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.

Hilbert schemes and stable pairs: GIT and derived category wall crossings / Stoppa, Jacopo; Thomas Richard, Paul. - In: BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE. - ISSN 0037-9484. - 139:3(2011), pp. 297-339. [10.24033/bsmf.2610]

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

Stoppa, Jacopo;
2011-01-01

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.
2011
139
3
297
339
http://arxiv.org/abs/0903.1444
Stoppa, Jacopo; Thomas Richard, Paul
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/14594
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