### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 264-286 |

Number of pages | 23 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 65 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2012 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*65*(2), 264-286. https://doi.org/10.1002/cpa.20384

**Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces.** / Harvey, David; Hassett, Brendan; Tschinkel, Yuri.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 65, no. 2, pp. 264-286. https://doi.org/10.1002/cpa.20384

}

TY - JOUR

T1 - Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces

AU - Harvey, David

AU - Hassett, Brendan

AU - Tschinkel, Yuri

PY - 2012/2

Y1 - 2012/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=81855176079&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=81855176079&partnerID=8YFLogxK

U2 - 10.1002/cpa.20384

DO - 10.1002/cpa.20384

M3 - Article

VL - 65

SP - 264

EP - 286

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 2

ER -