Hilbert schemes and stable pairs: GIT and derived category wall crossings

J. Stoppa; R. P. Thomas

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arXiv · 0903.1444

Hilbert schemes and stable pairs: GIT and derived category wall crossings

J. Stoppa·R. P. Thomas

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Abstract

We show that the Hilbert scheme of curves and Le Potier's moduli space of stable pairs with one dimensional support have a common GIT construction. The two spaces correspond to chambers on either side of a wall in the space of GIT linearisations. We explain why this is not enough to prove the "DT/PT wall crossing" conjecture relating the invariants derived from these moduli spaces when the underlying variety is a 3-fold. We then give a gentle introduction to a small part of Joyce's theory for such wall crossings, and use it to give a short proof of an identity relating the Euler characteristics of these moduli spaces. When the 3-fold is Calabi-Yau the identity is the Euler-characteristic analogue of the DT/PT wall crossing conjecture, but for general 3-folds it is something different, as we discuss.

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BibTeX RIS

J. Stoppa, R. P. Thomas. 2009-10-12. Hilbert schemes and stable pairs: GIT and derived category wall crossings. https://arxiv.org/abs/0903.1444

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