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Trigonal Curve with Trigonal Deformation of Maximal Rank

Jiacheng Zhang

TL;DR

The paper develops a toric analogue of the Favale-Pirola and González-Alonso-Torelli approach to infinitesimal variation of Hodge structure for nondegenerate ample curves, using the toric Jacobian ring to study deformations within a linear system. It proves the existence of trigonal curves with trigonal deformation of rank $g$ for odd genus values $g\in\{5,7,9,11,13,15\}$ via explicit models on $\mathcal{H}_1$ and $Bl_p\mathbb{P}^2$, and provides a computable criterion to detect first-order rank-$g$ deformations by bounding decomposable elements in graded components of the toric Jacobian ideal. The methodology combines toric geometry, residue dualities, and IVHS via the toric Jacobian framework, and includes concrete computations (e.g., using Macaulay2) to verify the bounds and rank conditions for a family of Hirzebruch surfaces. While the Hirzebruch case yields explicit odd-genus trigonal examples up to $g=15$, the authors also discuss extensions and limitations to higher Picard rank toric surfaces, outlining a general criterion that applies when certain ampleness and negativity conditions on $K_{\ extSigma}$ hold. The results advance understanding of how trigonal structure can be preserved under deformations on toric surfaces and provide a practical tool for identifying high-rank trigonal deformations within linear systems.

Abstract

By extending methods of Favale-Pirola arXiv:2108.02157 and González-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.

Trigonal Curve with Trigonal Deformation of Maximal Rank

TL;DR

The paper develops a toric analogue of the Favale-Pirola and González-Alonso-Torelli approach to infinitesimal variation of Hodge structure for nondegenerate ample curves, using the toric Jacobian ring to study deformations within a linear system. It proves the existence of trigonal curves with trigonal deformation of rank for odd genus values via explicit models on and , and provides a computable criterion to detect first-order rank- deformations by bounding decomposable elements in graded components of the toric Jacobian ideal. The methodology combines toric geometry, residue dualities, and IVHS via the toric Jacobian framework, and includes concrete computations (e.g., using Macaulay2) to verify the bounds and rank conditions for a family of Hirzebruch surfaces. While the Hirzebruch case yields explicit odd-genus trigonal examples up to , the authors also discuss extensions and limitations to higher Picard rank toric surfaces, outlining a general criterion that applies when certain ampleness and negativity conditions on hold. The results advance understanding of how trigonal structure can be preserved under deformations on toric surfaces and provide a practical tool for identifying high-rank trigonal deformations within linear systems.

Abstract

By extending methods of Favale-Pirola arXiv:2108.02157 and González-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 for 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 within the linear system.

Paper Structure

This paper contains 14 sections, 24 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Review of Toric Geometry of Smooth Toric Surface and Hodge Theory of its ample divisor
  3. Basic Construction
  4. Cox Ring of a smooth projective toric surface
  5. Two Tales of Toric Jacobian Rings
  6. IVHS of Ample Sections in Toric Surface
  7. A Criterion
  8. Hirzebruch Surface
  9. GIT Quotient
  10. Picard Group
  11. A Linear System
  12. Estimate
  13. Computation
  14. A word on arbitrary complete smooth toric surface

Key Result

Theorem 1.3

There exists a nondegenerate(for definition see Definition nondegenerate) section $f$ of $dD_1+3D_2$ with $D_1$ the fiber divisor and $D_2$ the exceptional divisor where $5\le d\le 10$ on $\mathcal{H}_1$ or $Bl_p\mathbb{P}^2$ such that there is a first order deformation of $f$ on $\mathcal{H}_1$ wit

Theorems & Definitions (58)

  • Definition 1.1
  • Conjecture 1.2: González-Alonso-Torelli
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • proof
  • Theorem 2.2: Cox
  • proof
  • Theorem 2.3: Generalized Euler Identity
  • ...and 48 more