}}$, we may assume that $I=\langle x_{1},x_{2}\rangle$. In
terms of abelian varieties, such a flag type can be obtained from a flag type

$None$
$E^{4}=Y_{3}\buildrel{\rho_{3}}\over{\longrightarrow}Y_{2}\buildrel{\rho_{2}}\over{\longrightarrow}Y_{1}\buildrel{\rho_{1}}\over{\longrightarrow}Y_{0}$

with quasi-polarization $\eta_{3}:Y_{3}\to Y_{3}^{t}$ with
$\ker\eta_{3}=E^{4}[F^{3}]$ if the composition $\rho_{2}\rho_{3}:E^{4}\to
Y_{1}$ factors through

$1_{E^{2}}\times F_{E^{2}}:E^{4}\longrightarrow E^{2}\times E^{2}/E^{2}[F]\,.$

By identifying $E^{2}\times E^{2}/E^{2}[F]$ with $E^{4}$ and thus factoring
$\rho_{2}\rho_{3}$, we put $Z_{2}=E^{2}\times E^{2}/E^{2}[F]\cong E^{4}$ and
$Z_{1}=Y_{1}$ and then associate to it the flag

$None$
$E^{4}=Z_{2}\buildrel{\zeta_{2}}\over{\longrightarrow}Z_{1}\buildrel{\zeta_{1}}\over{\longrightarrow}Z_{0}\,,$

where $\deg(\zeta_{2})=p^{3}$ and $\deg(\zeta_{1})=p$ and $\theta_{2}:Z_{2}\to
Z_{2}^{t}$ is a quasi-polarization with kernel equal to $E^{4}[p]$.

This can be described by Dieudonné modules: consider the Dieudonné module
$N_{2}=\langle x_{1},x_{2},Fx_{3},Fx_{4}\rangle$ with
$x_{1},x_{2},x_{3},x_{4}$ the skeleton of $M_{3}$. It satisfies
$N_{2}^{t}=F^{2}N_{2}$. We choose a submodule $N_{1}$ generated by
$u=ax_{1}+bx_{2}+cFx_{3}+dFx_{4}$ and $FM_{2}$ with $\langle u,Fu\rangle=0$.
By viewing $u$ as an element of $N_{2}/FM_{2}$ and the coefficients $a,b,c,d$
in $k$ we obtain an equation

$None$ $ad^{p}-a^{p}d+bc^{p}-b^{p}c=0\,.$

Then $\dim N_{2}/N_{1}=3$ and $\dim N_{1}/N_{1}^{t}=2$. We then can choose a
Dieudonné submodule $N_{0}$ with $N_{1}^{t}\subset N_{0}\subset N_{1}$ with
$\dim N_{1}/N_{0}=1$. The filtration $N_{0}\subset N_{1}\subset N_{2}$
corresponds to (13). The moduli of $N_{2}\supset N_{1}$ defines a surface
$\mathcal{G}_{1}$ in projective space ${\mathbb{P}}^{3}$ given by (14) and
choosing $N_{0}$ defines a ${\mathbb{P}}^{1}$-bundle
$\mathcal{G}_{0}\to\mathcal{G}_{1}$.

We now discuss how to map $\mathcal{G}_{1}$ to $\mathcal{F}_{1}$. Given $u$ we
choose $v_{0}$ as a multiple of $u$. This determines a submodule $M_{2}$ of
$M_{3}$, generated by $FM_{3}$ and $v_{0}$, that contains $N_{1}$ and we set
$M_{1}=N_{1}$. Note that $M_{2}$ is generated also by $ax_{1}+bx_{2}$ and
$FM_{3}$. Then we can choose two generators $v,w$ for $M_{1}$ modulo $FM_{2}$
and assuming that $a\neq 0$ we may choose

$v=a_{5}v_{0}+cFx_{3}+dFx_{4}=ax_{1}+bx_{2}+cFx_{3}+dFx_{4},\quad w=Fx_{2}\,.$

In terms of the coordinates in Section 10 we have

$a=a_{1}a_{5},b=a_{2}a_{5},c=a_{7},d=a_{8}\,.$

The fibre of $\mathcal{G}_{1}\to\mathcal{F}_{1}$ over a point
$v_{0}=(1:t:0:0)$ of $\mathcal{F}_{2}$ consists of all $(a:b:c:d)$ with
$d^{p}-a^{p-1}d+tc^{p}-t^{p}a^{p-1}c=0$; it is defined by a Lefschetz pencil
on $\mathcal{G}_{1}$ defined by $b=ta$. We refer to the paper [14] for such a
Lefschetz fibering. The general fibre is a rational curve with one singularity
given by $a=0$.

Recall that the automorphism group of $\mathcal{F}_{2}$ acts transitively on
the set of lines of $\mathcal{F}_{2}$ defined over ${\mathbb{F}}_{p^{2}}$. For
each line $L$ defined over ${\mathbb{F}}_{p^{2}}$ on the surface
$\mathcal{F}_{2}$ we find a surface isomorphic to $\mathcal{G}_{1}$ that is
contained in the inverse image $\pi_{1}^{-1}(L)$.

The fibration $\mathcal{G}_{0}\to\mathcal{G}_{1}$ has a natural section $S$ by
taking $Z_{0}=Z_{2}/Z_{2}[F]$. Note that then $N_{0}=FN_{2}\subset N_{1}$.
This implies that $Z_{0}$, determined by $N_{0}$, is constant for all choices
of $N_{1}$. It also implies that this section is blown down under the natural
morphism $\mathcal{G}_{0}\to S_{4}\subset\mathcal{A}_{4}$ that associates to a
flag type quotient (13) the isomorphism class of $Z_{0}$. We summarize:

Let $M=A_{1,1}^{4}$ with quasi-polarization such that $M^{t}=F^{3}M$.

######  Proposition 13.1.

For each totally isotropic subspace of $M_{3}/FM_{3}$ there is a threefold
$\mathcal{G}_{0}$ that is a ${\mathbb{P}}^{1}$-bundle over a surface given by
(14) with a section and a morphism $\mathcal{G}_{0}\to\mathcal{F}_{0}$ whose
image is a locus of supersingular abelian $4$-folds with $a\geq 2$. Under
$\mathcal{G}_{0}\to\mathcal{F}_{2}$ it maps to a line on $\mathcal{F}_{2}$.
Under the morphism $\mathcal{G}_{0}\to\mathcal{A}_{4}\otimes{\mathbb{F}}_{p}$
the section of $\mathcal{G}_{0}\to\mathcal{G}_{1}$ is blown down.

