We study the non-klt locus of singularities of pairs. We show that
given a pair (X, B) and a projective morphism X → Z with connected fibres such
that −(KX +B) is nef over Z, the non-klt locus of (X, B) has at most two connected
components near each fibre of X → Z. This was conjectured by Hacon and Han.
In a different direction we answer a question of Mark Gross on connectedness
of the non-klt loci of certain pairs. This is motivated by constructions in Mirror
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 the moduli spaces is the Hilbert scheme of points. In order to rigorously enumerate the finite Quot
scheme, we study the moduli space of limit stable pairs in which the target has rank one on a smooth complex projective surface. We obtain an embedding of this moduli space into a smooth space that induces a perfect obstruction theory. This obstruction theory yields a virtual fundamental class that can be computed explicitly. Because the moduli space coincides with the Quot scheme when they have dimension 0, this makes the desired count of the finite Quot scheme in Bertram et al. rigorous. As another application, we obtain a universality result for tautological integrals on the moduli space of stable pairs.