# Moduli of abelian varieties near the locus of products of elliptic curves

Samuel Grushevsky  Department of Mathematics and Simons Center for Geometry
and Physics, Stony Brook University, Stony Brook, NY 11794-3651
<EMAIL_ADDRESS>and  Riccardo Salvati Manni  Dipartimento di
Matematica, Piazzale Aldo Moro, 2, I-00185 Roma, Italy
<EMAIL_ADDRESS>
###### Abstract.

We study various naturally defined subvarieties of the moduli space
${\mathcal{A}}_{g}$ of complex principally polarized abelian varieties (ppav)
in a neighborhood of the locus of products of $g$ elliptic curves.

In this neighborhood, we obtain a local description for the locus of
hyperelliptic curves, reproving the recent result of Shepherd-Barron [SB21]
that the hyperelliptic locus is locally given by tridiagonal matrices. We
further reprove and generalize to arbitrary genus the recent result of
Agostini and Chua [AC21] showing that the locus of jacobians of genus 5 curves
with a theta-null is an irreducible component of the locus of ppav with a
theta-null such that the singular locus of the theta divisor at the
corresponding two-torsion point has tangent cone of rank at most 3. We further
show that the locus of ppav such that the gradient vanishes, for some odd
theta characteristic, locally has codimension $g$ near the diagonal. Finally,
we obtain new results on the locus where the rank of the Hessian of the theta
function at a two-torsion point that lies on the theta divisor is equal to 2.

Research of the first author is supported in part by NSF grant DMS-21-01631

## Introduction

Most known constructions of geometrically meaningful subvarieties of the
moduli space ${\mathcal{A}}_{g}$ of complex principally polarized abelian
varieties (ppav) are either via the Jacobian or Albanese map, or by imposing
certain conditions on the theta divisor and its singularities. Two most
classical such constructions are of course the locus
${\mathcal{J}}_{g}^{\circ}$ of Jacobians of smooth genus $g$ curves, and the
theta-null divisor $\vartheta_{\rm null}$ — the locus of those ppav that have
a vanishing theta constant, or equivalently for which the theta divisor
contains an even two-torsion point. Geometrically, one can further consider
Jacobians of hyperelliptic curves, intermediate Jacobians of cubic threefolds,
etc. Working with the theta divisor, one can impose conditions on the
dimension of its singular locus, existence of points or higher multiplicity,
or on the local structure of singularities.

In this paper we present a unified approach to determining the local structure
and some irreducible components of the subvarieties of ${\mathcal{A}}_{g}$
defined in these ways, which we apply in various situations. We thus reprove
recent results of Shepherd-Barron [SB21] characterizing the locus of
hyperelliptic Jacobians locally near the locus of products of elliptic curves.
We show that the locus of ppav with an (odd) two-torsion point of multiplicity
three on the theta divisor is smooth, locally of codimension $g$, as expected,
near the diagonal. We further show that the locus of Jacobians with a
vanishing theta null is an irreducible component of the locus of ppav with a
theta-null such that the Hessian matrix of theta has rank 3 — thus extending
to arbitrary genus the results of Agostini and Chua [AC21] in genus 5, and
generalizing our work in genus 4. We further show that the locus of products
with an elliptic curve is an irreducible component of the locus where the rank
of the Hessian as above is equal to 2.

Our method is inspired by our recent work [FGSM21] with Hershel Farkas, where
the geometric study in the neighborhood of the diagonal allowed us to give an
explicit solution to the weak Schottky problem. Our approach consists of
investigating the geometry of the various loci near the locus of diagonal
period matrices, i.e. geometrically near
${\mathcal{A}}_{1}\times\dots\times{\mathcal{A}}_{1}\subset{\mathcal{A}}_{g}$,
and using Taylor expansions of theta functions and infinitesimal geometry
there.

We denote ${\mathbb{H}}_{g}$ the Siegel upper half-space, denote
${\mathcal{A}}_{g}$ the moduli space of ppav, and denote
$p:{\mathbb{H}}_{g}\to{\mathcal{A}}_{g}$ the universal covering map, which is
the quotient by the action of $\operatorname{Sp}(2g,{\mathbb{Z}})$. We use
${\mathcal{J}}_{g}\subset{\mathcal{A}}_{g}$ to denote the closure of the locus
${\mathcal{J}}_{g}^{\circ}$ of Jacobians of smooth genus $g$ curves, and
denote by
${\mathcal{HJ}}_{g}^{\circ}\subset{\mathcal{HJ}}_{g}\subset{\mathcal{J}}_{g}$
respectively the locus of Jacobians of hyperelliptic genus $g$ curves and its
closure. We denote
${\mathbb{HJ}}_{g}^{\circ}\subset{\mathbb{HJ}}_{g}\subset{\mathbb{J}}_{g}\subset{\mathbb{H}}_{g}$
their respective preimages in the Siegel space, and denote
${\mathbb{HJ}}_{g}^{\circ}$ and ${\mathbb{J}}_{g}^{\circ}$ the open subsets of
hyperelliptic Jacobians, and Jacobians, of smooth curves.

It is a classical result of Mumford [Mum07] that the geometrically defined
locus ${\mathbb{HJ}}_{g}\subset{\mathbb{H}}_{g}$ is defined from the point of
view of the geometry of the theta divisor as the locus where a certain
configuration of theta constants with characteristics vanishes (we’ll review
this below in detail). We denote
${\mathcal{R}}_{g}:=({\mathcal{A}}_{1}\times{\mathcal{A}}_{g-1})\cup({\mathcal{A}}_{2}\times{\mathcal{A}}_{g-2})\cup\dots\subset{\mathcal{A}}_{g}$
the locus of decomposable (classically called reducible) ppav — this of course
includes ppav that have more than two factors, which may lie in more than one
component of the above union.

Finally, we denote
${\mathcal{D}}_{g}:={\mathcal{A}}_{1}\times\dots\times{\mathcal{A}}_{1}\subset{\mathcal{R}}_{g}$
the locus of products of elliptic curves, and denote
${\mathbb{D}}_{g}\subset{\mathbb{H}}_{g}$ its preimage in the universal cover.
One irreducible component of ${\mathbb{D}}_{g}$ is the locus
${\mathbb{I}}_{g}:={\mathbb{H}}_{1}\times\dots\times{\mathbb{H}}_{1}\subset{\mathbb{H}}_{g}$
of diagonal period matrices. Recently, Shepherd-Barron [SB21] determined the
local structure of ${\mathbb{HJ}}_{g}$ near ${\mathbb{I}}_{g}$. Our first
result is an alternative proof of this side result of his (the main thrust,
and the main results of [SB21], are on elliptic surfaces, which are beyond the
scope of our work):

######  Theorem 1 (Shepherd-Barron [SB21, Theorem 14.6]).

For every irreducible component ${\mathbb{X}}$ of ${\mathbb{HJ}}_{g}$
containing ${\mathbb{I}}_{g}\subset{\mathbb{H}}_{g}$, to first order at any
point of ${\mathbb{I}}_{g}$, ${\mathbb{X}}\subset{\mathbb{H}}_{g}$ is defined
by the vanishing of the entries $\tau_{ij}$ of the period matrix $\tau$ where
$(i,j)$ runs over the set of pairs that are not edges of the corresponding
alkane. In particular, the branch corresponding to the linear alkane equals,
to first order, the locus of tridiagonal matrices, i.e. matrices with non-zero
entries only on the main diagonal and on the two diagonals directly above and
below it.

######  Remark 2.

We have taken care to phrase the result above carefully on the Siegel space.
Note that there is a delicate point here: while
${\mathcal{HJ}}_{g}\subset{\mathcal{A}}_{g}$ is an irreducible algebraic
variety, its preimage ${\mathbb{HJ}}_{g}\subset{\mathbb{H}}_{g}$ has many
irreducible components, a number of which contain ${\mathbb{I}}_{g}$. The
result above describes the tangent space at a point of ${\mathbb{I}}_{g}$ to
every irreducible component of ${\mathbb{HJ}}_{g}$ containing
${\mathbb{I}}_{g}$.

Recall that $\vartheta_{\rm null}\subset{\mathcal{A}}_{g}$ denotes the locus
where some even theta constant $\theta\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right](\tau,0)$ vanishes. Following [GSM07], we denote
$\vartheta_{\rm null}^{k}\subset\vartheta_{\rm null}$ the locus where the rank
of the corresponding Hessian matrix
$(\partial_{z_{a}}\partial_{z_{b}}\theta\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right](\tau,z))|_{z=0}$ is at most $k$. Since the
theta function for a block-diagonal period matrix factorizes as the product of
theta functions, one immediately sees that
${\mathcal{A}}_{g_{1}}\times{\mathcal{A}}_{g_{2}}\subset\vartheta_{\rm
null}^{2}$, for any $g_{1}+g_{2}=g$ with $g_{1},g_{2}>0$ (see also the
explicit expansions in Section 3). This is to say that
${\mathcal{R}}_{g}\subset\vartheta_{\rm null}^{2}$, and we prove that at least
the largest irreducible component of ${\mathcal{R}}_{g}$ is also an
irreducible component of $\vartheta_{\rm null}^{2}$.

######  Theorem 3.

The locus ${\mathcal{A}}_{1}\times{\mathcal{A}}_{g-1}$ is an irreducible
component of $\vartheta_{\rm null}^{2}$.

From Riemann’s theta singularity theorem for Jacobians of curves one deduces
the inclusion ${\mathcal{J}}_{g}\cap\vartheta_{\rm null}\subset\vartheta_{\rm
null}^{3}$. Our next result is an alternative proof and a generalization to
arbitrary genus of the recent result of Agostini and Chua [AC21]. They prove
that in genus 5 there exists an irreducible component of ${\mathbb{J}}_{5}\cap
p^{-1}(\vartheta_{\rm null})$ that is also an irreducible component of
$p^{-1}(\vartheta_{\rm null}^{3})$ (while we recall that in genus 4 the
equality ${\mathcal{J}}_{4}\cap\vartheta_{\rm null}=\vartheta_{\rm null}^{3}$
was conjectured by H. Farkas [Far06] and proven by us in [GSM08]).

######  Theorem 4.

For any genus $g\geq 3$, ${\mathcal{J}}_{g}\cap\vartheta_{\rm null}$ is an
irreducible component of $\vartheta_{\rm null}^{3}$ .

Finally, we recall that the theta-null divisor has a natural “odd” counterpart
${\mathcal{G}}_{\rm null}\subset{\mathcal{A}}_{g}$, which is the locus of all
ppav such that the gradient
$(\partial_{z_{i}}\theta\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right](\tau,z))|_{z=0}$ vanishes, for some odd theta
characteristic $\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]$. It was conjectured in [GSM09] that the locus
${\mathcal{G}}_{\rm null}$ is purely of codimension $g$ in
${\mathcal{A}}_{g}$. In [GH12] this conjecture was proven completely for all
$g\leq 5$. We now prove that this also holds for every component intersecting
the diagonal or containing the hyperelliptic locus:

######  Theorem 5.

The locus $\vartheta_{\rm
null}\times{\mathcal{A}}_{1}\,\subset\,{\mathcal{A}}_{g-1}\times{\mathcal{A}}_{1}$
is an irreducible component of ${\mathcal{G}}_{\rm null}$.

Moreover, any irreducible component of the locus ${\mathcal{G}}_{\rm null}$
containing the diagonal is locally smooth along the diagonal, and has
codimension $g$ in ${\mathcal{A}}_{g}$.

######  Remark 6.

We note that the first part of the above theorem is the $k=1$ case of [GSM09,
Thm. 6].

The dimensionality in the second statement can in fact be reduced to the
argument in [GSM09], which was proven by a detailed degeneration argument.
Indeed, [GSM09, Prop. 12] describes the boundary of ${\mathcal{G}}_{\rm null}$
in the partial toroidal compactification, which turns out to be described
geometrically as a union of two components which involve respectively the
singular locus of the universal theta divisor in genus $g-1$ (which always has
expected dimension), and of the locus ${\mathcal{G}}_{\rm null}$ in genus
$g-1$. In [GSM09, Thm. 13] this is used to deduced that the codimension of
${\mathcal{G}}_{\rm null}$ is precisely $g$ if all of its components, and all
components of such loci for lower genera, intersect the boundary of the
partial toroidal compactification. However, what is really used in the proof
of that theorem is that for a given irreducible component of
${\mathcal{G}}_{\rm null}$, it intersects the partial toroidal boundary, and
if the intersection involves ${\mathcal{G}}_{\rm null}$ dimension in one less,
then that also intersects the partial toroidal boundary, etc.

Since the diagonal ${\mathcal{D}}_{g}\subset{\mathcal{A}}_{g}$ clearly
intersects the generic boundary stratum
${\mathcal{A}}_{g-1}\subset\partial{\mathcal{A}}_{g}^{\operatorname{Sat}}$ of
the Satake compactification, just by sending one of the
$t_{i}\in{\mathbb{H}}_{1}$ to $i\infty=\partial{\mathcal{A}}_{1}$ constructs
such a degeneration, this means that it also intersects the partial toroidal
boundary. Moreover its intersection with partial toroidal boundary will again
involve the (preimage in the universal family) of the diagonal
${\mathcal{D}}_{g-1}$, which will again intersect the partial toroidal
boundary, and thus the inductive proof of [GSM09, Thm. 13] applies to any
irreducible component of ${\mathcal{G}}_{\rm null}$ containing
${\mathcal{D}}_{g}$. The local smoothness statement of the theorem above is
new.

We note that it appears much harder to try to apply such degeneration
arguments for the study of the Hessian rank loci in 3 and 4. While the
degeneration of the derivatives of theta constants to the boundary of the
partial compactification is well-known, the Hessian matrix would involve
second order derivatives of the theta constants in genus $g-1$, but also first
order derivatives of theta functions in genus $g-1$ evaluated at the point of
the abelian variety that gives the semiabelian extension data, and thus the
rank condition appears much harder to work with by induction in genus.

In Section 1 we recall the basic notions about theta functions and the action
of the symplectic group on the set of theta characteristics. In Section 2 we
recall the classical characterizations of the hyperelliptic and decomposable
loci ${\mathcal{HJ}}_{g}$ and ${\mathcal{R}}_{g}$ within ${\mathcal{A}}_{g}$
in terms of vanishing of theta constants. In Section 3 we set up convenient
notation for writing down the expansions of the theta function and its
derivatives near the locus ${\mathbb{I}}_{g}$ of diagonal period matrices. One
new technical result that we prove is the description of the action of the
stabilizer group of the theta function with characteristic $m$ on the set of
irreducible components of ${\mathbb{D}}_{g}$. In Section 4 we reprove
Shepherd-Barron’s result on the infinitesimal structure of
${\mathcal{HJ}}_{g}$ near ${\mathcal{D}}_{g}$. In Section 5 we prove 5 on the
locus ${\mathcal{G}}_{\rm null}$. Finally, in Section 6 we prove Theorems 3
and 4 on the Hessian rank loci $\vartheta_{\rm null}^{2}$ and $\vartheta_{\rm
null}^{3}$.

### Acknowledgments

We are grateful to Daniele Agostini, Lynn Chua, and Nick Shepherd-Barron for
sharing with us their preprints [AC21] and [SB21], respectively, and their
interesting ideas, and thus reigniting our investigation of this subject. We
are indebted to Hershel Farkas, a collaboration with whom on [FGSM21] led us
to investigate and appreciate the importance of expansions of theta functions
near the diagonal. The second author is grateful to Enrico Arbarello and
Edoardo Sernesi for valuable conversations and correspondence in the past
years.

##  1\. Notation: theta functions and level covers

We denote by ${\mathbb{H}}_{g}:=\\{\tau\in\operatorname{Mat}_{g\times
g}({\mathbb{C}})\mid\tau=\tau^{t},\operatorname{Im}\tau>0\\}$ the Siegel upper
half-space of complex symmetric matrices with positive definitive imaginary
part. It is a homogeneous space for the action of
$\operatorname{Sp}(2g,{\mathbb{R}})$, where an element

$\sigma=\left(\begin{matrix}A&B\\\ C&D\end{matrix}\right)\in
Sp(2g,{\mathbb{R}})$

acts via

$\sigma\cdot\tau:=(A\tau+B)(C\tau+D)^{-1}\,.$

We denote by $\Gamma_{g}:=\operatorname{Sp}(2g,{\mathbb{Z}})$ the Siegel
modular group, and let $\Gamma_{g}(n):=\\{\sigma\in\Gamma_{g}:\sigma\equiv
1_{2g}\mod n\\}$ (where from now on we denote by $1_{k}$ the $k\times k$
identity matrix) denote the principal congruence subgroup of $\Gamma_{g}$. The
quotient ${\mathcal{A}}_{g}={\mathbb{H}}_{g}/\Gamma_{g}$ is the moduli space
of complex principally polarized abelian varieties (ppav), and
${\mathcal{A}}_{g}(n)={\mathbb{H}}_{g}/\Gamma_{g}(n)$ is the moduli space of
ppav with a choice of a full symplectic level $n$ structure. Recall that
${\mathcal{J}}_{g}$ and ${\mathcal{HJ}}_{g}$ denote the closures in
${\mathcal{A}}_{g}$ of the loci of Jacobians and of hyperelliptic Jacobians,
respectively.

