The Generalized Torelli Problem through the geometry of the Gauss map
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Date
2024
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Abstract
Given a non-hyperelliptic curve C of genus g and 1<n<g-1, in this work 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 linear system of dimension k+1 and degree n+k+1, for 0<k<n<g-1, we solve the problem of reconstructing each linear system of dimension k and degree n+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 (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 linear systems of dimension k and degree n+k, for each 0<k<n+1<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 the n-dimensional projective space.We also prove the existence of points in W_n such that their images on the Kummer variety of their Jacobian aren't in general position, i.e., they lie on a multisecant. For this purpose we use the Gunning multisecant formula. We then study some relations between these multisecant points and the Gauss map on W_n, in both cases non-hyperelliptic and hyperelliptic.
Description
Tesis (Doctor en Matemática)--Pontificia Universidad Católica de Chile, 2024.