Prime exceptional divisors on holomorphic symplectic varieties and monodromy reflections

Abstract

Let $X$ be a projective irreducible holomorphic symplectic manifold. The second integral cohomology of $X$ is a lattice with respect to the Beauville–Bogomolov pairing. A divisor $E$ on $X$ is called a prime exceptional divisor if $E$ is reduced and irreducible and of negative Beauville–Bogomolov degree.

Let $E$ be a prime exceptional divisor on $X$. We first observe that associated to $E$ is a monodromy involution of the integral cohomology $H^{\ast }(X,\mathbb{Z})$, which acts on the second cohomology lattice as the reflection by the cohomology class $\left[E\right]$ of $E$.

We then specialize to the case where $X$ is deformation equivalent to the Hilbert scheme of length $n$ zero-dimensional subschemes of a $K3$ surface, $n\ge 2$. We determine the set of classes of exceptional divisors on $X$. This leads to a determination of the closure of the movable cone of $X$.