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Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces
Author(s) -
Harvey David,
Hassett Brendan,
Tschinkel Yuri
Publication year - 2012
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20384
Subject(s) - mathematics , hilbert scheme , pure mathematics , homology (biology) , cohomology ring , k3 surface , projective space , cohomology , symplectic geometry , intersection theory , diophantine equation , singular homology , mapping class group , algebra over a field , mathematical analysis , projective test , surface (topology) , moduli space , geometry , equivariant cohomology , differential algebraic equation , ordinary differential equation , biochemistry , chemistry , gene , differential equation
We seek to characterize homology classes of Lagrangian projective spaces embedded in irreducible holomorphic‐symplectic manifolds, up to the action of the monodromy group. This paper addresses the case of manifolds deformation‐equivalent to the Hilbert scheme of length‐3 subschemes of a K3 surface. The class of the projective space in the cohomology ring has prescribed intersection properties, which translate into Diophantine equations. Possible homology classes correspond to integral points on an explicit elliptic curve; our proof entails showing that the only such point is two‐torsion. © 2011 Wiley Periodicals, Inc.