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Siegel upper half-space, $\mathfrak{h}_g$, consists of symmetric $g\times g$ complex matrices with positive-definite imaginary part. From an element $Z\in \mathfrak{h}_g$ we can construct a theta function $\theta(Z, \cdot): \mathbb{C}^g\rightarrow \mathbb{C}$ which may be thought of as a section of a polarizing line bundle on a certain principally polarized abelian variety whose period matrix is $\Omega = (I\ \ Z)$ (see, for example, Griffiths & Harris). If now we let the first argument vary as well, we obtain a holomorphic function $\mathfrak{h}_g\times \mathbb{C}^g\rightarrow \mathbb{C}$ which induces a function on $\mathfrak{h}_g$ given by $Z\rightarrow \theta(Z,0)$. Such a function is called a theta null. For a little more detail and a bit of context, check out this question on moduli spaces of curves

I am interested in the topology of the vanishing loci of theta nulls in dimensions $g\geq 2$. Specifically, I'd like to know more about their connectivity. Is much known in this area, besides whatever is known for analytic hypersurfaces in general? Any information would be greatly appreciated and, of course, references are welcome.

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    $\begingroup$ It appears to me that almost nothing is really known about theta functions $\Theta(\Omega, z)$. Even the (full!) functional equation wrt the $Sp(2g, \mathbb{Z})$ action remains opaque (cf. Mumford's Tata Lectures III). For instance, for given $\Omega \in \mathfrak{h}_{g}$ set $Z[\Omega]$ to be the zero locus of $\Theta(\Omega, \cdot)$ in $\mathbb{C}^{g}$, ie. the theta divisor. Then I don't think anyone has a decent description of the orbits $Z[\Omega.\gamma]$ for $\gamma \in Sp(2g, \mathbb{Z})$. Ie. is the theta divisor even equivariant? $\endgroup$
    – JHM
    Commented Jul 1, 2012 at 23:11
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    $\begingroup$ Your 'theta-null' $\Theta(\Omega, 0)$ is the usual theta function one would associate to the quadratic form $x \mapsto {}^t x \Omega x$. Cf. Conway/Sloane's "Sphere Packings, Lattices, and Groups" to see how theta functions of lattices can actually be used. For instance, C/S derive almost all the basic properties on the Leech lattice via its theta function. $\endgroup$
    – JHM
    Commented Jul 1, 2012 at 23:17

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For $g>2$ the locus $M_g^1$ of "vanishing theta nulls" = set of $X$ in $M_g$ with a spin bundle such that $h^0(L) > 1$ is shown to be a connected divisor in:

MONTSERRAT TEIXIDOR I BIGAS The divisor of curves with a vanishing theta-null Compositio Mathematica, tome 66, no 1 (1988), p. 15-22.

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