We denote by $p:{\mathbb{H}}_{g}\to{\mathcal{A}}_{g}$ and
$p_{n}:{\mathbb{H}}_{g}\to{\mathcal{A}}_{g}(n)$ the quotient maps, and by
abuse of notation will also denote the same way their restrictions to various
submanifolds such as ${\mathbb{J}}_{g}:=p^{-1}({\mathcal{J}}_{g})$ or
${\mathbb{HJ}}_{g}:=p^{-1}({\mathcal{HJ}}_{g})$. For a subvariety
${\mathcal{X}}\subset{\mathcal{A}}_{g}$, we will also write ${\mathcal{X}}(n)$
to denote its preimage on a level cover:
${\mathcal{X}}(n):=p_{n}(p^{-1}({\mathcal{X}}))\subset{\mathcal{A}}_{g}(n)$.
Very importantly, we note that since $p:{\mathbb{H}}_{g}\to{\mathcal{A}}_{g}$
is a Galois cover, for any irreducible subvariety
${\mathcal{X}}\subset{\mathcal{A}}_{g}$, for any two irreducible components
${\mathbb{X}}^{\prime}$ and ${\mathbb{X}}^{\prime\prime}$ of
$p^{-1}({\mathcal{X}})\subset{\mathbb{H}}_{g}$, there must exist an element
$\gamma\in\Gamma_{g}$ mapping ${\mathbb{X}}^{\prime}$ to
${\mathbb{X}}^{\prime\prime}$.

We call a ppav decomposable if it is isomorphic to a product of two lower-
dimensional ppav. Analytically, $\tau$ is decomposable if and only if there
exists $\sigma\in\Gamma_{g}$, such that

$\sigma\cdot\tau=\left(\begin{smallmatrix}\tau_{1}&0\\\ 0&\tau_{2}\\\
\end{smallmatrix}\right),\quad{\rm
with}\,\tau_{i}\in{\mathbb{H}}_{g_{i}},\,\,g=g_{1}+g_{2},\,\,g_{1},g_{2}>0\,.$

(classically, such ppav are called reducible). We denote by

${\mathcal{R}}_{g}:=({\mathcal{A}}_{1}\times{\mathcal{A}}_{g-1})\cup({\mathcal{A}}_{2}\times{\mathcal{A}}_{g-2})\cup\dots\subset{\mathcal{A}}_{g}$

the locus of decomposable ppav, and denote
${\mathbb{R}}_{g}:=p^{-1}({\mathcal{R}}_{g})\subset{\mathbb{H}}_{g}$ its
preimage in the Siegel space. We recall that
${\mathcal{D}}_{g}={\mathcal{A}}_{1}\times\dots\times{\mathcal{A}}_{1}\subset{\mathcal{A}}_{g}$
denotes the locus of products of elliptic curves, and
${\mathbb{D}}_{g}:=p^{-1}({\mathcal{D}}_{g})\subset{\mathbb{H}}_{g}$ denotes
its preimage, of which ${\mathbb{I}}_{g}$ is an irreducible component. We thus
have
${\mathcal{D}}_{g}\subset{\mathcal{HJ}}_{g}\subset{\mathcal{J}}_{g}\subset{\mathcal{A}}_{g}$
and ${\mathcal{D}}_{g}\subset{\mathcal{R}}_{g}\subset{\mathcal{A}}_{g}$.

The goal of this paper is to describe these loci locally near
${\mathcal{D}}_{g}$, and the main tool will be by analyzing the Taylor
expansions of theta functions near ${\mathbb{I}}_{g}$. Recall that the theta
function with characteristics $\varepsilon,\delta\in{\mathbb{Z}}_{2}^{g}$ is
the function of $\tau\in{\mathbb{H}}_{g}$ and $z\in{\mathbb{C}}^{g}$ given by

$\theta\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right](\tau,z):=\sum\limits_{p\in{\mathbb{Z}}^{g}}\exp\pi
i\left[{}^{t}(p+\tfrac{\varepsilon}{2})\tau(p+\tfrac{\varepsilon}{2})+2^{t}(p+\tfrac{\varepsilon}{2})(z+\tfrac{\delta}{2})\right]\,.$

We will write theta characteristics also as
$m=\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]\in{\mathbb{Z}}_{2}^{2g}$; we will usually write
$\varepsilon,\delta$ as rows (or sometimes columns, if notationally more
convenient) of $g$ zeroes and ones, and operate with them over
${\mathbb{Z}}_{2}$ unless stated otherwise. In particular, $m$ is called even
or odd depending on whether the scalar product $\varepsilon\cdot\delta$ is
zero or one as an element of ${\mathbb{Z}}_{2}$. As a function of $z$, the
theta function is even or odd, respectively. The theta constants are the
values of theta functions at $z=0$, and theta gradients are the values of the
$z$-gradient of the theta function, evaluated at $z=0$. We will drop the $z$
variable from notation in both cases, and write

$\theta_{m}(\tau):=\theta_{m}(\tau,0)\in{\mathbb{C}};\qquad\operatorname{grad}\theta_{m}(\tau):=\left\\{\tfrac{\partial}{\partial
z_{a}}\theta_{m}(\tau,z)|_{z=0}\right\\}_{a=1,\ldots,g}\in{\mathbb{C}}^{g}\,.$

Note that theta constants vanish identically for $m$ odd, while theta
gradients vanish identically for $m$ even. Theta constants and theta gradients
are examples of scalar- (resp. vector-) valued Siegel modular forms, i.e. are
sections of a suitable line (resp. rank $g$ vector) bundle on a suitable cover
of ${\mathcal{A}}_{g}$. In fact these are modular forms with non-trivial
multiplier with respect to $\Gamma_{g}(2)$. Moreover, $\Gamma_{g}$ acts on
theta characteristics, considered as elements of ${\mathbb{Z}}_{2}^{2g}$, via
an affine-linear action of its quotient
$\operatorname{Sp}(2g,{\mathbb{Z}}_{2})=\Gamma_{g}/\Gamma_{g}(2)$. This action
is given explicitly by

(1) $\sigma\circ\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]:=\left(\begin{smallmatrix}D&-B\\\
-C&A\end{smallmatrix}\right)\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]+\left[\begin{smallmatrix}{\rm
diag}(c\,^{t}d)\\\ {\rm diag}(a\,^{t}b)\end{smallmatrix}\right]\,.$

We refer to [Igu72] for further details, and note that $\Gamma_{g}(2)$ is
precisely the subgroup of $\Gamma_{g}$ that fixes every characteristic.

We recall from [Igu72, SM94] that the orbits of $\Gamma_{g}$ on tuples of
characteristics are fully characterized by parity of characteristics, by the
a/syzygy properties of triples of characteristics, and by linear relations
with an even number of terms.

We will not use the details of this except to note that the zero loci of
$\theta_{m}(\tau)$ and $\operatorname{grad}\theta_{m}(\tau)$,

$\displaystyle\theta_{\rm null}\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]$
$\displaystyle:=\\{\theta\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right](\tau)=0\\}\subset{\mathbb{H}}_{g}\quad{\rm
and}$ $\displaystyle\operatorname{grad}_{\rm
null}\left[\begin{smallmatrix}\varepsilon\\\ \delta\end{smallmatrix}\right]$
$\displaystyle:=\\{\operatorname{grad}\theta\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right](\tau)=0\\}\subset{\mathbb{H}}_{g}\,,$

are invariant under the action of $\Gamma_{g}(2)$, and thus are preimages of
well-defined subvarieties in ${\mathcal{A}}_{g}(2)$.

For any $2\leq k\leq g$ we define $\theta_{\rm
null}^{k}\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]\subset\theta_{\rm
null}\left[\begin{smallmatrix}\varepsilon\\\ \delta\end{smallmatrix}\right]$
to be the locus where the rank of the Hessian matrix
$\left(\partial_{z_{a}}\partial_{z_{b}}\theta\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right](\tau,z)|_{z=0}\right)_{1\leq a,b\leq g}$ is at
most $k$; by abuse of notation, we will use this notation for both a
subvariety of ${\mathcal{A}}_{g}(2)$ and an analytic subset of
${\mathbb{H}}_{g}$, when no confusion can arise.

Since the action of $\Gamma_{g}/\Gamma_{g}(2)$ permutes theta characteristics
and the loci $\theta_{\rm null}\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]$ and respectively $\operatorname{grad}_{\rm
null}\left[\begin{smallmatrix}\varepsilon\\\ \delta\end{smallmatrix}\right]$
transitively, it follows that their images

$\vartheta_{\rm null}:=p(\theta_{\rm
null}\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right])\subset{\mathcal{A}}_{g}\quad{\rm
and}\quad{\mathcal{G}}_{\rm null}:=p(\operatorname{grad}_{\rm
null}\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right])\subset{\mathcal{A}}_{g}$

are independent of the choices of even or odd characteristic
$\left[\begin{smallmatrix}\varepsilon\\\ \delta\end{smallmatrix}\right]$,
respectively. Geometrically, $\vartheta_{\rm null}$ is the locus of ppav whose
theta divisor has a singularity (necessarily of even multiplicity, at least 2)
at an even two-torsion point, while ${\mathcal{G}}_{\rm null}$ is the locus of
ppav whose theta divisor has a singularity (necessarily of odd multiplicity,
at least 3) at an odd two-torsion point of the abelian variety. The loci
$\vartheta_{\rm null}^{k}:=p(\theta_{\rm
null}^{k}\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right])\subset{\mathcal{A}}_{g}$ are similarly
independent of the choice of characteristic
$\left[\begin{smallmatrix}\varepsilon\\\ \delta\end{smallmatrix}\right]$.

For low genera these loci have a simple geometric interpretation, which is
part of the motivation for studying them:

$\begin{array}[]{clll}g=2:&&\vartheta_{\rm
null}={\mathcal{R}}_{2}={\mathcal{A}}_{1}\times{\mathcal{A}}_{1}&{\mathcal{G}}_{\rm
null}=\emptyset\\\ g=3:&\vartheta_{\rm
null}^{2}={\mathcal{R}}_{3}&\vartheta_{\rm
null}={\mathcal{HJ}}_{3}&{\mathcal{G}}_{\rm
null}={\mathcal{A}}_{1}\times{\mathcal{A}}_{1}\times{\mathcal{A}}_{1}\\\
g=4:&\vartheta_{\rm null}^{2}={\mathcal{R}}_{4}&\vartheta_{\rm
null}^{3}={\mathcal{J}}_{4}\cap\vartheta_{\rm null}&{\mathcal{G}}_{\rm
null}={\mathcal{A}}_{1}\times{\mathcal{HJ}}_{3}\\\ g=5:&&&{\mathcal{G}}_{\rm
null}=({\mathcal{A}}_{1}\times\vartheta_{\rm
null})\cup{\mathcal{I}}{\mathcal{J}}\,,\end{array}$

where in genera 4 and 5 the locus $\vartheta_{\rm null}$ does not admit such a
quick geometric description (while genus 4 curves with a theta-null are
canonical curves that lie on a singular quadric, there is no similarly easy
description for principally polarized abelian fourfolds with a theta-null),
and ${\mathcal{I}}{\mathcal{J}}$ denotes the closure of the locus of
intermediate Jacobians of cubic threefolds; see [GH12] for more discussion of
the cases $g=4,5$.

The locus $\vartheta_{\rm null}$ was studied classically, and the first result
in this study is that $\vartheta_{\rm null}$ is always an irreducible divisor
in ${\mathcal{A}}_{g}$ [Fre91, p. 88], while in [GSM07] we conjectured that
${\mathcal{G}}_{\rm null}$ is always of pure codimension $g$ in
${\mathcal{A}}_{g}$, and proved this for every irreducible component of
${\mathcal{G}}_{\rm null}$ that intersects the boundary of the partial
compactification of ${\mathcal{A}}_{g}$.

One can easily see that ${\mathcal{R}}_{g}\subset\vartheta_{\rm null}^{2}$,
while Riemann theta singularity theorem for Jacobians implies the inclusion
${\mathcal{J}}_{g}\cap\vartheta_{\rm null}\subset\vartheta_{\rm null}^{3}$. In
[GSM08] we proved the conjecture of H. Farkas that in genus 4 the equality
${\mathcal{J}}_{4}^{\circ}\cap\vartheta_{\rm null}=\vartheta_{\rm
null}^{3}\setminus\vartheta_{\rm null}^{2}$ holds. One of our main results is
4, extending this genus 4 statement, and the recent genus 5 result of Agostini
and Chua to arbitrary genus, showing that
${\mathcal{J}}_{g}^{\circ}\cap\vartheta_{\rm null}$ is an irreducible
component of $\theta_{\rm null}^{3}$ for any genus.

One technical point that pervades our work is whether we work on
${\mathcal{A}}_{g}$, ${\mathbb{H}}_{g}$, or (as the above discussion shows is
often useful) on ${\mathcal{A}}_{g}(2)$. Note for example that while
$\vartheta_{\rm null}\subset{\mathcal{A}}_{g}$ is irreducible, $\vartheta_{\rm
null}(2)=p_{2}(p^{-1}(\vartheta_{\rm null}))=\cup\vartheta_{\rm
null}\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]\subset{\mathcal{A}}_{g}(2)$ has
$2^{g-1}(2^{g}+1)$ irreducible components, indexed by characteristics.
However, by [MSM21] for any $g\geq 3$ the analytic spaces $\theta_{\rm
null}\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]\subset{\mathbb{H}}_{g}$ are irreducible for
each $\left[\begin{smallmatrix}\varepsilon\\\ \delta\end{smallmatrix}\right]$.

##  2\. The decomposable and hyperelliptic loci

In this section we recall the known characterizations of ${\mathcal{R}}_{g}$
and ${\mathcal{HJ}}_{g}$ in terms of vanishing of certain sets of theta
constants, and study the combinatorics of the relevant characteristics. This
is equivalent to describing the irreducible components of the corresponding
loci on the level covers. For further use we denote

(2)
$\langle\varepsilon,\delta\rangle:=\sum_{a=1}^{g}\varepsilon_{a}\cdot\delta_{a}\in{\mathbb{Z}}$

the scalar product of $\varepsilon$ and $\delta$, considered in ${\mathbb{Z}}$
(unlike the usual pairing $\varepsilon\cdot\delta\in{\mathbb{Z}}_{2}$). We
further denote by ${\mathcal{E}}$ the set of all even characteristics, and for
any even $\ell\in{\mathbb{Z}}_{\geq 0}$ denote by
${\mathcal{E}}_{\ell}\subset{\mathcal{E}}$ the set of all even characteristics
such that $\langle\varepsilon,\delta\rangle=\ell$, and denote
${\mathcal{E}}^{*}:={\mathcal{E}}\setminus{\mathcal{E}}_{0}$. We will
similarly decompose the set ${\mathcal{O}}$ of odd characteristics as
${\mathcal{O}}=\sqcup_{1\leq\ell\leq g,\ \ell\textrm{ odd
}}{\mathcal{O}}_{\ell}$, and denote
${\mathcal{O}}^{*}:={\mathcal{O}}\setminus{\mathcal{O}}_{1}$ to exclude the
“simplest” odd characteristics.

###  2.1. The decomposable locus, and the diagonal

Recall that the theta function near a block-diagonal period matrix factorizes
as follows:

(3) $\theta\left[\begin{smallmatrix}\varepsilon_{1}\varepsilon_{2}\\\
\delta_{1}\delta_{2}\end{smallmatrix}\right]\left(\left(\begin{smallmatrix}\tau_{1}&0\\\
0&\tau_{2}\end{smallmatrix}\right),\,\begin{smallmatrix}z_{1}\\\
z_{2}\end{smallmatrix}\right)=\theta\left[\begin{smallmatrix}\varepsilon_{1}\\\
\delta_{1}\end{smallmatrix}\right](\tau_{1},z_{1})\cdot\theta\left[\begin{smallmatrix}\varepsilon_{2}\\\
\delta_{2}\end{smallmatrix}\right](\tau_{2},z_{2})$

for any $\tau_{i}\in{\mathbb{H}}_{g_{i}}$ and $z_{i}\in{\mathbb{C}}^{g_{i}}$
with $g_{1}+g_{2}=g$. By applying this formula recursively, we see that for a
diagonal period matrix $\tau_{0}=\left(\begin{smallmatrix}t_{1}&0&\dots&0\\\
0&t_{2}&\dots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\dots&t_{g}\end{smallmatrix}\right)\in{\mathbb{I}}_{g}$ the value of the
theta constant is given by

$\theta\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right](\tau_{0})=\theta\left[\begin{smallmatrix}\varepsilon_{1}\\\
\delta_{1}\end{smallmatrix}\right](t_{1})\cdot\ldots\cdot\theta\left[\begin{smallmatrix}\varepsilon_{g}\\\
\delta_{g}\end{smallmatrix}\right](t_{g})\,.$

We recall the characterization of block-diagonal period matrices:

######  Proposition 7 ([SM94, Theorem 5]).

A period matrix $\tau\in{\mathbb{H}}_{g}$ lies in the $\Gamma_{g}(2)$ orbit of
the locus ${\mathbb{H}}_{g_{1}}\times{\mathbb{H}}_{g_{2}}$ if and only if
$\theta\left[\begin{smallmatrix}\varepsilon_{1}&\delta_{1}\\\
\varepsilon_{2}&\delta_{2}\end{smallmatrix}\right](\tau)=0$ for all pairs of
odd characteristics $\left[\begin{smallmatrix}\varepsilon_{1}\\\
\delta_{1}\end{smallmatrix}\right]\in{\mathbb{Z}}_{2}^{2g_{1}}$ and
$\left[\begin{smallmatrix}\varepsilon_{2}\\\
\delta_{2}\end{smallmatrix}\right]\in{\mathbb{Z}}_{2}^{2g_{2}}$.

