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A CLASSIFICATION OF LAGRANGIAN PLANES IN HOLOMORPHIC SYMPLECTIC VARIETIES
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 16 / Issue 4 / September 2017
- Published online by Cambridge University Press:
- 09 September 2015, pp. 859-877
- Print publication:
- September 2017
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- Article
- Export citation
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Classically, an indecomposable class
$R$ in the cone of effective curves on a K3 surface 
$X$ is representable by a smooth rational curve if and only if 
$R^{2}=-2$. We prove a higher-dimensional generalization conjectured by Hassett and Tschinkel: for a holomorphic symplectic variety 
$M$ deformation equivalent to a Hilbert scheme of 
$n$ points on a K3 surface, an extremal curve class 
$R\in H_{2}(M,\mathbb{Z})$ in the Mori cone is the line in a Lagrangian 
$n$-plane 
$\mathbb{P}^{n}\subset M$ if and only if certain intersection-theoretic criteria are met. In particular, any such class satisfies 
$(R,R)=-\frac{n+3}{2}$, and the primitive such classes are all contained in a single monodromy orbit.
