Trigonal Curve with Trigonal Deformation of Maximal Rank

Abstract

By extending methods of Favale-Pirola \cite{FavalePirola2022} and González-Alonso-Torelli\cite{TorelliGonzalezAlonso2024} to toric surfaces via the toric Jacobian ring, we give a computational criterion to determine whether a nondegenerate curve given by ample section of a Hirzebruch surface has first order deformation of rank $g$ within the linear system. We are able to show there exist trigonal curves with trigonal deformations of rank $g$ for $g=2k+1$ and $k\ge 2$ by computing an explicit example.

21 pages. Major changes following suggestions of the referee. Added more references. Final version. Comments are welcomed!