On Lagrangian fibrations by Jacobians I

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.

arXiv comment28 pages, 1 figure, the article has been extensively rewritten (Section 2 is completely new and numerous other lemmas and remarks have been added to Sections 3 and 4), the `mild singularities' hypothesis on the curves has been replaced by the simpler hypothesis that the curves are Gorenstein, the lower bound of 4n+2 for the degree of the discriminant locus is also new

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