###  13.2. Loci of the second type

The locus of $a$-number $\geq 2$ of the second type inside an irreducible
component $S$ of the supersingular locus $S_{4}$ is characterized by the
condition $M_{1}\subset FM_{3}$. This condition is determined on
$\mathcal{F}_{1}$.

The condition $M_{1}\subset FM_{3}$ can be paraphrased by saying that the
natural homomorphism

$None$ $M_{1}/(F,V)M_{2}\to M_{2}/FM_{3}$

induced by $M_{1}\hookrightarrow M_{2}$ is zero. Let $\mathcal{L}$ be the
sheaf corresponding to the module $M_{1}/(F,V)M_{2}$ and $U_{2}$ the one
corresponding to $VM_{2}/pM_{3}$. The invertible sheaf $U_{2}$ lives on
$\mathcal{F}_{2}$, and $\mathcal{L}$ lives on $\mathcal{F}_{1}$ and is
invertible only outside $\pi_{1}^{-1}(\mathcal{F}_{0}({\mathbb{F}}_{p^{2}}))$.
Thus we work on the open set $\mathcal{F}_{1}^{0}$ that is the complement of
$\pi_{1}^{-1}(\mathcal{F}_{2}({\mathbb{F}}_{p^{2}}))$. The homomorphism (15)
defines a homomorphism of sheaves

$\psi:\mathcal{L}\to\pi_{1}^{*}(U_{2}^{(p)})\,.$

The locus $\mathcal{H}_{1}$ can now be defined as the Zariski closure in
$\mathcal{F}_{1}$ of the zero locus $D(\psi)$ in $\mathcal{F}_{1}^{0}$ of the
map $\psi$.

######  Lemma 13.2.

The cycle class of the Zariski closure in $\mathcal{F}_{1}$ of the zero locus
$D(\psi)$ of $\psi$ equals

$[\overline{D(\psi)}]=p\,\ell_{1}-(p^{2}+1)\ell_{2}+e\,,$

where $e$ is a class with support in the fibres of $\pi_{1}$ over
$\mathcal{F}_{2}({\mathbb{F}}_{p^{2}})$.

###### Proof.

We work on the open set $\mathcal{F}_{1}^{0}$ that is the complement of
$\pi_{1}^{-1}(\mathcal{F}_{2}({\mathbb{F}}_{p^{2}}))$. Consider the exact
sequence

$0\to VM_{2}/FM_{2}\cap VM_{2}\to M_{1}/FM_{2}\to M_{1}/(F,V)M_{2}\to 0\,.$

If $M_{2}$ is generated by $FM_{3}$ and
$v_{0}=a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}+a_{4}x_{4}$ with
$(a_{1}:a_{2}:a_{3}:a_{4})$ determining a point not in
$\mathcal{F}_{2}({\mathbb{F}}_{p^{2}})$ then $FM_{2}\cap VM_{2}=pM_{3}$ and

$VM_{2}/FM_{2}\cap VM_{2}=VM_{2}/pM_{3}.$

This translates into a short exact sequence of
$\mathcal{O}_{\mathcal{F}_{1}^{0}}$-modules

$0\to\pi_{1}^{*}(U_{2})\to U_{1}^{(p)}\to\mathcal{L}\to 0$

with $U_{1}$ the locally free $\mathcal{O}_{\mathcal{F}_{1}}$-module
determined by $VM_{1}/pM_{2}$. We view $\psi$ as a section of
$\pi_{1}^{*}(U_{2}^{(p)})\otimes\mathcal{L}^{-1}$ on $\mathcal{F}_{1}^{0}$
with class $(p+1)[U_{2}]-p[U_{1}]$. From Section 12 we use the identities
$[U_{2}]=[4]-[Q_{2}]$, $[U_{1}]=[{\rm
Lie}(Y_{2})^{\vee}]-[Q_{1}]=[4]-[Q_{2}]+[Q_{2}^{(p)}]-[Q_{1}]$, hence
$c_{1}((p+1)[U_{2}]-p[U_{1}])=p\ell_{1}-(p^{2}+1)\ell_{2}$. When taking the
closure of the degeneracy locus of $\psi$ we have to take into account a class
with support in the fibres over $\mathcal{F}_{2}({\mathbb{F}}_{p^{2}})$ and
the result follows. ∎

######  Corollary 13.3.

An irreducible component of the locus with $a\geq 2$ of the second type maps
dominantly to $\mathcal{F}_{2}$ or is contained in the fibres over
$\mathcal{F}_{2}({\mathbb{F}}_{p^{2}})$.

###### Proof.

If an irreducible component is not contained in the fibres over
$\mathcal{F}_{2}({\mathbb{F}}_{p^{2}})$ then it is contained in
$\overline{D(\psi)}$ and the class is given by Lemma 13.2. The intersection
number of $\overline{D(\psi)}$ with a generic fibre of $\pi_{1}$ equals the
degree of $\ell_{1}$ on such a fibre, that is, $1$. That means that it
intersects the generic fibre of $\pi_{1}$ and is irreducible. ∎

The resulting abelian variety of a flag type with $M_{1}\subset FM_{3}$ is
defined by the filtration of Dieudonné modules $FM_{3}\supset M_{1}\supset
M_{0}$. By forgetting $M_{1}$ between $FM_{3}$ and $M_{0}$, we have polarized
flag types

$\textstyle{E^{4}=Z_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{1}}$$\scriptstyle{\eta_{1}}$$\textstyle{Z_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{E^{4}=Z_{1}^{t}}$$\textstyle{Z_{0}^{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

with $\ker(\eta_{1})=E^{4}[F]$ and $\ker(\rho_{1})\cong\alpha_{p}^{2}$. The
choice of $\ker(\rho_{1})$ in $E^{4}[F]$ defines a point in the Grassmann
variety $G={\rm Gr}(2,4)$. Note that $G$ can be identified with a quadric in
${\mathbb{P}}^{5}$ in terms of Plücker coordinates. If we choose a basis
$x_{1},x_{2},x_{3},x_{4}$ of the Dieudonné module of $E^{4}$ with
$(F-V)x_{i}=0$ and quasi-polarization with

