On Lagrangian fibrations by Jacobians I

TL;DR

This paper investigates conditions under which a Lagrangian fibration by Jacobians arises from a family of Gorenstein curves, establishing bounds on the discriminant locus and characterizing when it forms a Beauville-Mukai integrable system.

Contribution

It provides new lower bounds on the discriminant degree and characterizes when the Jacobian fibration is a Beauville-Mukai system based on discriminant degree.

Findings

Discriminant degree in P^n is at least 4n+2.

X is a Beauville-Mukai system if discriminant degree exceeds 4n+20.

Established conditions linking discriminant degree to integrable system structure.

Abstract

Let Y->P^n be a flat family of integral Gorenstein curves, such that the compactified relative Jacobian X=\bar{J}^d(Y/P^n) is a Lagrangian fibration. We prove that the degree of the discriminant locus Delta in P^n is at least 4n+2, and we prove that X is a Beauville-Mukai integrable system if the degree of Delta is greater than 4n+20.