Involutions and linear systems on holomorphic symplectic manifolds

Details Involutions and linear systems on holomorphic symplectic manifolds Involutions and linear systems on holomorphic symplectic manifolds

2004-03-30

arxiv.org/abs/math/0403519v1

Mathematics

Algebraic Geometry

42 pages

Scientific paper

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 $\PP^5$. 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 $(K3)^{[2]}$.

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