$\langle x_{i},p\,x_{5-j}\rangle=\delta_{ij},\quad\langle
x_{i},Fx_{j}\rangle=0\,,$

then $M_{0}$ can be generated by two vectors $a=\sum_{i=1}^{4}a_{i}x_{i}$,
$b=\sum_{i=1}^{4}b_{i}x_{i}$ and the condition $\langle a,b\rangle\in W$ says

$a_{1}b_{4}-a_{4}b_{1}+a_{2}b_{3}-a_{3}b_{2}\equiv\,0\,(\bmod\,p)$

and this defines a hyperplane section $\mathcal{Q}=H\cap G$ of the Grassmann
variety. Indeed, with the Plücker coordinates
$\lambda_{ij}=a_{i}b_{j}-a_{j}b_{i}$ the variety $G$ is given by
$\lambda_{12}\lambda_{34}-\lambda_{13}\lambda_{24}+\lambda_{14}\lambda_{23}=0$
and $H$ by $\lambda_{14}+\lambda_{23}=0$.

We summarize.

######  Lemma 13.4.

The image in $S_{4}$ of an irreducible component of the locus of second type
with $a\geq 2$ can be identified with a hyperplane section $\mathcal{Q}$ of
the Grassmann variety ${\rm Gr}(2,4)$.

We will analyze the case of loci of the second type contained in the fibres
over ${\mathbb{F}}_{p^{2}}$-rational points on $\mathcal{F}_{2}$ in the next
section.

##  14\. The fibres over $\mathcal{F}_{2}({\mathbb{F}}_{p^{2}})$

Here we study the fibre under $\mathcal{F}_{0}\to\mathcal{F}_{2}$ of a
rational point $\xi\in\mathcal{F}_{2}({\mathbb{F}}_{p^{2}})$. Since
$\mathcal{F}_{0}\to\mathcal{F}_{1}$ is a ${\mathbb{P}}^{1}$-bundle it suffices
to study the fibre under $\mathcal{F}_{1}\to\mathcal{F}_{2}$.

The automorphism group ${\rm Aut}(\mathcal{F}_{2})$ can be identified with the
general unitary group ${\rm GU}_{4}(p^{2})$ of $4$-dimensional space over
${\mathbb{F}}_{p^{2}}$ that fixes the Hermitian form

$\xi_{1}\bar{\xi}_{4}-\xi_{4}\bar{\xi}_{1}+\xi_{2}\bar{\xi}_{3}-\xi_{3}\bar{\xi}_{2}$

where $\bar{\xi}=\xi^{p^{2}}$. By a theorem of Witt this group acts
transitively on isotropic subspaces of dimension $1$ and $2$. This implies
that it acts transitively on the set of lines of $\mathcal{F}_{2}$ and on the
set of ${\mathbb{F}}_{p^{2}}$-rational points, see [14, Appendix]. We thus may
restrict to analyzing the fibre over the point $(1:0:0:0)$ of
$\mathcal{F}_{2}$. This corresponds to the case with $M_{2}\subset M_{3}$
generated by $v_{0}=x_{1}$ and $FM_{3}$. The fibre $\pi_{1}^{-1}(\xi)$
corresponds to the choices of $M_{1}$. It can be given by a choice of basis

$v=a_{5}v_{0}+a_{6}Fx_{2}+a_{7}Fx_{3}+a_{8}Fx_{4},\quad
w=a_{9}Fx_{2}+a_{10}Fx_{3}+a_{11}Fx_{4},$

satisfying the equations $g_{1}$, $g_{2}$ and $g_{3}$ of Section 10

$a_{8}^{p}-a_{5}^{p-1}a_{8}=0,\quad a_{11}^{p}=0,\quad a_{11}=0\,.$

We distinguish whether $a_{5}\neq 0$ or $a_{5}=0$.

Case i). $a_{5}\neq 0$. We may assume $a_{5}=1$ and find
$a_{8}\in{\mathbb{F}}_{p}$. We can change basis of $M_{3}$ by
$(x_{1},x_{2},x_{3},x_{4})\mapsto(x_{1}+a_{8}Fx_{4},x_{2},x_{3},x_{4})$ and
then may assume that $a_{8}=0$. Then $M_{1}$ is generated inside $M_{2}$ by
$FM_{2}=\langle Fx_{1},F^{2}x_{2},F^{2}x_{3},F^{2}x_{4}\rangle$ and
$v=x_{1}+a_{6}Fx_{2}+a_{7}Fx_{3}$ and $w=a_{9}Fx_{2}+a_{10}Fx_{3}$. We now
construct a flag of Dieudonné modules

$F^{2}M_{3}^{\prime}\subset M_{1}^{t}\subset M_{1}\subset FM_{3}^{\prime}$

with $M_{3}^{\prime}=\langle F^{-1}x_{1},x_{2},x_{3},Fx_{4}\rangle$ and show
that we can extend it to a flag type

$None$ $M_{1}\subset F^{2}M_{3}^{\prime}\subset M_{2}^{\prime}\subset
M_{3}^{\prime}$

so that we can associate to it a point of a locus of the second type as
treated in the Section 13.2 with respect to a changed basis $\langle
F^{-1}x_{1},x_{2},x_{3},Fx_{4}\rangle$ of $M_{3}$. To prove our claim we have
to construct $M_{2}^{\prime}$ with $(F,V)M_{2}^{\prime}\subset M_{1}$. We take
$v_{0}^{\prime}=\alpha_{1}F^{-1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+Fx_{4}$
and impose the following conditions

  1. (1)

$\langle v_{0}^{\prime},Fv_{0}^{\prime}\rangle\in W$, that is,
$\alpha_{1}-\alpha_{1}^{p^{2}}+\alpha_{2}\alpha_{3}^{p^{2}}-\alpha_{2}^{p^{2}}\alpha_{3}=0$,

  2. (2)

$Fv_{0}^{\prime}\in M_{1}$, equivalently, there exists $\beta$ with
$Fv_{0}^{\prime}=\alpha_{1}^{p}+\beta w+F^{2}x_{4}$, that is,
$\alpha_{2}^{p}=\alpha_{1}^{p}a_{6}+\beta a_{9}$ and
$\alpha_{3}^{p}=\alpha_{1}^{p}a_{7}+\beta a_{10}$,

  3. (3)

$Vv_{0}^{\prime}\in M_{1}$, equivalently, there exists $\gamma$ with
$Vv_{0}^{\prime}=\alpha^{1/p}v+\gamma w+F^{2}x_{4}$, that is,
$\alpha_{2}^{1/p}=\alpha_{1}^{1/p}a_{6}+\gamma a_{9}$ and
$\alpha_{3}^{1/p}=\alpha_{1}^{1/p}a_{7}+\gamma a_{10}$.

