The Generalized Torelli Problem through the geometry of the Gauss map

2024/06/17 by Sebastián Rahausen, Rahausen, Sebastián
Engineering · Mathematics · #14C20 #14C34 #14H40 #14H51 #Algebraic Geometry (math.AG) #Control and Dynamics of Mobile Robots #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2406.11197

openalex publication_date 2024/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a non-hyperelliptic curve C∈\mathscrM_g and 2≤ n≤ 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 \mathfrakg_n+k+1^k+1, for 1≤ k≤ n-1≤ g-3, we solve the problem of reconstructing each \mathfrakg_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 (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 \mathfrakg_n+k^k, 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.

Related

• Higher Gaussian maps on the hyperelliptic locus and second fundamental form
• Second fundamental form and higher Gaussian maps
• The Torelli map restricted to the hyperelliptic locus
• On the Bielliptic and bihyperelliptic loci
• On separable higher Gauss maps