The Generalized Torelli Problem through the geometry of the Gauss map

Abstract

Given a non-hyperelliptic curve $C\in {\mathcal{M}}_g$ and $2\le n\le g-2$, 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 $1\le k\le n-1\le g-3$, 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 $1\le k\le n\le g-1$, 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 ${\mathbb{P}}^n$.

28 pages. References updated. Comments are welcome