For generic $a_{i}$ (that is, $a_{7}a_{9}-a_{6}a_{10}$ and $a_{9}a_{10}$ not
in ${\mathbb{F}}_{p^{2}}$) we find a solution. We then set

$M_{2}^{\prime}=Av_{0}^{\prime}+FM_{3}^{\prime}$

and then by (2) we have $\langle v,w,FM_{2}^{\prime}\rangle=\langle
v,w,FM_{2}\rangle=M_{1}$. Thus we have a filtration (16) and it gives a point
of a locus $\mathcal{H}_{1}$ with respect to the module $M_{3}^{\prime}$.

Case ii). Here $a_{5}=0$. Then by $g_{2}$ we have $a_{11}=0$ and find that
$M_{1}$ is generated by $v=a_{6}Fx_{2}+a_{7}Fx_{3}$,
$w=a_{9}Fx_{2}+a_{10}Fx_{3}$ and $F^{2}M_{3}$, hence $M_{1}=\langle
Fx_{1},Fx_{2},Fx_{3},F^{2}x_{4}\rangle$. So $M_{1}$ is fixed and this case
thus yields one point. Moreover $M_{1}^{t}=\langle
Fx_{1},F^{2}x_{2},F^{2}x_{3},F^{2}x_{4}\rangle$.

We thus see that the supersingular abelian variety corresponding to a generic
point of an irreducible component $\mathcal{E}$ of the fibre over a rational
point $\xi\in\mathcal{F}_{2}({\mathbb{F}}_{p^{2}})$ can be viewed as the
supersingular abelian variety defined by a generic point of a locus with
$a\geq 2$ of the second kind with $M_{1}\subset FM_{3}^{\prime}$.

This means that there is an irreducible component $S^{\prime}$ of $S_{4}$ with
model $\mathcal{F}_{0}^{\prime}$ and a locus $\mathcal{H}_{0}^{\prime}$
mapping dominantly to $\mathcal{F}_{2}^{\prime}$ such that image of
$\mathcal{E}$ and $\mathcal{H}_{0}^{\prime}$ coincide in
$S_{4}\subset\mathcal{A}_{4}$.

We summarize.

######  Proposition 14.1.

Let $S$ be a component of $S_{4}$ and $\mathcal{F}_{0}$ be the model
constructed in Section 10. The fibre in $\mathcal{F}_{0}$ over a rational
point $\xi\in\mathcal{F}_{2}({\mathbb{F}}_{p^{2}})$ consists of $p$
irreducible components. The image of each of these in $S_{4}$ is a hyperplane
section of the Grassmann variety ${\rm Gr}(2,4)$ and can be seen as the image
of a locus of $a\geq 2$ of the second type in another component $S^{\prime}$
of $S_{4}$.

##  15\. Superspecial points of $S_{4}$

The number of points of $S_{4}$ representing isomorphism classes of
superspecial abelian varieties counted in the stacky sense was given in
formula (2) in Section 2 and equals

$\Sigma_{4}=(p-1)(p^{2}+1)(p^{3}-1)(p^{4}+1)\,v(4)\,.$

Each superspecial principally polarized abelian variety of dimension $4$
defines a ${\mathbb{F}}_{p^{2}}$-rational point of
$S_{4}\subset\mathcal{A}_{4}$. By Proposition 3.1 we have
$N_{4}=(p^{2}-1)(p^{6}-1)\,v(4)$ irreducible components (again counted in the
stacky sense) of $S_{4}$. Each irreducible component is the image of
$\mathcal{F}_{0}$ under a degree $p$ morphism to its image in $S_{4}$ that
induces a bijection between geometric points.

######  Lemma 15.1.

We have
$\\#\mathcal{F}_{0}({\mathbb{F}}_{p^{2}})=(p^{2}+1)^{3}(p^{3}+1)(p^{4}+1)$.

###### Proof.

We have $\\#\mathcal{F}_{2}({\mathbb{F}}_{p^{2}})=(p^{2}+1)(p^{4}+1)$, see for
example [14], hence
$\\#\tilde{\mathcal{F}}_{0}({\mathbb{F}}_{p^{2}})=(p^{2}+1)^{2}(p^{4}+1)$ and
these points are the ${\mathbb{F}}_{p^{2}}$-rational points on the exceptional
curves of $\tilde{\mathcal{F}}_{2}$. The fibre in $\mathcal{F}_{1}$ over a
${\mathbb{F}}_{p^{2}}$-rational point of $\tilde{\mathcal{F}}_{2}$ consists of
a union of $p$ lines through one point. So we find
$\\#\mathcal{F}_{1}({\mathbb{F}}_{p^{2}})=(p^{2}+1)^{2}(p^{4}+1)(p^{3}+1)$.
Since $\mathcal{F}_{0}$ is a ${\mathbb{P}}^{1}$-bundle over $\mathcal{F}_{1}$
the formula follows. ∎

Let $J$ be the set of irreducible components of $S_{4}$ and for $j\in J$ we
let $\mathcal{F}_{0}^{j}$ be the corresponding smooth model. The disjoint
union of these smooth models has

$\\#(\bigsqcup_{j\in
J}\mathcal{F}_{0}^{j})({\mathbb{F}}_{p^{2}})=N_{4}\,(p^{2}+1)^{3}(p^{3}+1)(p^{4}+1)$

