Gaussian maps on trigonal curves
Antonio Lacopo
TL;DR
The paper addresses how higher Gaussian maps $\,\mu_{2k}$ behave on trigonal curves, focusing on cyclic trigonal curves to obtain rank bounds for restricted maps and a complete description of $\ker(\mu_2)$. It develops a detailed decomposition $H^0(M)=W_1\oplus W_2$ for cyclic trigonal curves, derives explicit rank formulas for $\mu_{2k-1}$ on $\Lambda^2 W_i$, and establishes lower bounds for $\text{rk}(\mu_{2k})$ on general trigonal curves, along with an explicit kernel description of $\mu_2$ in the cyclic case. The work then connects these algebraic results to the geometry of the Torelli map via the second fundamental form $\rho$, proving nonexistence of extra asymptotic directions in certain Schiffer-variation subspaces and deducing dimension bounds for totally geodesic subvarieties in $\mathcal{A}_g$ contained in the Torelli locus. Overall, the results advance understanding of the Torelli geometry of trigonal curves, provide explicit kernel and rank data for Gaussian maps, and constrain the deformation directions arising from higher Schiffer variations.
Abstract
In this paper we study higher even Gaussian maps of the canonical bundle for cyclic trigonal curves. More precisely, we study suitable restrictions of these maps determining a lower bound for the rank, and more generally, a lower bound for the rank for the general trigonal curve. We also manage to give the explicit description of the kernel of the second Gaussian map. Finally, we use these results to show the non existence of "extra" asymptotic directions for cyclic trigonal curves in some spaces generated by higher Schiffer variations.