Incidence Hilbert schemes and infinite dimensional Lie algebras

Details Incidence Hilbert schemes and infinite dimensional Lie algebras Incidence Hilbert schemes and infinite dimensional Lie algebras

2006-06-20

arxiv.org/abs/math/0606504v1

Mathematics

Algebraic Geometry

Scientific paper

Given a projetive surface $S$, using correspondences, we construct an infinite dimensional Lie algebra that acts on the direct sum $\Wfock$ of the cohomology groups of the incidence Hilbert schemes $S^{[n,n+1]}$ over all $n$. The algebra is related to an extension of an infinite dimensional Heisenberg algebra. The space $\Wfock$ is a highest weight representation of this algebra. Our result provides a representation-theoretic interpretation of Cheah's generating function of Betti numbers of the incidence Hilbert schemes. As a consequence, an additive basis of the cohomology group of the incidence Hilbert scheme is obtained.

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