${\mathbb{F}}_{p^{2}}$-rational points mapping to $\Sigma_{4}$ superspecial
points of $S_{4}$. The variety $\mathcal{F}_{0}$ contains $(p^{2}+1)(p^{4}+1)$
loci $\mathcal{G}_{0}^{n}$ of the first kind, each isomorphic to
$\mathcal{G}_{0}$. We have
$\\#\mathcal{G}_{1}({\mathbb{F}}_{p^{2}})=(p^{2}+1)(p^{3}+1)$ (see [14]) and
$\\#\mathcal{G}_{0}({\mathbb{F}}_{p^{2}})=(p^{2}+1)^{2}(p^{3}+1)$ since
$\mathcal{G}_{0}$ is a ${\mathbb{P}}^{1}$-bundle over $\mathcal{G}_{1}$. On
$\mathcal{F}_{1}$ these loci $\mathcal{G}_{1}$ of the first kind are disjoint
and we see

$\\#\mathcal{F}_{0}({\mathbb{F}}_{p^{2}})=(p^{2}+1)(p^{4}+1)\,\\#\mathcal{G}_{0}({\mathbb{F}}_{p^{2}})\,.$

On each component $\mathcal{G}_{0}^{n}$ a section of
$\mathcal{G}_{0}\to\mathcal{G}_{1}$ is blown down. This section has
$(p^{2}+1)(p^{3}+1)$ points rational over ${\mathbb{F}}_{p^{2}}$.

######  Lemma 15.2.

Each superspecial point of $S_{4}$ lies on $(p+1)(p^{3}+1)$ irreducible
components of $S_{4}$.

###### Proof.

The number of totally isotropic subspaces of dimension $2$ in a
$4$-dimensional unitary space over ${\mathbb{F}}_{p^{2}}$ with conjugation
given by Frobenius is equal to $(p+1)(p^{3}+1)$. A choice of an irreducible
component corresponds exactly to the choice of a totally isotropic subspace. ∎

We thus see that under the natural map

$\bigsqcup_{j\in J}\mathcal{F}_{0}^{j}\longrightarrow S_{4}$

the inverse image of each of the $\Sigma_{4}$ superspecial points of $S_{4}$
has

$(p+1)(p^{3}+1)\times(p^{2}+1)(p^{3}+1)\times(p^{2}+1)$

points, where the second factor corresponds to blowing down the section of
$\mathcal{G}_{0}\to\mathcal{G}_{1}$, and the third one comes from the fact
that each exceptional curve on $\tilde{\mathcal{F}}_{2}$ intersects $p^{2}+1$
proper images of the lines defined over ${\mathbb{F}}_{p^{2}}$, in agreement
with the formula

$N_{4}\,(p^{2}+1)^{3}(p^{3}+1)(p^{4}+1)=\Sigma_{4}\,(p+1)(p^{2}+1)^{2}(p^{3}+1)^{2}\,.$

##  16\. The cycle class of $S_{4}$ and intersection numbers

In this section we express the cycle class of the supersingular locus $S_{4}$
for dimension $g=4$ in terms of intersection numbers.

We know that the cycle class of $S_{4}$ lies in the tautological ring and is a
multiple of $\lambda_{4}\lambda_{2}$. This multiple can be determined by
intersection numbers. We identify the degree of a top-dimensional Chern class
with an intersection number.

######  Proposition 16.1.

We have $[S_{4}]=a\,\lambda_{4}\lambda_{2}$ with

$a=\frac{\lambda_{3}\lambda_{1}\,[S_{4}]}{v(4)}=\frac{\lambda_{1}^{4}[S_{4}]}{8\,v(4)}$

with $v(4)$ the proportionality constant defined in Section 2.

###### Proof.

We have
$\lambda_{3}\lambda_{1}[S_{4}]=a\,\lambda_{4}\lambda_{3}\lambda_{2}\lambda_{1}=a\,v(4)$.
In the tautological ring $R_{4}$ we have
$\lambda_{3}\lambda_{1}=\lambda_{1}^{4}/8$. ∎

We shall calculate the intersection number $[S]\cdot\lambda_{3}\lambda_{1}$
for each irreducible component $S$ of $S_{4}$. We will do this by pulling back
the Hodge bundle of $\mathcal{A}_{4}$ to $\mathcal{F}_{0}$ and calculating the
degrees of the top Chern classes of the Hodge bundle on $\mathcal{F}_{0}$.

##  17\. Determination of intersection numbers

Our goal is to calculate the intersection number $\lambda_{1}\lambda_{3}[S]$
for each irreducible component $S$ of the supersingular locus. For this we
calculate $\deg(\lambda_{3}\lambda_{1})$ on the $4$-dimensional variety
$\mathcal{F}_{0}$.

By Proposition 12.3, that describes the total Chern class of the Hodge bundle,
and by Corollary 12.4 this intersection number can be expressed in the
intersection numbers given by the monomials of degree $4$ in
$\ell_{0},\ell_{1},\ell_{2}$ evaluated at the fundamental class of
$\mathcal{F}_{0}$. Note that we write $\ell_{1}$ and $\ell_{2}$ for their
pullbacks to $\mathcal{F}_{0}$ and sometimes identify such a monomial
$\ell_{0}^{a}\ell_{1}^{b}\ell_{2}^{c}$ with
$\deg(\ell_{0}^{a}\ell_{1}^{b}\ell_{2}^{c})$.

######  Lemma 17.1.

The following intersection numbers vanish on $\mathcal{F}_{0}$:

$\ell_{0}^{4},\ell_{0}^{2}\ell_{1}^{2},\ell_{0}^{2}\ell_{1}\ell_{2},\ell_{0}^{2}\ell_{2}^{2},\ell_{0}\ell_{2}^{3},\ell_{1}^{4},\ell_{1}^{3}\ell_{2},\ell_{1}^{2}\ell_{2}^{2},\ell_{1}\ell_{2}^{3},\ell_{2}^{4}\,.$

###### Proof.

Since $\dim\mathcal{F}_{1}=3$ and $\ell_{0}^{2}$ is a pullback from
$\mathcal{F}_{1}$ by Corollary 12.4, and $\ell_{1}$ and $\ell_{2}$ are also
pullbacks from $\mathcal{F}_{1}$ we find that
$\ell_{0}^{4},\ell_{0}^{2}\ell_{1}^{2},\ell_{0}^{2}\ell_{1}\ell_{2},\ell_{0}^{2}\ell_{2}^{2}$
vanish.