By applying this proposition and using the factorization formula (3)
recursively, one obtains

######  Corollary 8.

A period matrix $\tau\in{\mathbb{H}}_{g}$ lies in the $\Gamma_{g}(2)$ orbit of
the locus ${\mathbb{I}}_{g}$ if and only if all theta constants
$\theta\left[\begin{smallmatrix}\varepsilon\\\ \delta\end{smallmatrix}\right]$
such that at least one column $\left[\begin{smallmatrix}\varepsilon_{a}\\\
\delta_{a}\end{smallmatrix}\right]$ of
$\left[\begin{smallmatrix}\varepsilon\\\ \delta\end{smallmatrix}\right]$ is
equal to $\left[\begin{smallmatrix}1\\\ 1\end{smallmatrix}\right]$ vanish at
$\tau$.

In our notation, this corollary can be reformulated as the statement that
$\tau\in\Gamma_{g}(2)\circ{\mathbb{I}}_{g}$ if and only if $\tau\in\theta_{\rm
m,\,null}$ for all $m\in{\mathcal{E}}^{*}$.

7 and 8 characterize the loci of block-diagonal and diagonal period matrices
in ${\mathbb{H}}_{g}$, and their images in ${\mathcal{A}}_{g}(2)$; to obtain
from these a characterization of the loci ${\mathcal{R}}_{g}$ and
${\mathcal{D}}_{g}$ in ${\mathcal{A}}_{g}$ one needs to consider the orbits of
the action of $\Gamma_{g}$ on the locus of block-diagonal period matrices. The
(setwise) stabilizers of such loci are known classically [Fre68]:

######  Proposition 9.

The (setwise) stabilizer
$\operatorname{Stab}_{\Gamma_{g}}({\mathbb{H}}_{g_{1}}\times{\mathbb{H}}_{g_{2}})$
is equal to the direct product
$Stab_{g_{1},g_{2}}:=\Gamma_{g_{1}}\times\Gamma_{g_{2}}$, except for the case
$g_{1}=g_{2}=g/2$, when the stabilizer is the semi-direct product
$Stab_{g/2,g/2}:=(\Gamma_{g/2}\times\Gamma_{g/2})\ltimes S_{2}$ with the
involution interchanging the two blocks.

######  Corollary 10.

The (setwise) stabilizer of the diagonal
$\operatorname{Stab}_{{\mathbb{I}}_{g}}\subset\Gamma_{g}$ is equal to the
wreath product $Stab_{{\mathbb{I}}_{g}}:=\Gamma_{1}\wr S_{g}$, i.e. is the
semidirect product of $(\Gamma_{1})^{\times
g}=\operatorname{SL}(2,{\mathbb{Z}})^{\times g}$ and the permutation group
$S_{g}$ of $g$ elements.

Here we think of the permutation group $S_{g}$ as embedded into $\Gamma_{g}$
as block matrices with two off-diagonal $g\times g$ blocks equal to zero, and
two on-diagonal $g\times g$ blocks equal to each other, and each being a
permutation matrix.

For our purposes, we are interested in finding an explicit manageable subgroup
acting transitively on the sets ${\mathcal{E}}$ and ${\mathcal{O}}$, and we
will find such a subgroup that contains
$\operatorname{Stab}_{{\mathbb{I}}_{g}}$, but is slightly larger. This is a
manifestation of the idea we’ll use later in the paper: instead of just
considering diagonal period matrices, we will allow some $2\times 2$ blocks,
and will enlarge the group correspondingly.

We thus set $G_{g}:=\Gamma_{2}\times(\Gamma_{1})^{\times(g-2)}\wr S_{g}$,
where we think of the first factor as block-diagonal period matrices with one
$2\times 2$ block and $g-2$ blocks of size $1\times 1$, and $S_{g}$ is
embedded into $\operatorname{Sp}(2g,{\mathbb{Z}})$ as before.

######  Lemma 11.

The group $G_{g}$ acts transitively on each of the two sets of characteristics
${\mathcal{E}}$ and ${\mathcal{O}}$.

###### Proof.

We do the even case, the odd case being completely analogous.

We first permute the coordinates so that all the $\ell$ columns of
characteristic that are equal to $\left[\begin{smallmatrix}1\\\
1\end{smallmatrix}\right]$ appear first. Then since $\Gamma_{1}$ acts
transitively on the set of 3 even characteristics in genus one, acting by
$\Gamma_{1}$ on the characteristics in each even column maps them all to
$\left[\begin{smallmatrix}0\\\ 0\end{smallmatrix}\right]$. Altogether, this
shows that for a fixed $\ell$, the stabilizer
$\operatorname{Stab}_{{\mathbb{I}}_{g}}\subset\Gamma_{g}$ acts transitively on
the set ${\mathcal{E}}_{\ell}$. Since
$\operatorname{Stab}_{{\mathbb{I}}_{g}}\subset G_{g}$, it is thus enough to
show that $G_{g}$ can change $\ell$ arbitrarily. For this, we observe that the
element

$\sigma_{0}:=\left(\begin{smallmatrix}1&1&1&0\\\ 1&1&0&1\\\ 0&1&1&0\\\
1&0&0&1\end{smallmatrix}\right)\in\Gamma_{2}$

sends $\left[\begin{smallmatrix}00\\\ 00\end{smallmatrix}\right]$ to
$\left[\begin{smallmatrix}11\\\ 11\end{smallmatrix}\right]$. Once the columns
of a characteristic are permuted so that the first $\ell$ ones are equal to
$\left[\begin{smallmatrix}1\\\ 1\end{smallmatrix}\right]$, we apply
$\sigma_{0}$ in the first two coordinates to make the first two columns equal
to $\left[\begin{smallmatrix}0\\\ 0\end{smallmatrix}\right]$, thus going from
a characteristic in ${\mathcal{E}}_{\ell}$ to a characteristic in
${\mathcal{E}}_{\ell-2}$. Repeating this process shows that the $G_{g}$ orbit
of any characteristic in ${\mathcal{E}}_{\ell}$ contains a characteristic in
${\mathcal{E}}_{0}$. ∎

###  2.2. The hyperelliptic locus

We recall from [Tsu91] and [MSM21] that the irreducible components of
${\mathbb{HJ}}_{g}\subset{\mathbb{H}}_{g}$ are in bijection with (and are in
fact preimages of) the irreducible components of
${\mathcal{HJ}}_{g}(2)\subset{\mathcal{A}}_{g}(2)$. We will describe one such
component explicitly, which will suffice since $\Gamma_{g}$ acts transitively
on the set of irreducible components of ${\mathbb{HJ}}_{g}$ or
${\mathcal{HJ}}_{g}(2)$.

We say that a set $m_{0},\ldots,m_{2g}$ of characteristics is called an
essential basis if any characteristic $m\in{\mathbb{Z}}_{2}^{2g}$ can be
written uniquely as a sum of an odd number of $m_{i}$’s. It follows from the
description of the action that an element of $\Gamma_{g}$ lies in
$\Gamma_{g}(2)$ (equivalently, fixes all characteristics) if and only if it
fixes every element of a chosen essential basis.

Recall that a special fundamental system of characteristics is a set of $g$
odd characteristics and $g+2$ even characteristics such that every triple of
characteristics is azygetic. The description of the orbits of the action of
$\Gamma_{g}$ on tuples of characteristics implies that $\Gamma_{g}$ acts
transitively on the set of special fundamental systems. Moreover, the
condition of being azygetic implies that any subsequence of a fundamental
system is a sequence of essentially independent characteristics,i.e. the sum
of an even number of characteristics is always different from $0$, see
[Igu80]. As a consequence, eliminating any characteristic from any special
fundamental system of characteristics gives an essential basis. We now fix the
following special fundamental system:

(4)
$I:=\left(o_{1},\dots,o_{g},e_{1},\dots,e_{g+2}\right)=\left(\setcounter{MaxMatrixCols}{14}\begin{smallmatrix}1&0&0&\dots&0&0&0&1&0&0&\ldots&0&0&0\\\
0&1&0&\dots&0&0&0&0&1&0&\ldots&0&0&0\\\ 0&0&1&\dots&0&0&0&0&0&1&\dots&0&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
0&0&0&\dots&1&0&0&0&0&0&\dots&1&0&0\\\
0&0&0&\dots&0&1&0&0&0&0&\dots&0&1&0\\\\[6.0pt] \hline\cr\\\\[6.0pt]
1&1&1&\dots&1&1&0&0&1&1&1&\dots&1&1\\\ 0&1&1&\dots&1&1&0&0&0&1&1&\dots&1&1\\\
0&0&1&\dots&1&1&0&0&0&0&1&\dots&1&1\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&\dots&1&1&0&0&0&0&0&\dots&1&1\\\
0&0&0&\dots&0&1&0&0&0&0&0&\dots&0&1\end{smallmatrix}\right)\,,$

where we have denoted the $g$ odd characteristics by $o_{j}$, and the $g+2$
even characteristics by $e_{j}$. We further denote

(5) $b^{g}:=o_{1}+\dots+o_{g}=\left[\begin{smallmatrix}1&1&\dots&1&1\\\
\tfrac{1-(-1)^{g}}{2}&\tfrac{1-(-1)^{g-1}}{2}&\dots&0&1\end{smallmatrix}\right]\equiv\left[\begin{smallmatrix}1&1&\dots&1&1\\\
g&g-1&\dots&2&1\end{smallmatrix}\right]\mod 2$

the sum of the odd characteristics in this special fundamental system.

If we exclude $e_{1}=\left[\begin{smallmatrix}0\dots 0\\\ 0\dots
0\end{smallmatrix}\right]$, the remaining $2g+1$ characteristics of the
special fundamental system $I$ form an essential basis. Thus any
characteristic $m\in{\mathbb{Z}}_{2}^{2g}$ can be written as a sum of an odd
number among these $2g+1$ characteristics. Since the sum of all
characteristics in $I$ is zero, the sum of any subset of characteristics in
$I$ is equal to the sum of the complementary subset of characteristics in $I$.
Thus altogether we see that every characteristic $m$ can be written uniquely
as a sum of at most $g$ among the characteristics
$o_{1},\dots,o_{g},e_{2},\dots,e_{g+2}$. Suppose $m$ is the sum of $k$ among
these characteristics; then it can be checked that the characteristic
$m+b^{g}$ is even if and only if $k\equiv g$ or $k\equiv g+1\mod 4$. We then
have the following

######  Proposition 12 ([Mum07], [Poo94], [Tsu91]).

There exists an irreducible component ${\mathbb{HJ}}_{g}^{I}$ of
${\mathbb{HJ}}_{g}$ that is defined by the equations

(6) $\theta_{m+b^{g}}(\tau)=0$

for all $m$ that are equal to a sum of strictly less than $g$ among the
characteristics $o_{1},\ldots,o_{g},e_{2},\ldots,e_{g+2}$.

######  Remark 13.

The actual result of Mumford is that if $\tau\in{\mathbb{H}}_{g}$ is such that
$\theta_{m+b^{g}}(\tau)\neq 0$ if and only if $k=g$, then
$\tau\in{\mathbb{HJ}}_{g}^{\circ}$. In the above proposition we do not require
the non-vanishing of those $\theta_{m+b^{g}}(\tau)$ where $m$ is the sum of
precisely $g$ elements of the special fundamental system. Thus clearly the
locus described in the proposition contains an irreducible component of
${\mathbb{HJ}}_{g}$. Furthermore, Poor [Poo94] proved that the vanishing
conditions (6) by themselves cut out an irreducible component of
${\mathbb{HJ}}_{g}\setminus{\mathbb{R}}_{g}$.

The action of $\Gamma_{g}$ is transitive on the set of all special fundamental
systems, and thus one has the following characterization of the hyperelliptic
locus:

######  Proposition 14 ([Mum07], [Poo94]).

An indecomposable period matrix
$\tau\in{\mathbb{H}}_{g}\setminus{\mathbb{R}}_{g}$ lies in ${\mathbb{HJ}}_{g}$
if and only if there exists a special fundamental system
$o^{\prime}_{1},\ldots,o^{\prime}_{g}$,
$e^{\prime}_{1},\dots,e^{\prime}_{g+2}$ such that defining $b^{\prime
g}:=o^{\prime}_{1}+\dots+o^{\prime}_{g}$, the theta constant
$\theta_{m+b^{\prime g}}$ vanishes at $\tau$ if and only if $m$ can be written
as a sum of strictly less than $g$ elements of the special fundamental system.

######  Remark 15.

About the odd counterpart, we observe that, when $g\geq 5$, the theta gradient
with characteristic
$m_{I}=o_{1}+o_{2}+o_{3}+o_{4}+o_{5}=\left[\begin{smallmatrix}111110\dots 0\\\
101010\dots 0\end{smallmatrix}\right]$ vanishes along the component
${\mathbb{HJ}}_{g}^{I}$. Indeed,

$m_{I}=b^{g}+o_{6}+\dots+o_{g}$

is the sum of $b^{g}$ and $g-5$ elements of the special fundamental system,
and this condition implies the vanishing of the gradient of the theta function
at the hyperelliptic point $\tau$, see [Igu80].

######  Remark 16.

We will show that 1 applies to ${\mathbb{HJ}}_{g}^{I}$ , though in fact the
irreducible component that Shepherd-Barron uses in [SB21] is a different one.
Note that ${\mathbb{I}}_{g}$ is contained in multiple irreducible components
of ${\mathbb{HJ}}_{g}$, see the discussion after the proof of 20.

In [Tsu91], Tsuyumine studies the intersection of irreducible components of
${\mathcal{R}}_{g}(2)$ and of ${\mathcal{HJ}}_{g}(2)$. He also shows that the
stabilizer of the component ${\mathcal{HJ}}_{g}^{I}$ of
${\mathcal{HJ}}_{g}(2)$ is isomorphic to the symmetric group $S_{2g+2}$.
Moreover he considers also the boundary components of ${\mathcal{HJ}}_{g}^{I}$
contained in ${\mathcal{R}}_{g}(2)$. While his analysis again is only for
decomposable ppav that are products of two indecomposable ones, his analysis
extends in full generality to yield the statement that all boundary components
related to a decomposition $g=g_{1}+\dots+g_{k}$ are conjugated via the
stabilizer subgroup at ${\mathcal{HJ}}_{g}^{I}$:

######  Lemma 17.

For any two irreducible components $Z$ and $W$ of ${\mathbb{D}}_{g}$ contained
in an irreducible component $X$ of ${\mathbb{HJ}}_{g}$, there exists
$\sigma\in Stab_{X}$ such that $\sigma(Z)=W$.

For any two irreducible components $X$ and $Y$ of ${\mathbb{HJ}}_{g}$
containing ${\mathbb{I}}_{g}$, there exists $\sigma\in
Stab_{{\mathbb{I}}_{g}}$ such that $\sigma(Y)=X$.

###### Proof.

For the first statement, recall that as already discussed in Section 2.2, the
irreducible components of ${\mathbb{HJ}}_{g}$ are in bijection with those of
${\mathcal{HJ}}_{g}(2)$. Hence the stabilizer of $X$ acts transitively on the
set of all its boundary components related to a decomposition $g=1+\dots+1$.

For the second statement, since the cover
${\mathbb{H}}_{g}\to{\mathcal{A}}_{g}$ is Galois, being the quotient by
$\Gamma_{g}$, we know that $\Gamma_{g}$ acts transitively on the set of
irreducible components of ${\mathbb{HJ}}_{g}$, and thus there exists some
$\sigma_{1}\in\Gamma_{g}$ such that $X=\sigma_{1}(Y)$. Denoting
${\mathbb{I}}^{\prime}_{g}:=\sigma_{1}({\mathbb{I}}_{g})$ the irreducible
component of ${\mathbb{D}}_{g}$ that ${\mathbb{I}}_{g}$ is mapped to, by the
first statement there exists $\sigma_{2}\in Stab_{X}$ such that
$\sigma_{2}({\mathbb{I}}^{\prime}_{g})={\mathbb{I}}_{g}$. Thus
$\sigma:=\sigma_{2}\circ\sigma_{1}$ satisfies
$\sigma({\mathbb{I}}_{g})=\sigma_{2}({\mathbb{I}}^{\prime}_{g})={\mathbb{I}}_{g}$
and maps $Y$ to $X$, as required. ∎

##  3\. Expansions of theta functions near the diagonal

Our main computational tool is working with Taylor expansions of defining
equations of our loci near ${\mathbb{I}}_{g}$. We will work in a sufficiently
small analytic neighborhood $U$ of ${\mathbb{I}}_{g}$; that is, we will fix
arbitrary generic $t_{1},\dots,t_{g}\in{\mathbb{H}}_{1}$, and assume that all
$\tau_{ab}$ with $a<b$ satisfy $|\tau_{ab}|<\varepsilon$ for some sufficiently
small $\varepsilon$ (small compared to all $t_{i}$). Since the diagonal period
matrix $\operatorname{diag}(t_{1},\dots,t_{g})$ lies in the open set
${\mathbb{H}}_{g}$, so doing this for every $t_{1},\dots,t_{g}$ we get an open
neighborhood ${\mathbb{I}}_{g}\subset U\subset{\mathbb{H}}_{g}$. We can thus
expand theta constants , theta gradients, etc. with respect to all the
variables $\tau_{ab}$ for $1\leq a<b\leq g$ at a fixed generic point
$\operatorname{diag}(t_{1},\dots,t_{g})\in{\mathbb{I}}_{g}$.

