The Generalized Torelli Problem through the geometry of the Gauss map

Given a non-hyperelliptic curve Cg and 2≤ n≤ g-2, we prove that the generic fiber of the Gauss map on Wₙ has one element and we characterize its multiple locus. Assuming that C doesn't have a gₙ₊ₖ₊₁ᵏ⁺¹, for 1≤ k≤ n-1≤ g-3, we solve the problem of reconstructing each gₙ₊ₖᵏ 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 (n+k)-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 gₙ₊ₖᵏ, for each 1≤ k≤ n≤ 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 Pⁿ.