Since the class $\ell_{2}$ is a pullback from $\mathcal{F}_{2}$, which is of
dimension $2$, we have $\ell_{2}^{3}=0$, implying that
$\ell_{2}^{4}=\ell_{0}\ell_{2}^{3}=\ell_{1}\ell_{2}^{3}=0$. Similarly,
$\ell_{2}$ and $\ell_{1}$ are induced from $\mathcal{F}_{1}$, which is of
dimension $3$, hence the monomials of degree $4$ in $\ell_{1}$ and $\ell_{2}$
vanish. ∎

Proposition 12.3 together with Lemma 17.1 imply the following relation.

######  Corollary 17.2.

We have on $\mathcal{F}_{0}$

$\deg(\lambda_{3}\lambda_{1})=\frac{1}{2}(p-1)^{4}\left(\ell_{0}^{3}\ell_{1}+\ell_{0}^{3}\ell_{2}+\ell_{0}\ell_{1}^{3}+3\,\ell_{0}\ell_{1}^{2}\ell_{2}+3\,\ell_{0}\ell_{1}\ell_{2}^{2}\right)\,.$

Thus we need the intersection numbers defined by the five monomials in
$\ell_{0},\ell_{1},\ell_{2}$ appearing in Corollary 17.2.

The intersection numbers
$(\ell_{0}\ell_{1}^{3},\ell_{0}\ell_{1}^{2}\ell_{2},\ell_{0}\ell_{1}\ell_{2}^{2})$
on $\mathcal{F}_{0}$ are equal to the intersection numbers
$(\ell_{1}^{3},\ell_{1}^{2}\ell_{2},\ell_{1}\ell_{2}^{2})$ on
$\mathcal{F}_{1}$ as the degree of $\ell_{0}$ on a generic fibre of $\pi_{0}$
is $1$.

######  Lemma 17.3.

We have $\deg\ell_{1}\ell_{2}^{2}=p^{2}(p^{2}+1)$ on $\mathcal{F}_{1}$.

###### Proof.

The space $\mathcal{F}_{2}$ can be identified with the surface in
${\mathbb{P}}^{3}$ over ${\mathbb{F}}_{p}$ given by the equation

$x_{1}x_{4}^{p^{2}}-x_{1}^{p^{2}}x_{4}+x_{2}x_{3}^{p^{2}}-x_{2}^{p^{2}}x_{3}=0$

and $\ell_{2}$ is represented by the pullback under $\pi_{1}$ of the
hyperplane class $h$ on $\mathcal{F}_{2}$. Therefore $h^{2}$ can be
represented by an effective zero cycle of degree $p^{2}+1$. The surface
$\mathcal{F}_{2}$ is unirational (see [14]), hence $h^{2}$ can be represented
by $p^{2}+1$ times a point. The morphism $\pi_{1}$ is inseparable of degree
$p$, hence the pullback of a point $\mathcal{F}_{2}$ is $p$ times a fibre of
$\mathcal{F}_{1}$. Since the degree of $\ell_{1}$ on a fibre of $\pi_{1}$ is
$p$ we get $\deg(\ell_{1}\ell_{2}^{2})=p\cdot p\cdot(p^{2}+1)$. ∎

######  Lemma 17.4.

We have on $\mathcal{F}_{0}$ the relation

$p\,\ell_{0}^{3}\ell_{1}-(p^{2}+1)\,\ell_{0}^{3}\ell_{2}+p\,\ell_{0}\ell_{1}^{3}-(p-1)^{2}\,\ell_{0}\ell_{1}^{2}\ell_{2}-(2p^{2}-p+2)\ell_{0}\ell_{1}\ell_{2}^{2}=0\,.$

###### Proof.

This follows from the fact that $\lambda_{4}$ vanishes in the Chow ring of
$\mathcal{A}_{g}$ as explained in Section 2 and the expression for
$\lambda_{4}$ as a polynomial in the $\ell_{i}$ by Proposition 12.3 and
Corollary 12.4. ∎

######  Corollary 17.5.

On $\mathcal{F}_{1}$ we have the relation

$p\,\ell_{0}^{2}\ell_{1}-(p^{2}+1)\,\ell_{0}^{2}\ell_{2}+p\,\ell_{1}^{3}-(p-1)^{2}\,\ell_{1}^{2}\ell_{2}-(2p^{2}-p+2)\ell_{1}\ell_{2}^{2}=0\,.$

###### Proof.

We know that $\mathcal{F}_{0}$ is a ${\mathbb{P}}^{1}$-bundle over
$\mathcal{F}_{1}$. Therefore each cycle class $\xi\in A_{k}(\mathcal{F}_{0})$
can be written uniquely as
$\xi=\pi_{0}^{*}(\xi_{0})+\pi_{0}^{*}(\xi_{1})\ell_{0}$ with $\xi_{0}\in
A_{k-1}(\mathcal{F}_{1})$ and $\xi_{1}\in A_{k}(\mathcal{F}_{1})$. In
particular, the map $\xi_{1}\mapsto\pi_{0}^{*}(\xi_{1})\ell_{0}$ is injective.
The result thus follows from Lemma 17.4. ∎

######  Lemma 17.6.

We have on $\mathcal{F}_{1}$ the relation

$2\,\ell_{0}^{2}\ell_{1}-(p-1)\ell_{0}^{2}\ell_{2}+(p-1)\ell_{1}^{2}\ell_{2}-2(p^{2}-p+1)\ell_{1}\ell_{2}^{2}=0\,.$

###### Proof.

We have the exact sequence of Dieudonné modules

$0\to A\to VM_{2}/pM_{2}\to VM_{2}/VM_{1}\to 0\,$

with ${\rm Lie}(Y_{2})^{\vee}=VM_{2}/pM_{2}$ and $Q_{1}=VM_{2}/VM_{1}$. The
total Chern class of the sheaf corresponding to $A$ has the form

