2004/03/30 by Kieran G. O'Grady, Kieran G. O’Grady, O'Grady, Kieran G. · 4 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #math.AG
paper · pdf · doi:10.48550/arxiv.math/0403519
42 pages
arxiv created 2004/03/30 · openalex publication_date 2004/03/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A K3 surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that a similar statement holds for the generic couple (X,H) with X a deformation of (K3)[n] and H an ample divisor of square 2 for Beauville's quadratic form. If n=2 then according to the conjecture X is a double cover of a (singular) sextic 4-fold in \PP5. It follows from the conjecture that a deformation of (K3)[n] carrying a divisor (not necessarily ample) of degree 2 has an anti-symplectic birational involution. We test the conjecture. In doing so we bump into some interesting geometry: examples of two anti-symplectic involutions generating an interesting dynamical system, a case of Strange duality and what is probably an involution on the moduli space of degree-2 quasi-polarized (X,H) where X is a deformation of (K3)[2].