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

Understand

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.