$c(A)=(1-\ell_{2})(1-p\,\ell_{2})^{-1}(1+\ell_{1}+c_{2}(Q_{1}))^{-1}\,.$

Since ${\rm rank}(A)=2$ the third Chern class should vanish; this gives a
relation on $\mathcal{F}_{1}$

$2\ell_{0}^{2}\ell_{1}-(p-1)\ell_{0}^{2}\ell_{2}+(p-1)\ell_{1}^{2}\ell_{2}-2(p^{2}-p+1)\ell_{1}\ell_{2}^{2}=0\,.$

∎

######  Lemma 17.7.

On $\mathcal{F}_{1}$ we have the relation

$p\,\ell_{0}^{2}\ell_{2}-p\,\ell_{1}^{2}\ell_{2}+2(p^{2}-p+1)\ell_{1}\ell_{2}^{2}=0\,.$

###### Proof.

Let $H$ be a hyperplane section of $\mathcal{F}_{2}$ with
$H\cap\mathcal{F}({\mathbb{F}}_{p^{2}})=\emptyset$. We work on
$\pi_{1}^{-1}(H)$. Here we have that $\dim M_{2}/(F,V)M_{2}=3$ and we thus
have a rank $3$ locally free sheaf $B$ on $H$ determined by
$M_{2}/(F,V)M_{2}$. Because of the exact sequence

$0\to VM_{2}/VM_{2}\cap FM_{2}\to M_{2}/FM_{2}\to M_{2}/(F,V)M_{2}\to 0$

we have the exact sequence

$0\to U_{2}\to{{\rm Lie}(Y_{2})^{(p)}}^{\vee}\to B\to 0\,,$

since $VM_{2}/VM_{2}\cap FM_{2}=VM_{2}/pM_{3}$. We thus find

$[B]=[4]-[Q_{2}^{(p)}]+[Q_{2}^{(p^{2})}]-[U_{2}]=[4]+[U_{2}^{(p)}]-[U_{2}^{(p^{2})}]-[U_{2}]\,.$

We also have the inclusions $(F,V)M_{2}\subset M_{1}\subset M_{2}$ on
$\pi_{1}^{-1}(H)$ and we thus have a locally free sheaf $L$ corresponding to
$M_{1}/(F,V)M_{2}$. In the Grothendieck group we have the corresponding
relation $[B]=[L]+[Q_{1}^{(p)}]$. Thus we find
$[L]=[4]+[U_{2}^{(p)}]-[U_{2}^{(p^{2})}]-[U_{2}]-[Q_{1}^{(p)}]$ and we see
that the total Chern class of $L$ is given by

$c(L)=\frac{(1-p\ell_{2})}{(1-p^{2}\ell_{2})(1-\ell_{2})}\,\frac{1}{(1+p\ell_{1}+p^{2}c_{2}(Q_{1}))}\,.$

But $L$ has rank $1$, so $c_{2}(L)=0$. With
$c_{2}(Q_{1})=(\ell_{0}^{2}+\ell_{1}^{2}-\ell_{2}^{2})/2$ this gives

$(p^{4}-p^{3}+\frac{3}{2}p^{2}-p+1)\ell_{2}^{2}-(p^{3}-p^{2}+p)\ell_{1}\ell_{2}-\frac{1}{2}p^{2}\ell_{0}^{2}+\frac{1}{2}p^{2}\ell_{1}^{2}=0\,.$

Recall now that the class of $H$ is $\ell_{2}$. Multiplying the preceding
relation by $\ell_{2}$ and using $\ell_{2}^{3}=0$ we find

$p\ell_{0}^{2}\ell_{2}-p\ell_{1}^{2}\ell_{2}+2(p^{2}-p+1)\ell_{1}\ell_{2}^{2}=0\,.$

∎

As remarked above we need five intersection numbers:

$\ell_{0}^{3}\ell_{1},\,\ell_{0}^{3}\ell_{2},\,\ell_{0}\ell_{1}^{3},\,\ell_{0}\ell_{1}^{2}\ell_{2},\,\ell_{0}\ell_{1}\ell_{2}^{2}\,.$

We know already the last one by Lemma 17.3. By multiplying the relations of
Lemma 17.6 and 17.7 by $\ell_{0}$ we find in total three relations coming from
Lemmas 17.4, 17.6 and 17.7 between these five intersection numbers.

######  Corollary 17.8.

We have $\deg(\ell_{0}^{3}\ell_{1})=p(p^{2}+1)(p^{2}-p+1)$.

###### Proof.

The sum of $p$ times the relation of 17.6 and $(p-1)$ times that of 17.7 gives
the relation
$2p\,\ell_{0}^{3}\ell_{1}-2(p^{2}-p+1)\ell_{0}\ell_{1}\ell_{2}^{2}=0$. ∎

Using the three relations and Lemma 17.3 our five intersection numbers depend
on one unknown.

######  Corollary 17.9.

With $x=\deg(\ell_{0}\ell_{1}^{2}\ell_{2})$ we find that

$\deg\left[\begin{matrix}\ell_{0}^{3}\ell_{1}\\\ \ell_{0}^{3}\ell_{2}\\\
\ell_{0}\ell_{1}^{3}\\\ \ell_{0}\ell_{1}^{2}\ell_{2}\\\
\ell_{0}\ell_{1}\ell_{2}^{2}\\\
\end{matrix}\right]=\left[\begin{matrix}p(p^{2}+1)(p^{2}-p+1)\\\
x-2p(p^{2}+1)(p^{2}-p+1)\\\ 2(p-1+1/p)x-(p^{2}+1)^{2}(2p^{2}-3p+2)\\\ x\\\
p^{2}(p^{2}+1)\\\ \end{matrix}\right]$

######  Remark 17.10.

We have on $\mathcal{F}_{0}$

$\deg\lambda_{1}^{4}=8\,(p-1)^{4}(p^{2}+p+1)\left(\frac{\deg(\ell_{0}\ell_{1}^{2}\ell_{2})}{p}-(p^{2}+1)(p-1)^{2}\right)\,.$

Since $\lambda_{1}$ is ample on $S_{4}$ this should be positive and this gives

$\deg(\ell_{0}\ell_{1}^{2}\ell_{2})>p(p^{2}+1)(p-1)^{2}\,.$

We now determine the last intersection number. Recall that the second Chern
class $c_{2}(Q_{1})$ satisfies
$c_{2}(Q_{1})=(\ell_{0}^{2}+\ell_{1}^{2}-\ell_{2}^{2})/2$. Furthermore, recall
the cycle class $[\overline{D(\psi)}]$ of a ‘horizontal’ $a\geq 3$-locus on
$\mathcal{F}_{1}$ given by

$[\overline{D(\psi)}]=p\,\ell_{1}-(p^{2}+1)\ell_{2}+e$

with $e$ a class with support in the exceptional fibres as given in Lemma
13.2.