The Taylor expansion of theta constants near ${\mathbb{I}}_{g}$ was recently
used in our work [FGSM21] with H. Farkas on the Schottky problem, and we now
recall it. We also give the formulas for the Taylor expansions of theta
gradients near ${\mathbb{I}}_{g}$, and for the Hessian of the theta function.
These are the formulas that will make all of our results work, and we
introduce various conventions to be able to keep track of the formulas in a
reasonable way.

First of all, we recall that by (3) the theta constant near a diagonal period
matrix in ${\mathbb{I}}_{g}$ factorizes. Furthermore, the $z$-derivatives of
the theta constant factorize the same way. Thus recalling the heat equation
satisfied by theta functions

$\frac{\partial\theta\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right](\tau,z)}{\partial_{\tau_{jj}}}=\frac{1}{4\pi
i}\frac{\partial^{2}\theta\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right](\tau,z)}{\partial z_{j}\partial
z_{j}};\quad\frac{\partial\theta\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right](\tau,z)}{\partial_{\tau_{jk}}}=\frac{1}{2\pi
i}\frac{\partial^{2}\theta\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right](\tau,z)}{\partial z_{j}\partial z_{k}}\
\textrm{for\ }j\neq k$

allows us to evaluate at a point of ${\mathbb{I}}_{g}$ the derivatives of
$\theta\left[\begin{smallmatrix}\varepsilon\\\ \delta\end{smallmatrix}\right]$
with respect to $\tau$. We will be expanding theta functions and their
derivatives in Taylor series near ${\mathbb{I}}_{g}$ with respect to the
variables $\tau_{ab}$ for all $1\leq a<b\leq g$, with each term of the
expansion being a function in the variables $\tau_{11},\dots,\tau_{gg}$, which
we will denote $t_{1},\dots,t_{g}$, to remember that they are elements of the
upper half-plane. To shorten the formulas in the rest of the text, we adopt
the following

######  Convention 1.

For $t_{j}\in{\mathbb{H}}_{1}$ and fixed
$\left[\begin{smallmatrix}\varepsilon\\\ \delta\end{smallmatrix}\right]$ we
denote by

$\vartheta_{j}:=\theta\left[\begin{smallmatrix}\varepsilon_{j}\\\
\delta_{j}\end{smallmatrix}\right](t_{j});\quad\vartheta_{j}^{\prime}:=\frac{\partial}{\partial
z}\theta\left[\begin{smallmatrix}\varepsilon_{j}\\\
\delta_{j}\end{smallmatrix}\right](t_{j},z)|_{z=0};\quad\vartheta_{j}^{\prime\prime}:=\dots$

the one-variable theta functions and their $z$-derivatives evaluated at
$z=0\in{\mathbb{C}}$ (which may be identically zero depending on the parity of
$\left[\begin{smallmatrix}\varepsilon_{j}\\\
\delta_{j}\end{smallmatrix}\right]$).

We write $O(\varepsilon^{N})$ to signify a sum of monomials of total degree at
least $N$ in all the variables $\tau_{ab}$, for all $1\leq a<b\leq g$.

Even with this notation, the formulas, as in [FGSM21], would get very
complicated, so we introduce further conventions to make them more readable.
For a characteristic of the form $\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]=\left[\begin{smallmatrix}1\dots 10\dots 0\\\
1\dots 10\dots 0\end{smallmatrix}\right]$ with the first $\ell$ columns equal
to $\left[\begin{smallmatrix}1\\\ 1\end{smallmatrix}\right]$, we will use
capital letters $J,K,\dots$ to denote columns where the characteristic is
$\left[\begin{smallmatrix}1\\\ 1\end{smallmatrix}\right]$, i.e. $1\leq
J\leq\ell$, and we will use small letters $j,k,\dots$ to denote columns where
the characteristic is $\left[\begin{smallmatrix}0\\\
0\end{smallmatrix}\right]$, i.e. $\ell+1\leq j\leq g$.

We denote $S_{2n}$ the permutation group on $2n$ elements, and by
$T_{2n}\subset S_{2n}$ the set of permutations that can be written as products
of $n$ disjoint transpositions. For $\sigma\in T_{2n}$, we denote by
$\mu\subset\sigma$ one of the transpositions
$\mu:\alpha_{\mu}\longleftrightarrow\beta_{\mu}$ whose product is $\sigma$.

Finally, for a set of an even number of possibly repeating indices
$a_{1},\dots,a_{2n}\in\\{1,\dots,g\\}$ we will denote

$[a_{1},\dots,a_{2n}]:=\frac{1}{(2\pi i)^{n}}\sum_{\sigma\in
T_{2n}}n_{\sigma}\prod_{\mu\subset\sigma}\tau_{a_{\alpha_{\mu}}\,a_{\beta_{\mu}}}\,,$

where the combinatorial coefficient $n_{\sigma}=a_{\sigma}\cdot
b_{\sigma}\cdot c_{\sigma}$ is computed as follows. The factor $a_{\sigma}$ is
equal to $0$ if there exists any $\mu\subset\sigma$ such that
$a_{\alpha_{\mu}}=a_{\beta_{\mu}}$, and is equal to $1$ otherwise. If there
are precisely $N$ elements $\sigma=\sigma_{1},\sigma_{2},\dots,\sigma_{N}\in
T_{2n}$ such that the resulting monomials are equal, then $b_{\sigma}$ is set
to be equal to $1/N$ — so that the result is essentially that each distinct
monomial is counted exactly once in the sum. Finally, for a given $\mu$ we
rewrite

$\prod_{\mu\subset\sigma}\tau_{a_{\alpha_{\mu}}\,a_{\beta_{\mu}}}=\prod_{1\leq
i\leq j\leq g}\tau_{ij}^{d_{ij}}$

by gathering the powers of the same $\tau_{ij}$ together, and let then
$c_{\sigma}:=1/\prod(d_{ij}!)$. The reason for this last factor is that this
is the coefficient with which

$\prod_{\mu\subset\sigma}\partial_{\tau_{a_{\alpha_{\mu}}\,a_{\beta_{\mu}}}}=\prod_{1\leq
i\leq j\leq g}\partial_{\tau_{ij}}^{d_{ij}}$

appears in the Taylor expansion. Note that
$[a_{1},\dots,a_{2n}]=O(\varepsilon^{n})$, as each summand is a degree $n$
monomial in the $\tau$’s. ∎

To unravel this very useful notation, we give some examples:

$[1,1]=0;\quad[1,2]=\frac{1}{2\pi i}\tau_{12};\quad[1,1,2,3]=\frac{1}{(2\pi
i)^{2}}\tau_{12}\cdot\tau_{13};$ $[1,2,3,4]=\frac{1}{(2\pi
i)^{2}}\left(\tau_{12}\cdot\tau_{34}+\tau_{13}\cdot\tau_{24}+\tau_{14}\cdot\tau_{23}\right),$
$[1,1,2,2,3,4]=\frac{1}{(2\pi
i)^{3}}\left(\frac{1}{2}\tau_{12}^{2}\tau_{34}+\tau_{12}\tau_{13}\tau_{24}+\tau_{12}\tau_{14}\tau_{23}\right)\,.$

The reason this notation is so useful for us is that in computing the terms of
the expansion of theta functions and their derivatives we use the heat
equation repeatedly. Each factor $\tau_{ab}$ arises when the corresponding
derivative $\partial_{\tau_{ab}}$ is taken, so then $\vartheta_{a}$ and
$\vartheta_{b}$ are differentiated, by the heat equation. Each summand of the
Taylor expansion of the theta constant in variables $\tau_{ab}$ for all $1\leq
a<b\leq g$ is thus of the form $\prod_{\alpha=1\dots
g}\left(\partial^{n_{\alpha}}\vartheta_{\alpha}\right)$ times the following
polynomial in $\tau$’s:

$[\underbrace{1,\dots,1}_{n_{1}},\underbrace{2,\dots,2}_{n_{2}},\dots,\underbrace{g,\dots,g}_{n_{g}}]$

in our new notation. For the expansion of the derivative
$\partial_{\tau_{ab}}\theta_{m}$, the polynomial multiplying
$\prod_{\alpha=1\dots g}\partial^{n_{\alpha}}\vartheta_{\alpha}$ is similar,
except that $a$ and $b$ will be included in the expression $n_{a}-1$ and
$n_{b}-1$ times, respectively.

As a warm-up, we write down in genus $4$ the expansion of
$\theta_{m}=\theta\left[\begin{smallmatrix}1100\\\
1100\end{smallmatrix}\right]$ near ${\mathbb{I}}_{4}$, using this notation.

(7)
$\displaystyle\theta_{m}=\theta_{m}(\operatorname{diag}(t_{1},t_{2},t_{3},t_{4}))+\sum_{1\leq
a<b\leq
g}\tau_{ab}\frac{\partial\theta_{m}}{\partial\tau_{ab}}(\operatorname{diag}(t_{1},t_{2},t_{3},t_{4}))+\dots$
$\displaystyle=\vartheta_{1}^{\prime}\vartheta_{2}^{\prime}\vartheta_{3}\vartheta_{4}$
$\displaystyle\cdot\left([1,2]+\frac{\vartheta_{1}^{\prime\prime\prime}}{\vartheta_{1}^{\prime}}[1,1,1,2]+\frac{\vartheta_{2}^{\prime\prime\prime}}{\vartheta_{2}^{\prime}}[2,2,1,2]+\frac{\vartheta_{3}^{\prime\prime}}{\vartheta_{3}}[3,3,1,2]+\frac{\vartheta_{4}^{\prime\prime}}{\vartheta_{4}}[4,4,1,2]\right)+O(\varepsilon^{3})\,.$

Of course all of the terms above can easily be written out explicitly, the
terms $[1,1,1,2]$ and $[2,2,1,2]$ are in fact zero, but note that our
conventions make the formula readable. We note in particular that the lowest
order term of this expansion is $[1,2]=O(\varepsilon)$, while in general for
$\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]\in{\mathcal{E}}_{\ell}$, the lowest order term
of the expansion would be of order $O(\varepsilon^{\ell/2})$. From now on, we
denote $s:=\ell/2$, for even characteristics.

######  Convention 2.

To make the formulas still nicer, we finally denote, for a given $\ell$ with
$1\leq\ell\leq g$ (for thinking about even characteristics in
${\mathcal{E}}_{\ell}$ or odd characteristics in ${\mathcal{O}}_{\ell}$)

$f_{\alpha}:=\begin{cases}\vartheta_{\alpha}^{\prime},&\mbox{if
}1\leq\alpha\leq\ell\\\ \vartheta_{\alpha},&\mbox{if }\ell+1\leq\alpha\leq
g.\end{cases}$

and denote

$\phi_{\alpha}:=\frac{f_{\alpha}^{\prime\prime}}{f_{\alpha}};\qquad\psi_{\alpha}:=\frac{f_{\alpha}^{\prime\prime\prime\prime}}{f_{\alpha}}-\phi_{\alpha}^{2}\,,$

where we now use the index $1\leq\alpha\leq g$. ∎

We are now ready to compute the two lowest order terms of the expansion of the
theta constant with characteristics $\left[\begin{smallmatrix}1\dots 10\dots
0\\\ 1\dots 10\dots 0\end{smallmatrix}\right]$ for arbitrary $g$ and arbitrary
$2\leq\ell=2s\leq g$:

(8)
$\theta_{m}=\left([1,\dots,\ell]+\sum_{\alpha}\phi_{\alpha}\cdot[\alpha,\alpha,1,\dots,\ell]\right)\cdot\prod_{\alpha}f_{\alpha}+O(\varepsilon^{s+2})$

For further use, we denote

(9) $X_{\ell}:=[1,\dots,\ell];\quad
Y_{\ell}:=\sum_{\alpha}\phi_{\alpha}\cdot[\alpha,\alpha,1,\dots,\ell]$

these two leading terms (for $\ell$ of any parity), so that the above becomes
$\theta_{m}=X_{\ell}+Y_{\ell}+O(\varepsilon^{s+2})$.

Similarly, we can obtain formulas for the expansions of the derivatives, where
recall we use indices $1\leq J,K\leq\ell$ and $\ell+1\leq j,k\leq g$. We first
deal with the $z$-derivatives, to be used in our investigation of the locus
${\mathcal{G}}_{\rm null}$. In this case the expansion of the theta gradient
$\operatorname{grad}\theta\left[\begin{smallmatrix}1\dots 10\dots 0\\\ 1\dots
10\dots 0\end{smallmatrix}\right]$ where the characteristic lies in
${\mathcal{O}}_{\ell}$, i.e. has an odd number $\ell=2s+1$ of columns equal to
$\left[\begin{smallmatrix}1\\\ 1\end{smallmatrix}\right]$, is

(10) $\frac{\partial\theta\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]}{\partial
z_{J}}(\tau)=\left([1,\dots,\widehat{J},\dots,\ell]+\sum_{\alpha}\phi_{\alpha}\cdot[\alpha,\alpha,1,\dots,\widehat{J},\dots,\ell]\right)\prod
f_{\alpha}+O(\varepsilon^{s+2})$

and

(11) $\displaystyle\frac{\partial\theta\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]}{\partial z_{j}}(\tau)$
$\displaystyle=\left(\phi_{j}\cdot[j,1,\dots,\ell]+\psi_{j}\cdot[j,j,j,1,\dots,\ell]\right.$
$\displaystyle\left.+\phi_{j}\sum_{\alpha}\phi_{\alpha}\cdot[\alpha,\alpha,j,1,\dots,\ell]\right)\prod
f_{\alpha}+O(\varepsilon^{s+3})\,,$

where as usual the hat denotes omission of the index. In what follows we will
actually only need to use the formulas above for $\ell=3$, but setting
$\ell=3$ does not simplify the formula above much, so we have chosen to give
the general expression.

For investigating the rank of the Hessian of the theta function we will need
to compute the second order derivatives of the theta function. While these
formulas can also be written for arbitrary $\ell$, in our arguments we will
only need $\ell=2$, and here this makes the formulas much shorter. So from now
on we take $m=m_{0}=\left[\begin{smallmatrix}110\dots 0\\\ 110\dots
0\end{smallmatrix}\right]$, and first compute for $J=1,2$

$\partial_{\tau_{JJ}}\theta_{m_{0}}=\frac{\prod
f_{\alpha}}{2}\cdot\left(\phi_{J}\cdot X_{2}+\phi_{J}\cdot Y_{2}+\psi_{J}\cdot
X_{2}\right)+O(\varepsilon^{3})\,,$

where the extra factor of $\frac{1}{2}$ is due to $\frac{1}{4\pi i}$ appearing
instead of $\frac{1}{2\pi i}$ in the heat equation, for differentiating with
respect to $\tau_{JJ}$. Next, we compute

$\partial_{\tau_{12}}\theta_{m_{0}}=\prod
f_{\alpha}\cdot\left(1+\sum\phi_{\alpha}\cdot[\alpha,\alpha]+\psi_{1}\cdot[1,1]+\psi_{2}\cdot[2,2]\right)+O(\varepsilon^{2})\,;$
$\partial_{\tau_{1j}}\theta_{m}=\phi_{j}\cdot[j,2]\cdot\prod_{\alpha}f_{\alpha}+O(\varepsilon^{2})\,;$
$\partial_{\tau_{2j}}\theta_{m}=\phi_{j}\cdot[j,1]\cdot\prod_{\alpha}f_{\alpha}+O(\varepsilon^{2})\,;$
$\partial_{\tau_{jk}}\theta_{m}=\phi_{j}\cdot\phi_{k}\cdot[j,k,1,2]\cdot\prod_{\alpha}f_{\alpha}+O(\varepsilon^{3})\,,$

and finally

$\displaystyle\partial_{\tau_{jj}}\theta_{m}$ $\displaystyle=\frac{\prod
f_{\alpha}}{2}\cdot\left([1,2]+\psi_{j}\cdot[j,j,1,2]+\phi_{j}\cdot\sum_{\alpha}\phi_{\alpha}\cdot[\alpha,\alpha,1,2]\right)+O(\varepsilon^{3})$
$\displaystyle=\frac{\prod
f_{\alpha}}{2}\cdot\left(\phi_{j}\cdot(X_{2}+Y_{2})+\psi_{j}\cdot[j,j,1,2]\right)+O(\varepsilon^{3})\,,$

where of course in each of these formulas further terms in the expansion can
also be easily written down, though the formulas become very lengthy.

We give an example: the principal $4\times 4$ minor of the Hessian of
$\theta_{m_{0}}$, formed by the rows and columns numbered $1,2,j,k$ (for
$3\leq j<k\leq g$) is given by $\prod f_{\alpha}$ multiplied by

(12)
$\begin{smallmatrix}\frac{1}{2}\phi_{1}\cdot(X_{2}+Y_{2})&1+\sum_{\alpha,\beta}\phi_{\alpha}\phi_{\beta}\cdot[\alpha,\alpha,\beta,\beta]&\phi_{j}\cdot[2,j]+\phi_{j}\sum\phi_{\alpha}\cdot[\alpha,\alpha,2,j]&\phi_{k}\cdot[2,k]+\phi_{k}\cdot\sum\phi_{\alpha}\cdot[\alpha,\alpha,2,k]\\\
*&\frac{1}{2}\phi_{2}\cdot(X_{2}+Y_{2})&\phi_{j}\cdot[1,j]+\phi_{j}\sum\phi_{\alpha}\cdot[\alpha,\alpha,1,j]&\phi_{k}\cdot[1,k]+\phi_{k}\sum\phi_{\alpha}\cdot[\alpha,\alpha,1,k]\\\
*&*&\frac{1}{2}\phi_{j}\cdot(X_{2}+Y_{2})+\frac{1}{2}\psi_{j}\cdot[j,j,1,2]&\phi_{j}\cdot\phi_{k}\cdot[j,k,1,2]\\\
*&*&*&\frac{1}{2}\phi_{k}\cdot(X_{2}+Y_{2})+\frac{1}{2}\psi_{k}\cdot[k,k,1,2]\end{smallmatrix}$

where the $*$’s below the diagonal signify the fact that the matrix is
symmetric, and we have expanded the matrix up to dropping the
$\varepsilon^{3}$ terms.

