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Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces

D. Markushevich
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Abstract

A rational Lagrangian fibration f on an irreducible symplecitc variety V is a rational map which is birationally equivalent to a regular surjective morphism with Lagrangian fibers. By analogy with K3 surfaces, it is natural to expect that a rational Lagrangian fibration exists if and only if V has a divisor D with Bogomolov--Beauville square 0. This conjecture is proved in the case when V is the punctual Hilbert scheme of a generic algebraic K3 surface S. The construction of f uses a twisted Fourier--Mukai transform which induces a birational isomorphism of V with a certain moduli space of twisted sheaves on another K3 surface M, obtained from S as its Fourier--Mukai partner.

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D. Markushevich. 2007-03-06. Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces. https://doi.org/10.1007/s00229-006-0631-4

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