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**](https://mathoverflow.net/questions/1492/moduli-spaces-of-complex-curves-as-algebraic-varieties) 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. 



Topology of theta nulls
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.