##  4\. The local form of the hyperelliptic locus

In this section we describe the hyperelliptic locus near ${\mathbb{I}}_{g}$,
proving 20, which is a slightly stronger version of 1. We first check that
using the special fundamental system (4), the resulting irreducible component
${\mathbb{HJ}}_{g}^{I}$ of ${\mathbb{HJ}}_{g}\subset{\mathbb{H}}_{g}$ given by
12 contains the locus ${\mathbb{I}}_{g}$ of diagonal period matrices. This
follows from the following

######  Lemma 18.

If $m\in{\mathcal{E}}$ can be written as a sum of strictly less than $g$ among
the characteristics $o_{1},\dots,o_{g},e_{2},\dots,e_{g+2}$ of the special
fundamental system (4), then $m+b^{g}\in{\mathcal{E}}^{*}$.

###### Proof.

We proceed by induction on $g$, with the lemma being easily true for $g=1,2$.
Then observe that deleting the characteristics $o_{g}$ and $e_{g+2}$ from the
chosen special fundamental system, and forgetting the $g$-th column of each
characteristic gives the special fundamental system of genus $g-1$. Note now
that the last column of $b^{g}$ is equal to $\left[\begin{smallmatrix}1\\\
1\end{smallmatrix}\right]$. Thus unless the sum for $m$ includes one of the
characteristics that have a non-zero $g$’th column, the characteristic
$m+b^{g}$ also has the last column $\left[\begin{smallmatrix}1\\\
1\end{smallmatrix}\right]$, and thus does not lie in ${\mathcal{E}}_{0}$. The
only three characteristics among the special fundamental system that have a
non-zero $g$’th column are $o_{g},e_{g+1},e_{g+2}$. If $m$ is a sum of
characteristics including $o_{g}$, then we observe that
$b^{g}+o_{g}=b^{g-1}\oplus\left[\begin{smallmatrix}0\\\
0\end{smallmatrix}\right]$, and thus the statement follows from the inductive
assumption, by ignoring the $g$’th column of characteristics (since if any of
the first $g-1$ columns of a genus $g$ characteristic is equal to
$\left[\begin{smallmatrix}1\\\ 1\end{smallmatrix}\right]$, it is not in
${\mathcal{E}}_{0}$, and we are now using one less characteristic in the sum
for $m$). If $m$ is a sum of characteristics not including $o_{g}$, but
including exactly one of $e_{g+1}$ or $e_{g+2}$, we note that
$b^{g}+e_{g+1}=b^{g-1}\oplus\left[\begin{smallmatrix}0\\\
1\end{smallmatrix}\right]$ and
$b^{g}+e_{g+2}=b^{g-1}\oplus\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]$, and the same argument applies as for the previous
case of $o_{g}$. Finally, if both $e_{g+1}$ and $e_{g+2}$ are used in this sum
representation, we note that since the bottom characteristic of
$e_{g+1}+e_{g+2}$ is equal to zero vector in ${\mathbb{Z}}_{2}^{g}$, we can
again proceed by induction. ∎

######  Corollary 19.

All theta constants $\theta_{m+b^{g}}$ vanishing along the hyperelliptic
component ${\mathbb{HJ}}_{g}^{I}$ also vanish along ${\mathbb{I}}_{g}$.

We are now ready to prove the most precise version of our result on the
hyperelliptic locus.

######  Theorem 20.

For any irreducible component ${\mathbb{X}}$ of the hyperelliptic locus
${\mathbb{HJ}}_{g}\subset{\mathbb{H}}_{g}$, such that
${\mathbb{X}}\supset{\mathbb{I}}_{g}$, the tangent space to ${\mathbb{X}}$ at
any point of ${\mathbb{I}}_{g}$ is the set of period matrices satisfying the
set of equations $\\{\tau_{\pi(i)\pi(j)}=0\\}_{\forall 1\leq i,j\leq
g,|i-j|>1}$, where $\pi\in S_{g}$ is the permutation that is the image in
$S_{g}$ of the element $\sigma\in Stab_{{\mathbb{I}}_{g}}$ that sends
${\mathbb{HJ}}_{g}^{I}$ to ${\mathbb{X}}$.

###### Proof.

We first use 18 to prove 1, showing that the tangent space to
${\mathbb{HJ}}^{I}_{g}$ at any point of ${\mathbb{I}}_{g}$ is given by
equations $\tau_{jk}=0$ for all $|j-k|>1$.

Indeed, recall that $b^{g}=o_{1}+\dots+o_{g}$, and consider a characteristic
of the form

$m=b^{g}+o_{1}+\dots+\widehat{o_{j_{1}}}+\dots+\widehat{o_{j_{2}}}+\dots+\widehat{o_{j_{3}}}+\dots+o_{g}=o_{j_{1}}+o_{j_{2}}+o_{j_{3}}\,,$

with $1\leq j_{1}<j_{2}<j_{3}\leq g$. From the first expression for $m$ it
follows that $\theta_{m}$ vanishes identically on ${\mathbb{HJ}}^{I}_{g}$.
Writing the columns of $m$ as $\left[\begin{smallmatrix}\varepsilon_{i}\\\
\delta_{i}\end{smallmatrix}\right]$, we compute that

$\varepsilon_{j_{1}}=\varepsilon_{j_{2}}=\varepsilon_{j_{3}}=\delta_{j_{1}}=\delta_{j_{3}}=1$

and that all other $\varepsilon_{i}$ and $\delta_{i}$ are zero (note that this
includes $\delta_{j_{2}}=0$). Thus $m\in{\mathcal{E}}_{2}$, in agreement with
18, and in particular the lowest order term of the expansion of $\theta_{m}$
near ${\mathbb{I}}_{g}$ is linear, given explicitly by (8) (where we use
$\ell=2$, see the genus 4 example there) as

(13) $\theta_{m}(\tau)=\frac{1}{2\pi
i}\tau_{j_{1}j_{3}}\vartheta_{j_{1}}^{\prime}\vartheta_{j_{3}}^{\prime}\prod_{h\neq
j_{1},j_{3}}\vartheta_{h}$

in our conventions. Going over all possible choices of $1\leq
j_{1}<j_{2}<j_{3}\leq g$ means that in the above expressions there appear all
$\tau_{j_{1}j_{3}}$ such that $j_{3}-j_{1}>1$. Recalling that we are working
within the space of symmetric matrices, it follows that the tangent space to
${\mathbb{HJ}}^{I}$ at a point in ${\mathbb{I}}_{g}$ is contained in the locus
of tridiagonal matrices given by equations
$\\{\tau_{jk}=0\\}_{\forall|k-j|>1}$. Since the hyperelliptic locus
${\mathcal{HJ}}_{g}$ is of dimension $2g-1$, any irreducible component of its
preimage ${\mathbb{HJ}}_{g}$ is also of dimension $2g-1$. Thus the tangent
space to ${\mathbb{HJ}}^{I}_{g}$ at any point of ${\mathbb{I}}_{g}$ must be of
dimension at least $2g-1$. Since this tangent space is contained in the space
of tridiagonal matrices, which also has dimension $2g-1$, it must be equal to
it, and 1 is thus proven.

We now combine this with the study of the action of $\Gamma_{g}$ on
irreducible components of ${\mathbb{HJ}}_{g}$ to obtain the full statement. By
17, for any irreducible component ${\mathbb{X}}$ of ${\mathbb{HJ}}_{g}$
containing ${\mathbb{I}}_{g}$, there exists $\sigma\in
Stab_{{\mathbb{I}}_{g}}$ such that
$\sigma({\mathbb{HJ}}^{I}_{g})={\mathbb{X}}$. Thus the tangent space to
${\mathbb{X}}$ is the image of the tangent space to ${\mathbb{HJ}}^{I}_{g}$
under the action of $\sigma$. Since clearly the action of $\Gamma_{1}$ on each
column of the characteristic does not change the linear term of the expansion
(13), it follows that the action of $\sigma$ on the linear equations for the
tangent space to ${\mathbb{HJ}}^{I}_{g}$ along ${\mathbb{I}}_{g}$ is simply by
permuting the columns of the period matrix according to the image of the
permutation $\sigma$ under the surjection $Stab_{{\mathbb{I}}_{g}}\to S_{g}$.
∎

######  Remark 21.

From the proof of the theorem we see that multiple irreducible components of
${\mathbb{HJ}}_{g}$ contain ${\mathbb{I}}_{g}$, and many of them may have the
same tangent space along ${\mathbb{I}}_{g}$. This can already be seen in the
first interesting case of $g=3$. Recall that ${\mathbb{HJ}}_{3}$ has 36
irreducible components, each one being the zero locus of one of the 36 even
genus 3 theta constants. Nine of these 36 components, those with
characteristics in ${\mathcal{E}}^{*}={\mathcal{E}}_{2}$, contain
${\mathbb{I}}_{3}$: namely these correspond to characteristics

$\left[\begin{smallmatrix}110\\\
110\end{smallmatrix}\right],\,\,\left[\begin{smallmatrix}111\\\
110\end{smallmatrix}\right],\,\,\left[\begin{smallmatrix}110\\\
111\end{smallmatrix}\right],\,\,\left[\begin{smallmatrix}101\\\
101\end{smallmatrix}\right],\,\,\left[\begin{smallmatrix}111\\\
101\end{smallmatrix}\right],\,\,\left[\begin{smallmatrix}101\\\
111\end{smallmatrix}\right],\,\,\left[\begin{smallmatrix}011\\\
011\end{smallmatrix}\right],\,\,\left[\begin{smallmatrix}111\\\
011\end{smallmatrix}\right],\,\,\left[\begin{smallmatrix}011\\\
111\end{smallmatrix}\right]\,.$

The irreducible components of ${\mathbb{HJ}}_{3}$ corresponding to each of the
first three characteristics have local equation $\tau_{12}=0$ near
${\mathbb{I}}_{3}$; the next three components have local equation
$\tau_{13}=0$, and the final three cases have local equation $\tau_{23}=0$.

For the next, more interesting, case of $g=4$, a similar analysis can be
performed, noting that here 10 even theta constants vanish on every
irreducible component of ${\mathbb{HJ}}_{4}$. As a side remark, notice that no
matter how the columns are permuted, the three local defining equations for
the tangent space of a component of ${\mathbb{HJ}}_{4}$ near
${\mathbb{I}}_{4}$ cannot be of the form $\tau_{12}=\tau_{13}=\tau_{14}=0$.
Indeed, this corresponds to the fact that the locus of products
${\mathbb{H}}_{1}\times{\mathbb{H}}_{3}$ (which locally along
${\mathbb{I}}_{g}$ is given by precisely these three equations) is not
contained in ${\mathbb{HJ}}_{4}$, as
${\mathbb{HJ}}_{3}\subsetneq{\mathbb{H}}_{3}$.

In arbitrary genus, we see that the only element of $S_{g}$ that fixes the set
of equations $\\{\tau_{jk}=0\\}_{\forall\,|j-k|>1}$, is the product of
transpositions $\pi=(1,g)(2,g-1)\dots(\frac{g+3}{2},\frac{g}{2})$. Thus there
are altogether $g!/2$ possible different tangent spaces along
${\mathbb{I}}_{g}$ to the different irreducible components of
${\mathbb{HJ}}_{g}$ containing ${\mathbb{I}}_{g}$.

##  5\. The theta-null divisor and the vanishing gradient locus

We now proceed to investigate the loci $\vartheta_{\rm null}$ and
${\mathcal{G}}_{\rm null}$ locally near ${\mathcal{D}}_{g}$. The method we use
here, and then also for dealing with the Hessian rank loci, will be different,
and more general, than what we have done for the hyperelliptic locus.

The outline of our argument is as follows. First we use the expansion of theta
functions near ${\mathbb{I}}_{g}$, as computed in Section 3, to determine the
dimensions of the tangent spaces near ${\mathbb{I}}_{g}$ to the loci given by
equations $\theta_{m_{0}}(z)=0$ (resp.
$\operatorname{grad}\theta_{m_{0}}(z)=0$), where $m_{0}$ is the simplest even
(resp. odd) characteristic, i.e. a characteristic that lies in
${\mathcal{E}}_{2}$ (resp. ${\mathcal{O}}_{3}$) for which the expansions are
given. This is done by intersecting the Taylor expansions of the defining
equations with a suitable “transverse” subvariety, to simplify computations.

As one can already see from the formulas in Section 3, and from the proofs of
these statements, such expansions near ${\mathbb{I}}_{g}$ for characteristics
in ${\mathcal{E}}_{\ell}$ or ${\mathcal{O}}_{\ell}$ with $\ell\gg 2$ can
become very complicated — essentially as the corresponding locus contains the
diagonal ${\mathbb{I}}_{g}$ with high multiplicity. Thus to deal with
arbitrary characteristics, which is necessary to understand the loci
$\vartheta_{\rm null},{\mathcal{G}}_{\rm null}\subset{\mathcal{A}}_{g}$, we
will act by $\Gamma_{g}$, and will have to use the fact that the loci of
interest contain the “big” diagonal ${\mathbb{L}}_{g}^{e}$ or
${\mathbb{L}}_{g}^{o}$ defined below, which consists of block-diagonal
matrices that have two to four $1\times 1$ blocks, and the remaining blocks
are $2\times 2$. Since the setwise stabilizer $Stab_{{\mathbb{L}}_{g}}$
contains suitable subgroups of $G_{g}$, which by 11 permute all even (resp.
odd) characteristics, it will turn out that the statement for an arbitrary
characteristic $m$ will reduce to the statement for $m_{0}$ — and 25 is a
general statement to this effect.

###  5.1. Local structure of $\theta_{\rm m_{0},\,null}$ and
$\operatorname{grad}_{m_{0},\,\rm null}$ near the diagonal

We will start by explicit computations for the characteristics in
${\mathcal{E}}_{2}$ and ${\mathcal{O}}_{3}$. The even case is straightforward,
using the first term of the Taylor expansion.

######  Proposition 22.

Denote $m_{0}:=\left[\begin{smallmatrix}110\dots 0\\\ 110\dots
0\end{smallmatrix}\right]\in{\mathcal{E}}_{2}$. Then in a sufficiently small
neighborhood of ${\mathbb{I}}_{g}$, the locus

$\theta_{\rm
m_{0},\,null}=\\{\tau\in{\mathbb{H}}_{g}\colon\theta_{m_{0}}(\tau)=0\\}$

is smooth, of codimension $1$ in ${\mathbb{H}}_{g}$.

###### Proof.

Since we are dealing with one non-trivial equation, it is clear that the locus
is of codimension one, so the point of the statement is smoothness. For this,
we observe that by (8) the local defining equation admits the expansion

$\theta_{m_{0}}(\tau)=c\tau_{12}+O(\varepsilon^{2})$

for some non-zero $c$. Thus clearly the zero locus is smooth, of codimension
one. ∎

Of course by acting by $Stab_{2}$ we obtain the same statement for
$\theta_{\rm m,\,null}$ for any $m=\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]\in{\mathcal{E}}_{2}$ — and in this case the
local lowest order defining equation is $\tau_{ab}=0$, where $a$ and $b$ are
the two $\left[\begin{smallmatrix}1\\\ 1\end{smallmatrix}\right]$ columns of
the characteristics..

The odd case is much more elaborate, as there are $g$ components of the
gradient giving the $g$ defining equations of ${\mathcal{G}}_{\rm null}$, and
we want to check that these equations are locally independent. This is hard to
do directly, and for bounding the dimension of local irreducible components of
various loci near ${\mathbb{I}}_{g}$, we will use the following well-known
statement.

######  Lemma 23.

Let $X={\operatorname{Spec}}\left({\mathbb{C}}[x_{1},\dots,x_{N}]/\langle
F_{1},\dots,F_{k}\rangle\right)$ be an irreducible affine subscheme of
${\mathbb{A}}^{N}={\operatorname{Spec}}\,{\mathbb{C}}[x_{1},\dots,x_{N}]$, of
dimension $n=\dim X$. Suppose $x\in X\subset{\mathbb{A}}^{N}$, and let
${\mathfrak{M}}_{x}\subset{\mathcal{O}}_{{\mathbb{A}}^{N},x}$ be the maximal
ideal of the local ring. For any $h\geq 1$ let $N(h)$ denote the number of
algebraically independent among the images of $F_{1},\dots,F_{k}$ modulo
$({\mathfrak{M}}_{x})^{h}$. Then the inequality $n\leq N-N(h)$ holds for any
$h$.

We note that of course if $X$ is a not necessarily irreducible affine scheme,
the lemma shows that every irreducible component of $X$ that contains $x$ must
have dimension at most $N-N(h)$.

###### Proof.

We denote ${\mathfrak{m}}_{x}\subset{\mathcal{O}}_{X,x}$ the maximal ideal;
since $X$ is $n$-dimensional, we of course have
$\dim\left({\mathcal{O}}_{X,x}/({\mathfrak{m}}_{x}^{h})\right)=O(n^{h})$ as
$h\to\infty$.

