On Lagrangian fibrations by Jacobians II

TL;DR

This paper investigates conditions under which certain Lagrangian fibrations by Jacobians are equivalent to Beauville-Mukai integrable systems, focusing on specific dimensions and types of algebraic curves.

Contribution

It establishes new criteria identifying when these fibrations correspond to Beauville-Mukai systems based on dimension and curve properties.

Findings

X is a Beauville-Mukai system for n=3,4,5 with irreducible, non-hyperelliptic curves

X is a Beauville-Mukai system for n=3 with odd degree and hyperelliptic curves

The results extend understanding of Lagrangian fibrations in algebraic geometry

Abstract

Let Y->P^n be a flat family of reduced Gorenstein curves, such that the compactified relative Jacobian X=\bar{J}^d(Y/P^n) is a Lagrangian fibration. We prove that X is a Beauville-Mukai integrable system if n=3, 4, or 5, and the curves are irreducible and non-hyperelliptic. We also prove that X is a Beauville-Mukai system if n=3, d is odd, and the curves are canonically positive 2-connected hyperelliptic curves.