Vanishing thetanulls and hyperelliptic curves

Abstract

Let ${\mathcal{M}}_{g,2}$ be the moduli space of curves of genus $g$ with a level-2 structure. We prove here that there is always a non hyperelliptic element in the intersection of four thetanull divisors in ${\mathcal{M}}_{6,2}$. We prove also that for all $g⩾3$, each component of the hyperelliptic locus in ${\mathcal{M}}_{g,2}$ is a connected component of the intersection of $g-2$ thetanull divisors.

13 pages