Higher Gaussian maps on the hyperelliptic locus and second fundamental form

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 jₕ and its second fundamental form. We study isotropic subspaces of the tangent space T_ Hg, [C] to the moduli space Hg of hyperelliptic curves of genus g at a point $[C]$, with respect to the second fundamental form ρHE of jₕ. In particular, for any Weierstrass point p ∈ C, we construct a subspace Vₚ of dimension /2 of T_ Hg, [C] generated by higher Schiffer variations at p, such that the only isotropic tangent direction ζ ∈ Vₚ for the image of ρHE is the standard Schiffer variation ξₚ at the Weierstrass point p ∈ C.