Note now that the map
${\mathcal{O}}_{{\mathbb{A}}^{N},x}\to{\mathcal{O}}_{X,x}$ maps the ideal
$\langle F_{1},\dots,F_{k}\rangle$ to $0$, and maps ${\mathfrak{M}}_{x}$ onto
${\mathfrak{m}}_{x}$. Thus we have the bound

$\dim\left({\mathcal{O}}_{{\mathbb{A}}^{N},x}/\langle
F_{1},\dots,F_{k},({\mathfrak{M}}_{x})^{h}\rangle\right)\geq\dim\left({\mathcal{O}}_{X,x}/({\mathfrak{m}}_{x})^{h}\right)\,,$

while the map of the completions

$\widehat{{\mathcal{O}}_{{\mathbb{A}}^{N},x}}/\langle
F_{1},\dots,F_{k}\rangle\to\widehat{{\mathcal{O}}_{X,x}}$

is surjective. Since the function $N(h)$ is monotone, and bounded above by
$N$, it must have a limit, and from the above inequalities we thus obtain

$n\leq N-\lim_{h\to\infty}N(h)\leq N-N(h)$

for any $n$. ∎

We now obtain the dimension result for odd characteristics in
${\mathcal{O}}_{3}$.

######  Proposition 24.

Denote $m_{0}:=\left[\begin{smallmatrix}1110\dots 0\\\ 1110\dots
0\end{smallmatrix}\right]$. Then in a sufficiently small neighborhood of
${\mathbb{I}}_{g}$, the locus

$\operatorname{grad}_{m_{0},\,\rm
null}=\\{\tau\in{\mathbb{H}}_{g}\colon\operatorname{grad}\theta_{m_{0}}(\tau)=0\\}$

is of codimension $g$ in ${\mathbb{H}}_{g}$.

###### Proof.

We will apply 23 for $X=\operatorname{grad}_{m_{0},\,\rm null}$, and consider
the intersection of $X$ with the locus $Y$ given by equations

$\\{\tau_{jk}=0\quad\forall\,4\leq j<k\leq
g\\}\qquad\hbox{and}\qquad\\{\tau_{1j}=\tau_{2j}=\tau_{3j}\quad\forall\,4\leq
j\leq g\\}\,.$

These are altogether

$\frac{(g-3)(g-4)}{2}+2(g-3)=\frac{g(g-3)}{2}$

equations, which are clearly independent, and all of which are of course
satisfied on ${\mathbb{I}}_{g}$. Thus ${\mathbb{I}}_{g}\subset Y$, and
$\operatorname{codim}_{{\mathbb{H}}_{g}}Y=\tfrac{g(g-3)}{2}$. We claim that in
a sufficiently small neighborhood $U$ of ${\mathbb{I}}_{g}$ we have $X\cap
Y\cap U={\mathbb{I}}_{g}$. Since $\dim{\mathbb{I}}_{g}=g$, this would imply
that

$\dim(X\cap
U)\leq\dim{\mathbb{I}}_{g}+\operatorname{codim}_{{\mathbb{H}}_{g}}Y=g+\frac{g(g-3)}{2}=\frac{g(g-1)}{2}\,,$

and thus that the codimension of $X\cap U$ in $U$ is at least $g$. Since
$X\subset{\mathbb{H}}_{g}$ is given by $g$ equations, it follows that
$\operatorname{codim}_{{\mathbb{H}}_{g}}X\cap U=g$.

To see that $X\cap Y\cap U={\mathbb{I}}_{g}$, we simply plug in the defining
equations of $Y$ into the expansion (10) of the theta gradient computed above,
always excluding the common factor of $\prod f_{\alpha}$, which is non-zero at
a generic point of ${\mathbb{I}}_{g}$:

$\partial_{z_{1}}\theta_{m_{0}}\equiv[2,3]+\sum_{\alpha}\phi_{\alpha}\cdot[\alpha,\alpha,2,3]\equiv\tau_{23}+\phi_{1}\tau_{12}\tau_{13}+\sum_{j\geq
4}\phi_{j}\tau_{2j}\tau_{3j}\mod{\mathfrak{m}}^{3}\,,$

where we recall that $t_{1},\dots,t_{g}$ are considered fixed, so that
${\mathfrak{m}}=\langle\left\\{\tau_{ab}\right\\}_{1\leq a<b\leq g}\rangle$.
Now substituting into this the defining equations for $Y$ we obtain

$\partial_{z_{1}}\theta_{m}|_{Y}\equiv\tau_{23}+\phi_{1}\tau_{12}\tau_{13}+\sum_{j\geq
4}\phi_{j}\tau_{1j}^{2}\mod{\mathfrak{m}}^{3}\,,$

and of course the expressions for $\partial_{z_{2}}\theta_{m}|_{Y}$ and
$\partial_{z_{3}}\theta_{m}|_{Y}$ are completely analogous:

$\partial_{z_{2}}\theta_{m}|_{Y}\equiv\tau_{13}+\phi_{1}\tau_{12}\tau_{23}+\sum_{j\geq
4}\phi_{j}\tau_{1j}^{2}\mod{\mathfrak{m}}^{3}\,,$
$\partial_{z_{3}}\theta_{m}|_{Y}\equiv\tau_{12}+\phi_{1}\tau_{13}\tau_{23}+\sum_{j\geq
4}\phi_{j}\tau_{1j}^{2}\mod{\mathfrak{m}}^{3}\,.$

For the partial $z$-derivatives in the other directions we obtain from (11)

$\displaystyle\partial_{z_{j}}\theta_{m}$
$\displaystyle\equiv\phi_{j}\cdot([j,1,2,3]+\sum\phi_{\alpha}\cdot[\alpha,\alpha,j,1,2,3])+\psi_{j}\cdot[j,j,j,1,2,3]$
$\displaystyle\equiv\phi_{j}\cdot\left(\tau_{1j}\tau_{23}+\tau_{2j}\tau_{13}+\tau_{3j}\tau_{12}+\sum\phi_{\alpha}\cdot[\alpha,\alpha,j,1,2,3]\right)+\psi_{j}\tau_{1j}\tau_{2j}\tau_{3j}\mod{\mathfrak{m}}^{4}\,.$

We compute $[1,1,j,1,2,3]|_{Y}=\tau_{12}\tau_{13}\tau_{1j}$,
$[j,j,j,1,2,3]|_{Y}=\tau_{1j}^{3}$, and

$[k,k,j,1,2,3]|_{Y}=\tau_{1j}\tau_{2k}\tau_{3k}+\tau_{2j}\tau_{1k}\tau_{3k}+\tau_{3j}\tau_{1k}\tau_{2k}=3\tau_{1j}\tau_{1k}^{2}\,,$

so that we can finally evaluate

$\displaystyle\partial_{z_{j}}\theta_{m}|_{Y}$
$\displaystyle\equiv\phi_{j}\tau_{1j}\cdot\left(\tau_{23}+\tau_{13}+\tau_{12}+\phi_{1}\tau_{12}\tau_{13}+\phi_{2}\tau_{12}\tau_{23}+\phi_{3}\tau_{13}\tau_{23}\right)$
$\displaystyle+3\phi_{j}\tau_{1j}\sum_{k\geq
4}\phi_{k}\cdot\tau_{1k}^{2}+\psi_{j}\tau_{1j}^{3}\mod{\mathfrak{m}}^{4}\,.$

We are interested in the locus $Y\cap\operatorname{grad}_{\rm
null}\left[\begin{smallmatrix}\varepsilon\\\ \delta\end{smallmatrix}\right]$,
and thus we can substitute the expansions of the equations
$\partial_{z_{1}}\theta_{m}|_{Y}=\partial_{z_{2}}\theta_{m}|_{Y}=\partial_{z_{3}}\theta_{m}|_{Y}=0$
in the last equation, obtaining

$\displaystyle\partial_{z_{j}}\theta_{m}|_{Y}$
$\displaystyle\equiv\phi_{j}\tau_{1j}\cdot(\partial_{z_{1}}\theta_{m}|_{Y}+\partial_{z_{2}}\theta_{m}|_{Y}+\partial_{z_{3}}\theta_{m}|_{Y})$
$\displaystyle+\tau_{1j}^{3}(\psi_{j}-2\phi_{j}^{2})\mod{\mathfrak{m}}^{4}\,.$

Since $\psi_{j}-2\phi_{j}^{2}$ is a not identically zero function on
${\mathbb{H}}_{1}$, it does not vanish at a generic point of
${\mathbb{I}}_{g}$, and thus the vanishing of
$\operatorname{grad}\theta_{m_{0}}\mod{\mathfrak{m}}^{4}$ implies
$\tau_{1j}=0$ for all $4\leq j\leq g$, which together with the defining
equations for $Y$ implies that $\tau$ is diagonal except possibly for the
entries $\tau_{12},\tau_{23},\tau_{13}$. However, the vanishing of
$z_{1},z_{2},z_{3}$ derivatives $\mod{\mathfrak{m}}^{2}$ gives the equations
$\tau_{12}=\tau_{23}=\tau_{13}$. Thus altogether it follows that every
irreducible component of $\operatorname{grad}_{m_{0},\,\rm null}$ that
contains ${\mathbb{I}}_{g}$ has codimension at least $g$ in
${\mathbb{H}}_{g}$, and thus codimension precisely $g$. ∎

###  5.2. Theta-null and gradient loci for arbitrary characteristics

Note that the computations above for $m_{0}\in{\mathcal{O}}_{3}$ are already
quite involved. If we wanted to deal with the locus $\theta_{m}(\tau)=0$ or
$\operatorname{grad}\theta_{m}(\tau)=0$ for $\ell\gg 3$, the computation would
be daunting, as we would need to consider the Taylor expansion to order
$\ell/2$, see eg. (8). Thus instead we will use the action of $G_{g}$, and the
fact that the loci we are interested in contain the big diagonal, defined
below.

The loci we deal with will be defined in terms of the geometry of the theta
divisor. While just thinking of the locus of ppav such that the theta divisor
satisfies some geometric condition (eg has a singular point with some
properties) only defines a locus set-theoretically as a subset of
${\mathcal{A}}_{g}$, of course the loci we are interested in are in fact
algebraic. However, thinking of them as subvarieties of ${\mathcal{A}}_{g}$,
by arguing that the conditions that define them are algebraic is also
insufficient for our purposes. Indeed, for us it is important to consider
these as subschemes of ${\mathcal{A}}_{g}$ — as it is for Mumford’s
computation [Mum83] of the class of the Andreotti-Mayer divisor, and in
general for thinking about the Andreotti-Mayer loci.

The way we think of the scheme structure on these loci is as follows: recall
that the universal cover of the universal family
${\mathcal{X}}_{g}\to{\mathcal{A}}_{g}$ of ppav is
${\mathbb{H}}_{g}\times{\mathbb{C}}^{g}$, with the covering group being the
semidirect product
$\operatorname{Sp}(2g,{\mathbb{Z}})\rtimes{\mathbb{Z}}^{2g}$. The theta
function is a global holomorphic function on
${\mathbb{H}}_{g}\times{\mathbb{C}}^{g}$, and various geometric conditions on
the singularities of the theta divisor can then be interpreted as various
analytic equations on ${\mathbb{H}}_{g}\times{\mathbb{C}}^{g}$ involving the
theta function and its partial derivatives.

For the purposes of obtaining the results below, we will only be interested in
the geometry of the theta divisor at the two-torsion points of the ppav, and
these conditions can be defined analytically over ${\mathbb{H}}_{g}$ in terms
of theta constants with characteristics and their derivatives. The loci thus
defined naturally come with an analytic defining ideal on ${\mathbb{H}}_{g}$;
the defining ideal is invariant under the action of $\Gamma_{g}$ (or
$\Gamma_{g}(2)$ depending on the context), and thus images of these in
${\mathcal{A}}_{g}$ (or ${\mathcal{A}}_{g}(2)$) have natural defining
algebraic equations, and thus a natural scheme structure. This explains our
care in describing the following general setup.

For any $m=\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]\in{\mathcal{E}}$ we set $\theta_{\rm
m,\,null}:=\theta_{\rm null}{\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]}$, and for any
$m=\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]\in{\mathcal{O}}$ we set
$\operatorname{grad}\theta_{\rm m,\,null}:=\operatorname{grad}\theta_{\rm
null}{\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]}$. Recall that by definition $\theta_{\rm
null}=\cup_{m\in{\mathcal{E}}}\theta_{\rm m,\,null}$.

Given any analytic subspace ${\mathbb{X}}\subset{\mathbb{H}}_{g}$, which will
be contained in either $\theta_{\rm null}$ or $\operatorname{grad}_{\rm
null}$, depending on the context, we decompose it as
${\mathbb{X}}=\cup_{m}{\mathbb{X}}_{m}$, where
${\mathbb{X}}_{m}:={\mathbb{X}}\cap\theta_{\rm m,\,null}$ (or respectively
${\mathbb{X}}_{m}:={\mathbb{X}}\cap\operatorname{grad}\theta_{\rm m,\,null}$).
We note that ${\mathbb{X}}_{m}\cap{\mathbb{X}}_{n}$ may be non-empty.

Assume ${\mathbb{X}}\subset\theta_{\rm null}$ (or
${\mathbb{X}}\subset\operatorname{grad}_{\rm null}$) is an analytic subspace
of ${\mathbb{H}}_{g}$ satisfying $\Gamma_{g}\circ{\mathbb{X}}={\mathbb{X}}$
(as a set), so in particular $\Gamma_{g}$ acts transitively on the set of
${\mathbb{X}}_{m}$ for all $m\in{\mathcal{E}}$, and for any $m$ the setwise
stabilizer $\Gamma_{m}$ of ${\mathbb{X}}_{m}$ contains $\Gamma_{g}(2)$.
Denoting $X:=p({\mathbb{X}})\subset{\mathcal{A}}_{g}$ the image, observe that
$p^{-1}(X)={\mathbb{X}}$, and there exists a well-defined scheme structure on
$X$ induced by the defining equations of ${\mathbb{X}}$.

We will be interested in computing the dimension of irreducible components of
$X$ containing ${\mathcal{D}}_{g}$. This is related to computing the dimension
of irreducible components of ${\mathbb{X}}$ containing ${\mathbb{I}}_{g}$,
which we will approach via Taylor expansions in the neighborhood of
${\mathbb{I}}_{g}$. The difficulty is that a priori the scheme $X$ may have
embedded components containing ${\mathcal{D}}_{g}$, and thus thinking of $X$
as a subvariety may not suffice. Essentially the difficulty is that if
${\mathbb{I}}_{g}\subset{\mathbb{X}}_{m}$, and we can determine irreducible
components of ${\mathbb{X}}_{m}$ containing ${\mathbb{I}}_{g}$, it could be
that also ${\mathbb{I}}_{g}\subset{\mathbb{X}}_{n}$ for some other $n$, and
the image in ${\mathcal{A}}_{g}$ of an irreducible component of
${\mathbb{X}}_{n}$ containing ${\mathbb{I}}_{g}$ may be strictly contained in
the image in ${\mathcal{A}}_{g}$ of an irreducible component of
${\mathbb{X}}_{m}$ containing ${\mathbb{I}}_{g}$. We will deal with this by
explicitly imposing the additional assumption that a component contains the
big diagonal that we now define (this condition will hold for all those loci
that we are interested in).

For the even case (i.e. when we are interested in
${\mathbb{X}}\subset\theta_{\rm null}$), we define the big diagonal as

${\mathbb{L}}_{g}^{e}:={\mathbb{H}}_{1}\times{\mathbb{H}}_{1}\times{\mathbb{H}}_{2}\times\dots\times{\mathbb{H}}_{2}\quad{\rm
or}\quad{\mathbb{L}}_{g}^{e}:={\mathbb{H}}_{1}\times{\mathbb{H}}_{1}\times{\mathbb{H}}_{2}\times\dots\times{\mathbb{H}}_{2}\times{\mathbb{H}}_{1}\,,$

(where the presence of the last factor depends on the parity of $g$) as the
locus of period matrices that have one less than the maximal possible number
of $2\times 2$ blocks along the diagonal. The direct product
$L_{g}^{e}:=\Gamma_{1}\times\Gamma_{1}\times\Gamma_{2}\times\dots\times\Gamma_{2}$
or
$L_{g}^{e}:=\Gamma_{1}\times\Gamma_{1}\times\Gamma_{2}\times\dots\times\Gamma_{2}\times\Gamma_{1}$
(where the presence of the last factor depends on the parity of $g$) is
clearly contained in the stabilizer $Stab_{{\mathbb{L}}_{g}^{e}}$. Finally, in
the even case we set $m_{0}:=\left[\begin{smallmatrix}110\dots 0\\\ 110\dots
0\end{smallmatrix}\right]\in{\mathcal{E}}_{2}$, so that $\theta_{\rm
m_{0},\,null}\supset{\mathbb{I}}_{g}$.

Similarly for the odd case of ${\mathbb{X}}\subset\operatorname{grad}_{\rm
null}$ we define the big diagonal to be
${\mathbb{L}}_{g}^{o}:={\mathbb{H}}_{1}\times{\mathbb{L}}_{g-1}^{e}$, and note
that its stabilizer contains the direct product $L_{g}^{o}:=\Gamma_{1}\times
L_{g}^{e}$. In this case we set $m_{0}:=\left[\begin{smallmatrix}1110\dots
0\\\ 1110\dots 0\end{smallmatrix}\right]\in{\mathcal{O}}_{3}$, so that again
$\operatorname{grad}_{m_{0},\,\rm null}\supset{\mathbb{I}}_{g}$.

We will use these to investigate irreducible components locally near
${\mathbb{I}}_{g}$ by applying the following statement.

