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 $j_h$ 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 $\rho_{HE}$ of $j_h$. In particular, for any Weierstrass point $p \in C$, we construct a subspace $V_p$ of dimension $\lfloor\frac{g}{2} \rfloor$ of $T_{{\mathcal H}_g, [C]}$ generated by higher Schiffer variations at $p$, such that the only isotropic tangent direction $\zeta \in V_p$ for the image of $\rho_{HE}$ is the standard Schiffer variation $\xi_p$ at the Weierstrass point $p \in C$.