The Generalized Torelli Problem through the geometry of the Gauss map

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 $\mathbb{P}^n$.