######  Proposition 17.11.

We have $c_{2}(Q_{1})\cdot[\overline{D(\psi)}]=0$ and $c_{2}(Q_{1})\cdot e=0$.

###### Proof.

Since $Q_{1}$ is the tautological quotient of the
$O_{{\mathcal{F}}_{1}}$-module associated to $M_{2}/FM_{2}$ by the universal
rank $2$ subbundle $U_{1}$, the second Chern class can be realized as the
class of the locus where the fibre of $U_{1}$ contains a fixed vector. For
this we choose an element $v^{\prime}$ of $M_{2}/FM_{2}$ that has the property
that over each affine part of $\mathcal{F}_{2}$ with $a_{i}\neq 0$ (for
$i=1,\ldots,4$) it is of the form

$v^{\prime}=\alpha_{5}\,v_{0}+\alpha_{6}\,Fx_{2}+\alpha_{7}Fx_{3}+\alpha_{8}Fx_{4}$

with the property that the equation $g_{2}=0$, that is,

$a_{1}\,\alpha_{8}^{p}-a_{1}^{p}\alpha_{5}^{p-1}\alpha_{8}+a_{2}\alpha_{7}^{p}-a_{2}^{p}\alpha_{5}^{p-1}\alpha_{7}+a_{3}^{p}\alpha_{5}^{p-1}\alpha_{6}-a_{3}\alpha_{6}^{p}=0$

has no solutions with $(a_{1},a_{2},a_{3},a_{4})\in{\mathbb{F}}_{p^{2}}$ with
$a_{i}\neq 0$. Indeed, choosing $\alpha_{5}\neq 0$, $\alpha_{6}$ and
$\alpha_{7}$ there are only finitely many $\alpha_{8}$ satisfying this
equation. Then since $\alpha_{5}\neq 0$, we see that this locus has zero
intersection with $\overline{D(\psi)}$. We get
$c_{2}(Q_{1})\cdot[\overline{D(\psi)}]=0$. By the requirement that we put over
$\mathcal{F}_{2}({\mathbb{F}}_{p^{2}})$ we see that also $c_{2}(Q_{1})\cdot
e=0$. ∎

######  Corollary 17.12.

We have
$(\ell_{0}^{2}+\ell_{1}^{2}-\ell_{2}^{2})(p\ell_{1}-(p^{2}+1)\ell_{2})=0$.

###### Proof.

Recall that $c_{2}(Q_{1})=(\ell_{0}^{2}+\ell_{1}^{2}-\ell_{2}^{2})/2$ and
$[\overline{D(\psi)}]=p\ell_{1}-(p^{2}+1)\ell_{2}+e$ with $e$ a class with
support in the exceptional fibres. ∎

By combining Corollary 17.9 and Corollary 17.12 we can determine all the
intersection numbers.

######  Corollary 17.13.

We have on $\mathcal{F}_{0}$

$\deg\left[\begin{matrix}\ell_{0}^{3}\ell_{1}\\\ \ell_{0}^{3}\ell_{2}\\\
\ell_{0}\ell_{1}^{3}\\\ \ell_{0}\ell_{1}^{2}\ell_{2}\\\
\ell_{0}\ell_{1}\ell_{2}^{2}\\\
\end{matrix}\right]=p\,(p^{2}+1)\,\left[\begin{matrix}p^{2}-p+1\\\
-p^{2}+p-1\\\ -(p-1)^{2}\\\ p^{2}-p+1\\\ p\\\ \end{matrix}\right]$

Finally we are ready to calculate the coefficient $f_{4}(p)$ of Theorem 1.1.

######  Theorem 17.14.

The class of the supersingular locus
$S_{4}\subset\mathcal{A}_{4}\otimes{\mathbb{F}}_{p}$ in the Chow ring of
$\tilde{\mathcal{A}}_{4}\otimes{\mathbb{F}}_{p}$ equals

$[S_{4}]=(p-1)^{3}(p^{3}-1)(p^{4}-1)(p^{6}-1)\lambda_{4}\lambda_{2}\,.$

###### Proof.

For each irreducible component $S$ of $S_{4}$ we calculate the degree of
$\lambda_{3}\lambda_{1}$ on the model $\mathcal{F}_{0}$ of $S$. Indeed, we
have $[S_{4}]=a\lambda_{4}\lambda_{2}$ with
$a=\lambda_{3}\lambda_{1}[S_{4}]/v(4)$ by Proposition 16.1. A calculation
using Corollary 17.13 and taking into account the degree $p$ of the map
$\mathcal{F}_{0}\to S$ (see Lemma 12.1) yields that
$\deg(\lambda_{3}\lambda_{1})$ on $S$ equals $1/p$ times the degree on
$\mathcal{F}_{0}$ of

$\displaystyle(p^{2}-3p+1)\ell_{0}^{3}\ell_{1}+(2p^{2}-2p+2)\ell_{0}^{3}\ell_{2}+(p^{2}-3p+1)\ell_{0}\ell_{1}^{3}+$
$\displaystyle
4(p-1)^{2}\ell_{0}\ell_{1}^{2}\ell_{2}+(5p^{2}-7p+5)\ell_{0}\ell_{1}\ell_{2}^{2}$

and this equals $(p-1)^{4}(p^{2}+p+1)(p^{2}+1)$. Multiplying this with the
number of irreducible components $(p^{2}-1)(p^{6}-1)v(4)$ we find the
coefficient $a=(p-1)^{3}(p^{3}-1)(p^{4}-1)(p^{6}-1)$. ∎

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