The Generalized Torelli Problem through the geometry of the Gauss map

TL;DR

This paper advances the understanding of the Torelli problem by analyzing the geometry of the Gauss map on certain loci of algebraic curves, providing new reconstruction methods and characterizations for non-hyperelliptic and hyperelliptic cases.

Contribution

It introduces a novel approach to reconstructing linear series and dual hypersurfaces from the Gauss map, extending the classical Torelli problem to a generalized setting.

Findings

The generic fiber of the Gauss map on $W_n$ has one element.

Reconstruction of $ extit{g}^k_{n+k}$ linear series from the Gauss map is possible under certain conditions.

In hyperelliptic cases, the image of the Gauss map relates birationally to complete linear series and contains dual hypersurfaces.

Abstract

Given a non-hyperelliptic curve $C\in\mathscr{M}_g$ and $2\leq n\leq g-2$, we prove that the generic fiber of the Gauss map on $W_n$ has one element and we characterize its multiple locus. Assuming that $C$ doesn't have a $\mathfrak{g}_{n+k+1}^{k+1}$, for $1\leq k\leq n-1\leq g-3$, we solve the problem of reconstructing each $\mathfrak{g}_{n+k}^k$ and the dual hypersurface of the image of its associated morphism, through information encoded in the Gauss map. For this purpose we introduce the notion of $\left(n+k\right)$-intersection loci and we study their dimensions. In the hyperelliptic case we prove that the image of the Gauss map is a union of sets whose closures are birational to their complete $\mathfrak{g}_{n+k}^k$, for each $1\leq k\leq n\leq g-1$, and that these also contain a copy of the dual hypersurface of the image of its associated morphism. From the case $k=n$ we deduce that the closure of the image of the Gauss map is birational to $\mathbb{P}^n$.