######  Proposition 25.

Let ${\mathbb{X}}\subset\theta_{\rm null}$ (resp.
${\mathbb{X}}\subset\operatorname{grad}_{\rm null}$) be an analytic subspace
of ${\mathbb{H}}_{g}$ containing ${\mathbb{I}}_{g}$, such that
${\mathbb{X}}=\cup{\mathbb{X}}_{m}$ is invariant under $\Gamma_{g}$. Let
${\mathbb{Y}}\subset{\mathbb{X}}_{m_{0}}$ be an irreducible component of
${\mathbb{X}}_{m_{0}}$ containing ${\mathbb{I}}_{g}$.

If ${\mathbb{L}}_{g}^{e}\subset{\mathbb{Y}}$ (resp.
${\mathbb{L}}_{g}^{o}\subset{\mathbb{Y}}$), then for each
$m\in{\mathcal{E}}^{*}$ (resp. $m\in{\mathcal{O}}^{*}$) there exists an
element $\sigma_{m}\in G_{g}$ mapping $m_{0}$ to $m$, such that
$\sigma_{m}({\mathbb{Y}})$ is an irreducible component of ${\mathbb{X}}_{m}$
containing ${\mathbb{I}}_{g}$ .

###### Proof.

We give the argument for the even case; the argument for the odd case being
completely analogous, using ${\mathbb{L}}_{g}^{o}$ and $L_{g}^{o}$ instead of
${\mathbb{L}}_{g}^{e}$ and $L_{g}^{e}$.

We first observe that if $m\in{\mathcal{E}}_{2}$ (resp.
$m\in{\mathcal{O}}_{3}$), then, since $Stab_{{\mathbb{I}}_{g}}$ acts
transitively on ${\mathcal{E}}_{2}$ (resp. ${\mathcal{O}}_{3}$), there exists
$\sigma_{m}\in Stab_{{\mathbb{I}}_{g}}$ sending $m_{0}$ to $m$. Since
$\sigma_{m}({\mathbb{I}}_{g})={\mathbb{I}}_{g}$ (as a set), the image
$\sigma_{m}({\mathbb{Y}})$ contains ${\mathbb{I}}_{g}$. Since
$\sigma_{m}({\mathbb{X}}_{m_{0}})={\mathbb{X}}_{m}$ by the
$\Gamma_{g}$-invariance of ${\mathbb{X}}$ and by the definition of
${\mathbb{X}}_{m}$, it follows that $\sigma_{m}({\mathbb{Y}})$ is an
irreducible component of ${\mathbb{X}}_{m}$, containing ${\mathbb{I}}_{g}$.

To deal with the case of $m\in{\mathcal{E}}_{\ell}$ with $\ell\geq 4$, we
first observe that since $Stab_{{\mathbb{I}}_{g}}$ acts transitively on
${\mathcal{E}}_{\ell}$, it is enough to deal with the case of
$m=\left[\begin{smallmatrix}1\dots 10\dots 0\\\ 1\dots 10\dots
0\end{smallmatrix}\right]$ with $\ell$ columns equal to
$\left[\begin{smallmatrix}1\\\ 1\end{smallmatrix}\right]$. By the proof of 11
there exists an element $\sigma_{m}\in L_{g}^{e}\subset
Stab_{{\mathbb{L}}_{g}^{e}}$ such that $\sigma_{m}\cdot m_{0}=m$. Since
${\mathbb{L}}_{g}^{e}\subset{\mathbb{Y}}\subset{\mathbb{X}}_{m_{0}}$ by
assumption, and since ${\mathbb{X}}$ is $\Gamma_{g}$-invariant, it follows
that
$\sigma_{m}({\mathbb{L}}_{g}^{e})={\mathbb{L}}_{g}^{e}\subset\sigma_{m}({\mathbb{Y}})\subset{\mathbb{X}}_{m}$,
and in particular
$\sigma_{m}({\mathbb{Y}})\supset{\mathbb{L}}_{g}^{e}\supset{\mathbb{I}}_{g}$.
If $\sigma_{m}({\mathbb{Y}})$ were not an irreducible component of
${\mathbb{X}}_{m}$, i.e. if there existed an irreducible component
${\mathbb{W}}$ of ${\mathbb{X}}_{m}$ strictly containing
$\sigma_{m}({\mathbb{Y}})$, then by invariance of ${\mathbb{X}}$ under
$\Gamma_{g}$, the preimage $\sigma_{m}^{-1}({\mathbb{W}})$ would be an
irreducible component of ${\mathbb{X}}_{m_{0}}$ containing ${\mathbb{Y}}$,
giving a contradiction. ∎

The proposition can be rephrased as a statement on subvarieties of
${\mathcal{A}}_{g}$:

######  Corollary 26.

Let ${\mathcal{X}}\subset\vartheta_{\rm null}\subset{\mathcal{A}}_{g}$ (resp.
${\mathcal{X}}\subset{\mathcal{G}}_{\rm null}\subset{\mathcal{A}}_{g}$) be an
algebraic subvariety containing
$p({\mathbb{L}}_{g}^{e})\supset{\mathcal{D}}_{g}$ (resp.
$p({\mathbb{L}}_{g}^{o})\supset{\mathcal{D}}_{g}$). Denote
${\mathbb{X}}:=p^{-1}({\mathcal{X}})\subset{\mathbb{H}}_{g}$, and let
${\mathbb{Y}}\subset{\mathbb{X}}_{m}:={\mathbb{X}}\cap\theta_{\rm m,\,null}$
(resp. ${\mathbb{X}}\cap\operatorname{grad}_{m,\,\rm null}$) be an irreducible
component containing ${\mathbb{L}}_{g}^{e}$ (resp. ${\mathbb{L}}_{g}^{o}$).
Then $p({\mathbb{Y}})$ is an irreducible component of ${\mathcal{X}}$
containing ${\mathcal{D}}_{g}$.

What this proposition essentially rules out is the situation discussed above,
where ${\mathbb{I}}_{g}\subset{\mathbb{Y}}\subset{\mathbb{X}}_{m_{0}}$, but
where also ${\mathbb{I}}_{g}\subset{\mathbb{W}}\subset{\mathbb{X}}_{m}$ such
that $p({\mathbb{Y}})\subsetneq p({\mathbb{W}})\subset{\mathcal{A}}_{g}$.
Notice that since ${\mathbb{I}}_{g}\subset{\mathbb{X}}_{m}$ if and only if
$m\in{\mathcal{E}}^{*}$ (resp. ${\mathcal{O}}^{*})$, the characteristics
$m\in{\mathcal{E}}^{0}$ (resp. ${\mathcal{O}}^{1})$ do not occur in the above
discussion.

######  Remark 27.

The above proposition also holds in a more general context. For example, let
$M=(m_{1},\dots,m_{k})$ be a sequence of even characteristics, and let
${\mathcal{E}}_{M}$ be the set of all ordered $k$-tuples of characteristics
that form the $\Gamma_{g}$ orbit of $M$. Then we can also decompose
${\mathbb{X}}\subset{\mathbb{H}}_{g}$ as

${\mathbb{X}}=\cup_{(n_{1},\dots,n_{k})\in{\mathcal{E}}_{M}}{\mathbb{X}}_{n_{1},\dots,n_{k}}\,,$

where ${\mathbb{X}}_{n_{1},\dots,n_{k}}:={\mathbb{X}}\cap\theta_{n_{1},\,\rm
null}\cap\dots\cap\theta_{n_{k},\,\rm null}$. In this case we obtain similar
statements under the assumption
${\mathbb{L}}_{g}\subset{\mathbb{X}}_{(m_{1},\dots,m_{k})}$. In particular
this applies to the hyperelliptic case discussed in the previous section,
giving an alternative approach to the results there.

We will now apply this setup for the vanishing theta gradient loci.

###### Proof of 5.

We apply 25 for

${\mathbb{X}}=\operatorname{grad}_{\rm
null}=\cup_{m}\operatorname{grad}_{m,\,\rm null}\subset{\mathbb{H}}_{g}\,.$

To avoid confusion, denote $n_{0}:=\left[\begin{smallmatrix}110\dots 0\\\
110\dots 0\end{smallmatrix}\right]$ the even characteristic in genus $g-1$.
Note that the locus ${\mathbb{Y}}={\mathbb{H}}_{1}\times\theta_{\rm
n_{0},\,null}\subset{\mathbb{H}}_{1}\times{\mathbb{H}}_{g-1}$ is irreducible
and has codimension $g$ in ${\mathbb{H}}_{g}$, thus is an irreducible
component of $\operatorname{grad}_{m_{0},\,\rm null}$, containing
${\mathbb{L}}_{g}^{o}$. Then 26 implies that
$p({\mathbb{Y}})={\mathcal{A}}_{1}\times\vartheta_{\rm null}$ is an
irreducible component of ${\mathcal{G}}_{\rm null}$, which contains
${\mathcal{D}}_{g}$ (and in fact of course contains
$p({\mathbb{L}}_{g}^{o})$).

For components containing the hyperelliptic locus, suppose
${\mathbb{Y}}\subset\operatorname{grad}_{m,\,\rm null}$ is an irreducible
component of $\operatorname{grad}_{m,\,\rm null}$ containing some component
${\mathbb{X}}$ of the hyperelliptic locus ${\mathbb{HJ}}_{g}$, such that
${\mathbb{X}}\supset{\mathbb{I}}_{g}$. By 17, there exists
$\sigma\in\Gamma_{g}$ which lies in $Stab_{{\mathbb{I}}_{g}}$ and maps $m$ to
$m_{0}$. We can now apply 26 for
$\sigma({\mathbb{Y}})\subset\operatorname{grad}_{m_{0},\,\rm null}$, yielding
the result. ∎

##  6\. The hessian rank loci $\vartheta_{\rm null}^{2}$ and $\vartheta_{\rm
null}^{3}$

In this section we investigate the geometry of the loci with given rank of the
Hessian of the theta function near ${\mathbb{I}}_{g}$, proving Theorems 3 and
4. Using 26, at the end of the day it will suffice to study the hessian of
$\theta_{m_{0}}$ near ${\mathbb{I}}_{g}$.

###  6.1. The rank two locus

Our first goal is to prove 3, that
${\mathcal{A}}_{1}\times{\mathcal{A}}_{g-1}$ is an irreducible component of
$\vartheta_{\rm null}^{2}$. By applying 26, it will suffice to show that
${\mathbb{H}}_{1}\times{\mathbb{H}}_{g-1}$ is an irreducible component of
$\theta_{\rm m_{0},\,null}^{2}$. For this, similarly to how we dealt with the
locus ${\mathcal{G}}_{\rm null}$, working locally near ${\mathbb{I}}_{g}$ we
will compute the intersection of $\theta_{\rm m_{0},\,null}^{2}$ with the
locus $Z\subset{\mathbb{H}}_{g}$ given by the equations $\tau_{jk}=0$ for all
$2\leq j<k\leq g$ and $\tau_{1j}=\tau_{2j}$ for all $3\leq j\leq g$. Note that
$({\mathbb{H}}_{1}\times{\mathbb{H}}_{g-1})\cap Z={\mathbb{I}}_{g}$ by
definition. Since ${\mathbb{H}}_{1}\times{\mathbb{H}}_{g-1}\subset\theta_{\rm
m_{0},\,null}^{2}$, the following proposition will suffice to prove 3.

######  Proposition 28.

For a sufficiently small neighborhood $U$ of ${\mathbb{I}}_{g}$ the equality
$\theta_{\rm m_{0},\,null}^{2}\cap Z\cap U={\mathbb{I}}_{g}$ holds.

###### Proof.

Indeed, we will plug in the defining equations of $Z$ into the defining
equations of $\theta_{\rm m_{0},\,null}^{2}$ and check that they imply
$\tau_{1j}=0$ for all $1<j\leq g$. To see this, it will be sufficient to
consider the principal $3\times 3$ minors of the Hessian of $\theta_{m_{0}}$
that include the first and second rows. Using the expansion (12), for the
$3\times 3$ principal minor obtained by taking rows and columns $1,2,j$ for
some $3\leq j\leq g$, we compute the determinant to be

$\displaystyle D_{12j}:=\det(\ddots)$
$\displaystyle=\det\left(\begin{smallmatrix}\frac{1}{2}\phi_{1}\cdot
X_{2}&1&\phi_{j}\tau_{2j}+\phi_{j}\sum\phi_{\alpha}\tau_{2\alpha}\tau_{j\alpha}\\\
*&\frac{1}{2}\phi_{2}\cdot(X_{2}+Y_{2})&\phi_{j}\tau_{1j}+\phi_{j}\sum\phi_{\alpha}\tau_{2\alpha}\tau_{j\alpha}\\\
*&*&\frac{1}{2}\phi_{j}\cdot(X_{2}+Y_{2})+\frac{1}{2}\psi_{j}\tau_{1j}\tau_{2j}\end{smallmatrix}\right)$
$\displaystyle=-\frac{1}{2}\phi_{j}\cdot(X_{2}+Y_{2})+(2\phi_{j}^{2}-\psi_{j}/2)\tau_{1j}\tau_{2j}+O(\varepsilon^{3})\,.$

We also recall the expansion of $\theta_{m_{0}}$ itself, given by equation
(8). Using 23, we will work with the expansions of the theta constants and the
determinants of the $3\times 3$ minors of the Hessian up to
$O(\varepsilon^{3})$, intersected with $Z$. We thus compute

$\theta_{m_{0}}|_{Z}=\left([1,2]+Y_{2}\right)|_{Z}+O(\varepsilon^{3})=\tfrac{1}{2\pi
i}\tau_{12}+\tfrac{1}{(2\pi i)^{2}}\sum_{j\geq
3}\phi_{j}\tau_{1j}^{2}+O(\varepsilon^{3})\,.$

Substituting this into $D_{12j}$ and dropping the common and generically non-
zero factor of $\prod f_{\alpha}$ gives

$\displaystyle D_{12j}|_{Z\cap\theta_{\rm m_{0},\,null}}$
$\displaystyle=\left.-\tfrac{1}{2}\phi_{j}\cdot(\tau_{12}+Y_{2})+(2\phi_{j}^{2}-\tfrac{\psi_{j}}{2})\tau_{1j}\tau_{2j}\right|_{Z\cap\theta_{\rm
m_{0},\,null}}+O(\varepsilon^{3})$
$\displaystyle=O(\varepsilon^{3})+(2\phi_{j}^{2}-\psi_{j}/2)\tau_{1j}^{2}\,.$

Since the expression $2\phi_{j}^{2}-\psi_{j}/2$ is not identically zero in
$t_{j}$, for a generic value of $t_{j}$ the vanishing of
$D_{12j}|_{Z\cap\theta_{\rm m_{0},\,null}}$ implies
$\tau_{1j}^{2}=O(\varepsilon^{3})$ and then substituting this back, the
vanishing of $\theta_{m_{0}}|_{Z}$ implies $\tau_{12}=O(\varepsilon^{3})$, so
that altogether we get precisely the vanishing of
$\tau_{12},\tau_{13}=\tau_{23},\dots,\tau_{1g}=\tau_{2g}$ up to higher order,
as required. ∎

###### Proof of 3.

We observe that by the factorization of theta functions the big diagonal
${\mathbb{L}}^{e}\subset\theta_{\rm m_{0},\,null}^{2}\subset\theta_{\rm
m_{0},\,null}$. The above computation, using 23, shows that
${\mathbb{H}}_{1}\times{\mathbb{H}}_{g_{1}}$ is an irreducible component of
$\theta_{\rm m_{0},\,null}^{2}$ containing ${\mathbb{I}}_{g}$, and since the
defining equations of $\vartheta_{\rm null}^{2}$ are $\Gamma_{g}$ invariant,
by 26 it follows that ${\mathcal{A}}_{1}\times{\mathcal{A}}_{g-1}$ is an
irreducible component of $\vartheta_{\rm null}^{2}$. ∎

###  6.2. The rank three locus

We now deal with the locus $\vartheta_{\rm null}^{3}$; here our goal is to
show that the locus of Jacobians with a vanishing theta-null is an irreducible
component. Recall that the locus $\theta_{\rm
null}\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]\cap{\mathbb{J}}_{g}$ is purely of dimension
$3g-4$, and in fact by [MSM21], both ${\mathbb{J}}_{g}\subset{\mathbb{H}}_{g}$
and $\theta_{\rm null}\left[\begin{smallmatrix}\varepsilon\\\
\delta\end{smallmatrix}\right]\subset{\mathbb{H}}_{g}$ are irreducible;
moreover, the intersection $\vartheta_{\rm
null}\cap{\mathcal{J}}_{g}\subset{\mathcal{A}}_{g}$ is irreducible by [TiB88].
Thus we can apply 26, so that it will again suffice to work with the Taylor
expansions of the Hessian of $\theta_{m_{0}}$, and not with an arbitrary
characteristic.

The additional complication in this case is that, for high genus, the
dimension of $\vartheta_{\rm null}\cap{\mathcal{J}}_{g}$ is smaller than the
dimension of a number of irreducible components of ${\mathcal{R}}_{g}$ that
are contained in $\vartheta_{\rm null}^{3}$ and contain ${\mathcal{D}}_{g}$
and ${\mathbb{L}}_{g}^{o}$ (for example, for the component
${\mathcal{A}}_{g}\times{\mathcal{A}}_{g-2}$).

