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

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

2009-03-08

arxiv.org/abs/0903.1444v3

Mathematics

Algebraic Geometry

37 pages. Exposition much improved by referee's corrections

Scientific paper

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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