##### Department of Mathematics,

University of California San Diego

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### Lagrangian Hyperplanes in Holomorphic Symplectic Varieties

## Benjamin Bakker

#### Courant

## Algebraic Geometry Seminar

##### Abstract:

It is well known that the extremal rays in the cone of effective curve classes on a K3 surface are generated by rational curves $C$ for which $(C,C)=-2$; a natural question to ask is whether there is a similar characterization for a higher-dimensional holomorphic symplectic variety $X$. The intersection form is no longer a quadratic form on curve classes, but the Beauville-Bogomolov form on $X$ induces a canonical nondegenerate form $(\cdot,\cdot)$ on $H_2(X;\mathbb{R} )$ which coincides with the intersection form if $X$ is a K3 surface. We therefore might hope that extremal rays of effective curves in $X$ are generated by rational curves $C$ with $(C,C)=-c$ for some positive rational number $c$. In particular, if $X$ contains a Lagrangian hyperplane $\mathbb P^n\subset X$, the class of the line $\ell\subset\mathbb P^n$ is extremal. For $X$ deformation equivalent to the Hilbert scheme of $n$ points on a K3 surface, Hassett and Tschinkel conjecture that $(\ell,\ell)=-\frac{n+3}{2}$; this has been verified for $n<4$. In joint work with Andrei Jorza, we prove the conjecture for $n=4$, and discuss some general properties of the ring of Hodge classes on $X$.

Host: Dragos Oprea

### January 11, 2012

### 2:00 PM

### AP&M 7218

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