We demonstrate this issue in detail for the various components of interest in
low genus. For $g=5$ we have

$\dim({\mathcal{A}}_{3}\times{\mathcal{A}}_{2})=9<\dim({\mathcal{J}}_{5}\cap\vartheta_{\rm
null})=11=\dim({\mathcal{A}}_{4}\times{\mathcal{A}}_{1})\,,$

and indeed ${\mathcal{A}}_{3}\times{\mathcal{A}}_{2}$ is contained in the
closure of ${\mathcal{J}}_{5}\cap\vartheta_{\rm null}$ since
${\mathcal{J}}_{3}={\mathcal{A}}_{3}$. However, already for genus 6 we have

$\dim({\mathcal{A}}_{4}\times{\mathcal{A}}_{2})=10<\dim({\mathcal{J}}_{6}\cap\vartheta_{\rm
null})=14<\dim({\mathcal{A}}_{5}\times{\mathcal{A}}_{1})=16\,,$

and thus ${\mathcal{A}}_{4}\times{\mathcal{A}}_{2}$ must be contained in an
irreducible component of $\vartheta_{\rm null}^{3}$ that has dimension at
least 14; moreover, note that ${\mathcal{J}}_{6}\cap\vartheta_{\rm
null}\not\supset{\mathcal{A}}_{4}\times{\mathcal{A}}_{2}$.

This discussion makes the following statement more surprising in that we can
show that components of $\vartheta_{\rm null}^{3}$ not contained in the
decomposable locus have expected dimension.

######  Theorem 29.

For any genus $g$, the locus $\theta_{\rm
m_{0},\,null}^{3}\setminus{\mathbb{R}}_{g}$ locally near ${\mathbb{I}}_{g}$
has dimension $3g-4$.

As a consequence, all irreducible components of $\vartheta_{\rm null}^{3}$
containing ${\mathcal{D}}_{g}$ and not contained in the decomposable locus
${\mathcal{R}}_{g}$ must have dimension $\leq 3g-4$.

######  Corollary 30.

For any genus $g\geq 3$, the locus $\vartheta_{\rm
null}^{3}\setminus{\mathcal{R}}_{g}$ locally near ${\mathcal{D}}_{g}$ has
dimension equal to $3g-4$.

###### Proof.

Indeed, we observe that ${\mathbb{L}}_{g}^{e}\subset\theta_{\rm
m_{0},\,null}\cap{\mathbb{J}}_{g}\subset\theta_{\rm m_{0},\,null}^{3}$. The
locus $\vartheta_{\rm null}^{3}$ is by definition $\Gamma_{g}$-invariant, and
hence the conditions of 26 are satisfied. ∎

An immediate consequence is that the $(3g-4)$-dimensional irreducible locus
${\mathcal{J}}_{g}\cap\vartheta_{\rm null}=p(\theta_{\rm
m_{0},\,null}\cap{\mathbb{J}}_{g})$, contained in $\vartheta_{\rm null}^{3}$,
is an irreducible component of $\vartheta_{\rm null}^{3}$. This finishes the
proof of 4, once we obtain the local dimension statement.

###### Proof of 29.

As above, we will be working in a sufficiently small neighborhood
$U\supset{\mathbb{I}}_{g}$ of the diagonal, using the expansions (8) for
$\theta_{m_{0}}$ and the expansion (12) for the $4\times 4$ minors of its
Hessian. We first note that the vanishing of
$\theta_{m_{0}}=X_{2}+Y_{2}+O(\varepsilon^{3})$ implies that
$\tau_{12}=O(\varepsilon^{2})$ and moreover that
$X_{2}+Y_{2}=O(\varepsilon^{3})$.

Since we are interested in the locus where the rank of the Hessian is equal to
3, and not 2, and since the Hessian symmetric, there must exist a principal
$3\times 3$ minor of the Hessian with a non-zero determinant. Notice that the
$2\times 2$ principal minor of the Hessian formed by rows and columns $1$ and
$2$ becomes, after plugging in $X_{2}+Y_{2}=O(\varepsilon^{3})$ from the
vanishing of $\theta_{m_{0}}$, equal to
$\left(\begin{smallmatrix}O(\varepsilon^{3})&1+O(\varepsilon^{2})\\\
1+O(\varepsilon^{2})&O(\varepsilon^{3})\end{smallmatrix}\right)$, so that its
determinant is equal to $-1+O(\varepsilon^{2})$, and thus non-zero. Since this
is a non-degenerate principal $2\times 2$ minor, for the matrix to have rank
equal to $3$, it must be contained in a non-degenerate $3\times 3$ minor.
Moreover, since the matrix is symmetric, there must exists a principal such
non-degenerate $3\times 3$ minor, and by renumbering the coordinates, we thus
assume without loss of generality that this non-degenerate minor is made up by
rows and columns number $1,2,3$. Similarly to the proof of 3, for convenience
we will intersect, in a neighborhood $U\supset{\mathbb{I}}_{g}$, the locus
$\theta_{\rm m_{0},\,null}^{3}$, with the codimension $g-2$ subvariety
$Z\subset{\mathbb{H}}_{g}$ given by equations $\tau_{1j}=\tau_{2j}$ for all
$3\leq j\leq g$. Our goal is to prove that $U\cap Z\cap\theta_{\rm
m_{0},\,null}^{3}$ has dimension at most $(3g-4)-(g-2)=2g-2$. Indeed, since we
know that ${\mathbb{J}}_{g}\cap\theta_{\rm m_{0},\,null})$ has dimension
precisely $3g-4$, and thus $\dim(U\cap{\mathbb{J}}_{g}\cap\theta_{\rm
m_{0},\,null}\cap Z)\geq 2g-2$ is contained in $\theta_{\rm
m_{0},\,null}^{3}\cap Z$, this will then imply 29.

To bound from above the dimension of $U\cap Z\cap\theta_{\rm
m_{0},\,null}^{3}$, we will look at determinants $D_{jk}$ of the $4\times 4$
minors of the Hessian made up of rows $(123j)$ and columns $(123k)$, for any
$4\leq j\leq k\leq g$. We will see that on $U\cap Z\cap\theta_{\rm
m_{0},\,null}$, for $j=k$ the vanishing of $D_{jj}$ will require
$\tau_{1j}=\tau_{2j}$ to vanish to higher order, and then we will see that the
vanishing of $D_{jk}$ for $j<k$ determines $\tau_{jk}$ in terms of the other
variables, up to higher order. Thus altogether, up to higher order, the point
of $U\cap Z\cap\theta_{\rm m_{0},\,null}^{3}$ will be determined by the values
of the diagonal period matrix elements
$t_{1}=\tau_{11},\dots,t_{g}=\tau_{gg}$, together with $\tau_{13}=\tau_{23}$,
and $\tau_{34},\dots,\tau_{3g}$ (recall that $\tau_{12}$ is determined in
terms of other coordinates, up to higher order, from the vanishing of
$\theta_{m_{0}}$. Thus altogether by applying 23, we will see that the
dimension of $U\cap Z\cap\theta_{\rm m_{0},\,null}^{3}$ is equal the number of
these coordinates, i.e. $g+1+(g-3)=2g-2$. We now inspect these $4\times 4$
minors in detail.

First, recall from the proof of 3 that the determinant $D_{123}$ of the
$3\times 3$ minor of the Hessian formed by the first 3 rows and columns is
equal, up to higher order terms and generically non-vanishing factor, to
$\tau_{13}^{2}$. Since $D_{123}\neq 0$ by assumption, this means that
$\tau_{13}\neq 0$.

Now, for a principal $4\times 4$ minor $D_{jj}$, we plug in
$X_{2}+Y_{2}=O(\varepsilon^{3})$ into (12), and see that the lowest order
entries of the minor are as follows:

$\left(\begin{smallmatrix}O(\varepsilon^{3})&1+O(\varepsilon^{2})&\phi_{3}\cdot[2,3]+O(\varepsilon^{2})&\phi_{j}\cdot[2,j]+O(\varepsilon^{2})\\\
*&O(\varepsilon^{3})&\phi_{3}\cdot[1,3]+O(\varepsilon^{2})&\phi_{j}\cdot[1,j]+O(\varepsilon^{2})\\\
*&*&\frac{1}{2}\psi_{3}\cdot[3,3,1,2]+O(\varepsilon^{3})&\phi_{3}\cdot\phi_{j}\cdot[3,j,1,2]+\dots\\\
*&*&*&\frac{1}{2}\psi_{j}\cdot[j,j,1,2]+O(\varepsilon^{3})\end{smallmatrix}\right)\,.$

Thus the lowest order term that could appear in the determinant of this matrix
is of order $O(\varepsilon^{4})$, and we write it explicitly in terms of the
entries of the period matrix (noting, importantly, that in $[j,k,1,2]$ the
term $\tau_{jk}\tau_{12}$ is higher order, and using Maple to compute safely)

(14)
$D_{12jk}=-\tau_{13}^{2}\tau_{1j}^{2}(4\phi_{3}^{2}-\psi_{3})(4\phi_{j}^{2}-\psi_{j})/4+O(\varepsilon^{5})$

For generic values of $t_{3},t_{j}$ the expressions depending on them are non-
zero, and thus the vanishing of such a determinant implies, since
$\tau_{13}\neq 0$, that $\tau_{1j}=0$, up to higher order terms.

We now inspect the determinant of $D_{jk}$ of the $4\times 4$ minor formed by
rows $(123j)$ and columns $(123k)$ for $j<k$; all the terms can again be read
off from (12), so that the corresponding $4\times 4$ minor is as follows
(where to make the formula fit we dropped the $1/2\pi i$ factors in front of
each $\tau$, coming from the bracket expressions, and we recalled
$\tau_{1a}=\tau_{2a}$ for $a=3,j,k$). Note that the fourth row and column of
the minor are no longer symmetric.

$\left(\begin{smallmatrix}O(\varepsilon^{3})&1+O(\varepsilon^{2})&\phi_{3}\tau_{13}+O(\varepsilon^{2})&\phi_{k}\tau_{1k}+O(\varepsilon^{2})\\\
1+O(\varepsilon^{2})&O(\varepsilon^{3})&\phi_{3}\tau_{13}+O(\varepsilon^{2})&\phi_{k}\tau_{1k}+O(\varepsilon^{2})\\\
\phi_{3}\tau_{13}+O(\varepsilon^{2})&\phi_{3}\tau_{13}+O(\varepsilon^{2})&\frac{1}{2}\psi_{3}\cdot[3,3,1,2]+O(\varepsilon^{3})&\phi_{3}\cdot\phi_{k}\cdot[3,k,1,2]+\dots\\\
\phi_{j}\tau_{1j}+O(\varepsilon^{2})&\phi_{j}\tau_{1j}+O(\varepsilon^{2})&\phi_{3}\cdot\phi_{j}\cdot[3,j,1,2]+\dots&\phi_{j}\cdot\phi_{k}\cdot[j,k,1,2]+\dots\end{smallmatrix}\right)\,.$

Notice, however, that this formula does not really give the lowest order terms
of the expansion, as indeed by the vanishing of the determinants $D_{jj}$ and
$D_{kk}$ of the principal minors we know that
$\tau_{1j},\tau_{1k}=O(\varepsilon^{2})$. Thus in fact the entries
$(1,k),(2,k),(j,1),(k,1)$ of the minor containing these entries are themselves
of order $O(\varepsilon^{2})$, while the correction term to
$\phi_{j}\tau_{1j}$ is actually of higher order, as all the brackets involved
will contain $\tau_{1j}$ or $\tau_{12}$, and are thus of order at least one
higher than their degree in $\tau$’s.

What we want to determine is the dependence of $D_{jk}$ on $\tau_{jk}$, more
precisely we want to determine the lowest order term that contains
$\tau_{jk}$. By inspection, we see that

$[j,k,1,2]=\tau_{jk}\tau_{12}+2\tau_{1j}\tau_{2k}=\tau_{jk}\tau_{12}+O(\varepsilon^{4})$

appearing in the $(j,k)$ entry of the minor above is the only entry where
$\tau_{jk}$ appears. We recall that from the vanishing of $\theta_{\rm
m_{0},\,null}$, given by expansion (8), we have (again, up to all the $\pm
2\pi i$ factors)

$X_{2}+Y_{2}=O(\varepsilon^{3})=\tau_{12}+\sum_{a>2}\phi_{a}\tau_{1a}\tau_{2a}=\tau_{12}+\tau_{13}^{2}+\sum_{j>3}\tau_{1j}^{2}=\tau_{12}+\tau_{13}^{2}+O(\varepsilon^{4})\,,$

since $\tau_{1j}=O(\varepsilon^{2})$. Thus we see that
$\tau_{12}=-\tau_{13}^{2}+O(\varepsilon^{3})$, and is of order precisely
$\varepsilon^{2}$, as $\tau_{13}$ is non-zero due to the assumed non-vanishing
of the determinant $D_{123}$. Thus the $(j,k)$ entry of the minor above
contributes $\phi_{j}\phi_{k}(-\tau_{jk}\tau_{13}^{2}+O(\varepsilon^{4}))\cdot
D_{123}$ to $D_{jk}$, where we expanded $D_{jk}$ using the last row. By
assumption $D_{123}$ is non-zero, and in fact of order $O(\varepsilon^{2})$ as
discussed above. By inspection of the minor, the only other places where
$\tau_{jk}$ appears in the minor are when expanding brackets of 6 terms, and
then at least two of these terms would be of order $O(\varepsilon^{2})$, so we
have found that the only dependence of $D_{jk}$ modulo $O(\varepsilon^{6})$ on
$\tau_{jk}$ is the term $-\phi_{j}\phi_{k}\tau_{jk}\tau_{13}^{2}\cdot
D_{123}$. Thus requiring $D_{jk}$ to vanish modulo $O(\varepsilon^{6})$
expresses $\tau_{jk}$ in terms of the other variables, modulo
$O(\varepsilon^{2})$.

Thus altogether each $\tau_{1j}=\tau_{2j}$ must be of order
$O(\varepsilon^{2})$ by the vanishing of $D_{jj}$, while each $\tau_{jk}$ is
expressed in terms of the rest of the entries in the first 3 rows of the
period matrix, and the diagonal entries, by computing the $O(\varepsilon^{5})$
term of $D_{jk}$ and requiring it to vanish. Altogether we see that the local
dimension of the locus $\theta_{\rm m_{0},\,null}^{3}\cap U\cap Z$ is as
claimed. ∎

## References

  * [AC21] D. Agostini and L. Chua.  On the Schottky problem for genus-five Jacobians with a vanishing theta-null.  Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 22(1):333–350, 2021.
  * [Far06] H. Farkas.  Vanishing thetanulls and Jacobians.  In The geometry of Riemann surfaces and abelian varieties, volume 397 of Contemp. Math., pages 37–53. Amer. Math. Soc., Providence, RI, 2006.
  * [FGSM21] H. Farkas, S. Grushevsky, and R. Salvati Manni.  An explicit solution to the weak Schottky problem.  Algebr. Geom., 8(3):358–373, 2021.
  * [Fre68] E. Freitag.  Fortsetzung von automorphen Funktionen.  Math. Ann., 177:95–100, 1968.
  * [Fre91] E. Freitag.  Singular modular forms and theta relations, volume 1487 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 1991.
  * [GH12] S. Grushevsky and K. Hulek.  The class of the locus of intermediate Jacobians of cubic threefolds.  Invent. Math., 190(1):119–168, 2012.
  * [GSM07] S. Grushevsky and R. Salvati Manni.  Singularities of the theta divisor at points of order two.  Int. Math. Res. Not. IMRN, (15):Art. ID rnm045, 15, 2007.
  * [GSM08] S. Grushevsky and R. Salvati Manni.  Jacobians with a vanishing theta-null in genus 4.  Israel J. Math., 164:303–315, 2008.
  * [GSM09] S. Grushevsky and R. Salvati Manni.  The loci of abelian varieties with points of high multiplicity on the theta divisor.  Geom. Dedicata, 139:233–247, 2009.
  * [Igu72] J.-I. Igusa.  Theta functions, volume 194 of Grundlehren der Mathematischen Wissenschaften.  Springer-Verlag, New York, 1972.
  * [Igu80] J.-I. Igusa.  On Jacobi’s derivative formula and its generalizations.  Amer. J. Math., 102(2):409–446, 1980.
  * [Mah69] K. Mahler.  On algebraic differential equations satisfied by automorphic functions.  J. Austral. Math. Soc., 10:445–450, 1969.
  * [MSM21] G. Mondello and R. Salvati Manni.  Totally irreducible subvarieties of Siegel moduli spaces.  2021\.
  * [Mum83] D. Mumford.  On the Kodaira dimension of the Siegel modular variety.  In Algebraic geometry—open problems (Ravello, 1982), volume 997 of Lecture Notes in Math., pages 348–375, Berlin, 1983. Springer.
  * [Mum07] D. Mumford.  Tata lectures on theta. II.  Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston, MA, 2007\.  Jacobian theta functions and differential equations, With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura, Reprint of the 1984 original.
  * [Poo94] C. Poor.  The hyperelliptic locus.  Duke Math. J., 76(3):809–884, 1994.
  * [SB21] N. I. Shepherd-Barron.  Asymptotic period relations for Jacobian elliptic surfaces.  Proc. Lond. Math. Soc. (3), 122(4):479–520, 2021.
  * [SM94] R. Salvati Manni.  Modular varieties with level $2$ theta structure.  Amer. J. Math., 116(6):1489–1511, 1994.
  * [TiB88] M. Teixidor i Bigas.  The divisor of curves with a vanishing theta-null.  Compositio Math., 66(1):15–22, 1988.
  * [Tsu91] Sh. Tsuyumine.  Thetanullwerte on a moduli space of curves and hyperelliptic loci.  Math. Z., 207(4):539–568, 1991.