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4:00 pm Tuesday, March 20, 2012 Algebraic Geometry Seminar: Lagrangian Hyperplanes in Holomorphic Symplectic Varietiesby
Benjamin Bakker (NYU) in HB 227- 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 ( , ) on H_2(X; 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 of dimension n, the class L of the line in that hyperplane is extremal. For X deformation equivalent to the Hilbert scheme of n points on a K3 surface, Hassett and Tschinkel conjecture that (L,L)=-(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.
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