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

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.