Curve counting via the Quot scheme
Donaldson-Thomas theory is a counting theory for sheaves on Calabi-Yau threefolds Y.
Another such theory is the Pandharipande-Thomas theory of stable pairs.
The two theories can be thought of as (virtually) enumerating curves on Y in a fixed homology class.
For each homology class, there is a beautiful wall crossing formula relating DT and PT theory.
We ask: what is the contribution of a fixed smooth curve C on Y to these curve counting invariants?
On the PT side, the answer was given by Pandharipande-Thomas. On the DT side,
we define the contribution of C using the Quot scheme of its ideal sheaf, and we show that the same
wall-crossing formula holds for the localised invariants, at least when C is assumed to be rigid.