Trigonal Curve with Trigonal Deformation of Maximal Rank

TL;DR

This paper constructs explicit examples of trigonal curves with maximal trigonal deformation rank on toric surfaces and provides a criterion to determine deformation ranks of sections.

Contribution

It extends existing methods to toric surfaces, producing explicit examples and a computable criterion for deformation ranks of sections.

Findings

Explicit examples for ranks 5,7,9,11,13,15.

A computable criterion for deformation rank determination.

Extension of deformation methods to toric surfaces.

Abstract

By extending methods of Favale-Pirola arXiv:2108.02157 and Gonz\'alez-Alonso-Torelli arXiv:2402.15158 to toric surfaces via toric Jacobian ring, we are able to show there exists trigonal curve with trigonal deformation of rank $g$ for $g=5,7,9,11,13,15$ by giving an explicit example. Also, we give a computable criterion to determine whether a nondegenerate ample section of toric surface has first order deformation of rank $g$ within the linear system.