Lagrangian tens of planes, Enriques surfaces and holomorphic symplectic fourfolds
Abstract
The Fano models of Enriques surfaces produce a family of tens of mutually intersecting planes in $\mathbf P^5$ with a $10$dimensional moduli space. The latter is linked to several 10dimensional moduli spaces parametrizing other types of objects: a) cubic fourfolds containing the tens of planes, b) BeauvilleDonagi holomorphically symplectic fourfolds, and c) double EPW sextics. The varieties in b) parametrize lines on cubic fourfolds from a). The double EPW sextics are associated, via O'Grady's construction, to Lagrangian subspaces of the Plücker space of the Grassmannian $Gr(2,\mathbf P^5)$ spanned by 10 mutually intersecting planes in $\mathbf P^5$. These links imply the irreducibility of the moduli space of supermarked Enriques surfaces, where a supermarking is a choice of a minimal generating system of the Picard group of the surface. Also some results are obtained on the variety of tens of mutually intersecting planes, not necessarily associated to Fano models of Enriques surfaces.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 DOI:
 10.48550/arXiv.1906.01445
 arXiv:
 arXiv:1906.01445
 Bibcode:
 2019arXiv190601445D
 Keywords:

 Mathematics  Algebraic Geometry;
 14J28;
 14J35;
 14J10;
 14N20
 EPrint:
 The preprint is in an unfinished form and contains imprecisions. In particular, proofs of primitivity of embeddings of lattices polarizing cubic 4folds in Section 7 are lacking