Higher Gaussian maps on the hyperelliptic locus and second fundamental form

2024/06/25 by Faro, Dario, Frediani, Paola, Lacopo, Antonio
#14H10 #14H15 #32G20 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2406.17408

Abstract

In this paper we study higher even Gaussian maps of the canonical bundle on hyperelliptic curves and we determine their rank, giving explicit descriptions of their kernels. Then we use this descriptions to investigate the hyperelliptic Torelli map j_h and its second fundamental form. We study isotropic subspaces of the tangent space T_\mathcal H_g, [C] to the moduli space \mathcal H_g of hyperelliptic curves of genus g at a point [C], with respect to the second fundamental form ρ_HE of j_h. In particular, for any Weierstrass point p ∈ C, we construct a subspace V_p of dimension \lfloor(g)/(2) \rfloor of T_\mathcal H_g, [C] generated by higher Schiffer variations at p, such that the only isotropic tangent direction ζ∈ V_p for the image of ρ_HE is the standard Schiffer variation ξ_p at the Weierstrass point p ∈ C.

Related

• Second fundamental form and higher Gaussian maps
• The Torelli map restricted to the hyperelliptic locus
• The Generalized Torelli Problem through the geometry of the Gauss map
• Higher Gaussian maps for special classes of curves
• On the Bielliptic and bihyperelliptic loci