Schottky versus super Schottky in genus 4

RESUMO

The study of super Riemann surfaces and their moduli led Witten and Felder, Kazhdan and Polishchuk to ask what is the smallest $d$ such that the $d$-th power of the Schottky ideal is contained in the super Schottky ideal. Felder, Kazhdan and Polishchuk proved that $d=g$ in odd genus $g\geq5$ and that $d\in\{g-1,g\}$ in even genus $g\geq4$; Y.~Shen subsequently proved that $d=g$ in even genus $g\geq6$. Thus the only remaining open case was $g=4$. Here we close this gap by showing that in genus $4$ too, the smallest power of the Schottky ideal contained in the super Schottky ideal is $d=g=4$. The question is equivalent to one concerning the codifferential of the super period map. The quadratic odd contribution sends a conormal vector to a bivector, represented by a skew-symmetric $(2g-2)\times(2g-2)$ matrix. The question is to find such a vector for which, at a generic point, this matrix has maximal rank $2g-2$, or $6$ in our case. Our computation uses the standard exact sequences on $C\times C$ attached to the sheaves $\OO(a,b,c)$, together with a degeneration to a vanishing theta-null and some facts about the Szegő kernel. At a curve with a vanishing theta-null, a regularized version of the codifferential can be described by an explicit multiplication map, allowing an easy computation of the rank. This rank turns out to be $4$. A first-order calculation on an explicit deformation of a cyclic trigonal vanishing theta-null differentiates the global Gaussian-map identity for the regularized conormal bivector. Including the variation of the Gaussian map gives a nonzero first normal symbol. The resulting first-order form is nonzero on the null space of the limiting skew-symmetric matrix, so the rank jumps to $6$ on nearby curves, showing that $d=4$.