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Brill-Noether theory and Green's conjecture for general curves on simple abelian surfaces

Federico Moretti

TL;DR

The paper extends Lazarsfeld–Mukai bundle techniques from K3 surfaces to simple abelian surfaces to study Brill–Noether loci on curves in non-primitive linear systems. It introduces g(S,L) and N(S,L) to capture gonality and base-point data, constructs LM-bundle parameter spaces, and proves a linear growth bound dimW^1_d(|L|) ≤ d − g(S,L) + g for dominating components. Consequently, for general C ∈ |L| the gonality is governed by g(S,L) and Green's conjecture holds for general curves in |mN|, with further structural insight into non-reduced Brill–Noether loci and the abundance of minimal pencils. The results highlight both similarities and essential differences with the K3 case, notably in the non-constancy of Clifford index and the Prym-like phenomena arising from non-primitive systems on abelian surfaces.

Abstract

In this paper we compute the gonality and the dimension of the Brill-Noether loci $W^1_d(C)$ for curves in a non primitive linear system of a simple abelian surface, adapting vector bundles techniques à la Lazarsfeld originally introduced with $K3$ surfaces. As a corollary, we obtain general Green's conjecture for curves on abelian surfaces.

Brill-Noether theory and Green's conjecture for general curves on simple abelian surfaces

TL;DR

The paper extends Lazarsfeld–Mukai bundle techniques from K3 surfaces to simple abelian surfaces to study Brill–Noether loci on curves in non-primitive linear systems. It introduces g(S,L) and N(S,L) to capture gonality and base-point data, constructs LM-bundle parameter spaces, and proves a linear growth bound dimW^1_d(|L|) ≤ d − g(S,L) + g for dominating components. Consequently, for general C ∈ |L| the gonality is governed by g(S,L) and Green's conjecture holds for general curves in |mN|, with further structural insight into non-reduced Brill–Noether loci and the abundance of minimal pencils. The results highlight both similarities and essential differences with the K3 case, notably in the non-constancy of Clifford index and the Prym-like phenomena arising from non-primitive systems on abelian surfaces.

Abstract

In this paper we compute the gonality and the dimension of the Brill-Noether loci for curves in a non primitive linear system of a simple abelian surface, adapting vector bundles techniques à la Lazarsfeld originally introduced with surfaces. As a corollary, we obtain general Green's conjecture for curves on abelian surfaces.
Paper Structure (10 sections, 34 theorems, 37 equations)

This paper contains 10 sections, 34 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. The dimension of $\mathcal{W}^1_d(L)$
  3. Structure of Lazarsfeld--Mukai bundles
  4. Case $h^1(E)=0$.
  5. Case $h^1(E)>0$.
  6. A space parametrizing Lazarsfeld--Mukai bundles
  7. The dimension of $\mathcal{G}_{[n],l}$
  8. A bound for the dimension of $\mathcal{W}^1_d|L|$
  9. An upper bound for the gonality and linear growth condition
  10. On certain special loci in $\mathrm{Hilb}(S)$

Key Result

Theorem 1.1

Let $(S,N)$ be a $(1,e)$ polarized abelian surface with $\mathrm{NS}(S)=\mathbb{Z}N$ and $e\ge 2$. Let $C$ be a general curve of genus $g=m^2e+1$ in $|mN|$. If $m \le 2$ then $C$ is of maximal gonality. Otherwise the following hold: Moreover, the general curve carries at least $\frac{N^2(N^2-1)}{2}$ minimal pencils.

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Proposition 2.4
  • ...and 52 more