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

Jacopo Stoppa; Richard P. Thomas

Bulletin de la Société Mathématique de France (2011)

  • Volume: 139, Issue: 3, page 297-339
  • ISSN: 0037-9484

Abstract

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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.

How to cite

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Stoppa, Jacopo, and Thomas, Richard P.. "Hilbert schemes and stable pairs: GIT and derived category wall crossings." Bulletin de la Société Mathématique de France 139.3 (2011): 297-339. <http://eudml.org/doc/272609>.

@article{Stoppa2011,
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.},
author = {Stoppa, Jacopo, Thomas, Richard P.},
journal = {Bulletin de la Société Mathématique de France},
language = {eng},
number = {3},
pages = {297-339},
publisher = {Société mathématique de France},
title = {Hilbert schemes and stable pairs: GIT and derived category wall crossings},
url = {http://eudml.org/doc/272609},
volume = {139},
year = {2011},
}

TY - JOUR
AU - Stoppa, Jacopo
AU - Thomas, Richard P.
TI - Hilbert schemes and stable pairs: GIT and derived category wall crossings
JO - Bulletin de la Société Mathématique de France
PY - 2011
PB - Société mathématique de France
VL - 139
IS - 3
SP - 297
EP - 339
AB - 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.
LA - eng
UR - http://eudml.org/doc/272609
ER -

References

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