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Optimality of the Prym-Tyurin construction for $\mathcal{A}_6$

Philip Engel, Olivier de Gaay Fortman, Stefan Schreieder

TL;DR

The paper determines the minimal algebraic multiple $N_6$ of the minimal curve class on a very general $6$-dimensional PPAV to be 6, by bounding obstructions from graphic matroids via a mod $p$ Albanese framework and a symmetry-averaged computation. It introduces a concrete invariant $d(G)$ attached to graphs and shows how averaging over graph automorphisms reduces the search for obstructions to a tractable computation. The main technical novelty lies in translating matroidal obstructions into an explicit linear-algebra problem for graphic matroids and solving it for $K_7$ using a carefully designed fundamental domain and equivariant solver matrices. The result aligns lower bounds with known Prym–Tyurin constructions, yielding a complete picture for $g\le 6$ and outlining pathways toward understanding $N_7$ via further obstructions and constructions.

Abstract

We prove that on a very general principally polarized abelian 6-fold, the smallest multiple of the minimal curve class which can be represented by an algebraic cycle is 6.

Optimality of the Prym-Tyurin construction for $\mathcal{A}_6$

TL;DR

The paper determines the minimal algebraic multiple of the minimal curve class on a very general -dimensional PPAV to be 6, by bounding obstructions from graphic matroids via a mod Albanese framework and a symmetry-averaged computation. It introduces a concrete invariant attached to graphs and shows how averaging over graph automorphisms reduces the search for obstructions to a tractable computation. The main technical novelty lies in translating matroidal obstructions into an explicit linear-algebra problem for graphic matroids and solving it for using a carefully designed fundamental domain and equivariant solver matrices. The result aligns lower bounds with known Prym–Tyurin constructions, yielding a complete picture for and outlining pathways toward understanding via further obstructions and constructions.

Abstract

We prove that on a very general principally polarized abelian 6-fold, the smallest multiple of the minimal curve class which can be represented by an algebraic cycle is 6.

Paper Structure

This paper contains 12 sections, 13 theorems, 25 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Matroids and their solutions
  4. Solutions of matroids in graphs
  5. Graphic matroids and the minimal class
  6. Solutions of graphic matroids
  7. Symmetrizing solutions
  8. Proof of the main result
  9. Lexicographic vertex fundamental domain
  10. Equivariant $1$-chains of the Albanese graph
  11. Building the equivariant solution matrix
  12. Computational aspects of the proof of Theorem \ref{['average-no']}

Key Result

Theorem 1.2

Let $(X,\Theta)\in {\mathcal{A}}_6$ be a very general principally polarized abelian $6$-fold. The smallest positive multiple $N_6$ of the minimal curve class $\Theta^5/5!$ which can be represented by an algebraic cycle is exactly $6$.

Figures (1)

  • Figure 1: Orientation of $K_7$ as a regular tournament with step-1 edges (blue), step-2 edges (green), and step-3 edges (yellow).

Theorems & Definitions (42)

  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.11
  • ...and 32 more