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Journal of the European Mathematical Society

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Volume 20, Issue 7, 2018, pp. 1629–1654
DOI: 10.4171/JEMS/795

Published online: 2018-05-22

Homology of Hilbert schemes of points on a locally planar curve

Jørgen Vold Rennemo[1]

(1) All Souls College, Oxford, UK

Let $C$ be a proper, integral, locally planar curve, and consider its Hilbert schemes of points $C^[n]$. We define four creation/annihilation operators acting on the rational homology groups of these Hilbert schemes and show that the operators satisfy the relations of a Weyl algebra. The action of this algebra is similar to that defined by Grojnowski and Nakajima for a smooth surface. As a corollary, we compute the cohomology of $C^[n]$ in terms of the cohomology of the compactified Jacobian of $C$ together with an auxiliary grading on the latter. This recovers and slightly strenghtens a formula recently obtained in a different way by Maulik and Yun and independently Migliorini and Shende.

Keywords: Locally planar curves, Hilbert scheme, compactified Jacobian, Gopakumar– Vafa invariants, Weyl algebra

Vold Rennemo Jørgen: Homology of Hilbert schemes of points on a locally planar curve. J. Eur. Math. Soc. 20 (2018), 1629-1654. doi: 10.4171/JEMS/795