manuscripta mathematica

, Volume 120, Issue 2, pp 131–150

Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces


DOI: 10.1007/s00229-006-0631-4

Cite this article as:
Markushevich, D. manuscripta math. (2006) 120: 131. doi:10.1007/s00229-006-0631-4


A rational Lagrangian fibration f on an irreducible symplectic 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 Hilbert scheme of d points on a generic K3 surface S of genus g under the hypothesis that its degree 2g−2 is a square times 2d−2. 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.

Mathematics Subject Classification (2000)

14J60 14J40 

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Mathématiques - bât.M2Université Lille 1Villeneuve d'Ascq CedexFrance

Personalised recommendations