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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...
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