Higher Gaussian maps on the hyperelliptic locus and second fundamental form

TL;DR

This paper investigates higher Gaussian maps on hyperelliptic curves, explicitly describes their kernels, and explores the second fundamental form of the hyperelliptic Torelli map, revealing properties of isotropic subspaces related to Weierstrass points.

Contribution

It provides explicit descriptions of higher Gaussian maps' kernels on hyperelliptic curves and analyzes the second fundamental form of the hyperelliptic Torelli map.

Findings

Explicit kernels of higher Gaussian maps on hyperelliptic curves

Construction of isotropic subspaces generated by higher Schiffer variations

Identification of the unique isotropic tangent direction at Weierstrass points

Abstract

In this paper we study higher even Gaussian maps of the canonical bundle on hyperelliptic curves and we determine their rank, giving explicit descriptions of their kernels. Then we use this descriptions to investigate the hyperelliptic Torelli map $j_h$ and its second fundamental form. We study isotropic subspaces of the tangent space $T_{{\mathcal H}_g, [C]}$ to the moduli space ${\mathcal H}_g$ of hyperelliptic curves of genus $g$ at a point $[C]$, with respect to the second fundamental form $\rho_{HE}$ of $j_h$. In particular, for any Weierstrass point $p \in C$, we construct a subspace $V_p$ of dimension $\lfloor\frac{g}{2} \rfloor$ of $T_{{\mathcal H}_g, [C]}$ generated by higher Schiffer variations at $p$, such that the only isotropic tangent direction $\zeta \in V_p$ for the image of $\rho_{HE}$ is the standard Schiffer variation $\xi_p$ at the Weierstrass point $p \in C$.