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

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.

37 pages. Exposition much improved by referee's corrections

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Cited by in corpus (9)

• Derived categories of small toric Calabi-Yau 3-folds and counting invariants
• Counting invariant of perverse coherent sheaves and its wall-crossing
• Chamber Structure and Wallcrossing in The ADHM Theory of Curves I
• Crystal melting on toric surfaces
• Motivic Donaldson-Thomas invariants of the conifold and the refined topological vertex
• Curve counting theories via stable objects I. DT/PT correspondence
• Multiple cover formula of generalized DT invariants I: parabolic stable pairs
• Flops and Hilbert schemes of space curve singularities
• Curve counting invariants around the conifold point