Rank-one sheaves and stable pairs on surfaces

Finite Quot schemes were used by Bertram, Johnson, and the first author to study Le Potier's strange duality conjecture on del Pezzo surfaces when one of moduli spaces is Hilbert scheme points. In order rigorously enumerate finite scheme, we space limit stable pairs in which target has rank a smooth complex projective surface. We obtain an embedding this into that induces perfect obstruction theory. This theory yields virtual fundamental class can be computed explicitly. Because coincides with they have dimension 0, makes desired count Bertram et al. rigorous. As another application, universality result for tautological integrals pairs.