Gaussian maps on trigonal curves

TL;DR

This paper investigates higher Gaussian maps on cyclic trigonal curves, providing bounds on their rank, an explicit kernel description for the second map, and implications for asymptotic directions.

Contribution

It offers new bounds for Gaussian map ranks on trigonal curves, explicit kernel descriptions, and insights into asymptotic directions, advancing understanding of these geometric objects.

Findings

Lower bounds for Gaussian map ranks established

Explicit kernel of the second Gaussian map described

Non-existence of extra asymptotic directions shown

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.