Categorical resolutions of curves and Bridgeland stability

Abstract: Moduli spaces of semistable vector bundles on a curve have been studied extensively in the last half century. If the curve is smooth and projective, these spaces are well-understood: they are projective, irreducible, and have mild singularities.

If the curve is singular, the moduli spaces of torsion-free sheaves are much harder to describe. Partial descriptions are available for nodal curves, following Oda–Seshadri and Bhosle, whereas much less is known for other types of curve singularities.

In this talk, we will describe an approach to these types of questions using categorical resolutions of singularities. Following Kuznetsov and Lunts, we produce a well-behaved triangulated category that helps us describe sheaves on the singular curve. We then construct a family of Bridgeland stability conditions on this category, which interpolate between slope-stability on the singular curve and its (geometric) resolution. Finally, we will discuss applications to nodal and tacnodal curves.