2026 Theses Doctoral

Stability conditions and singular varieties

Vilches Reyes, Nicolas Ignacio

We study Bridgeland stability conditions and moduli spaces of semistable objects on categories associated to singular curves and surfaces. For singular curves, we construct stability conditions on the Kuznetsov–Lunts categorical resolution of singularities. These stability conditions interpolate between slope-stability on the original curve and on its normalization. For surfaces, we study the degenerations of the Arcara–Bertram stability condition on the derived category of a surface, and their connection to birational contractions. Depending on the singularities of the contraction, we either prove the existence or nonexistence of such limits.

Finally, we study various moduli spaces of Bridgeland semistable objects, using the stability conditions constructed above. In the case of curves, we recover various classical descriptions of compactified Jacobians by Oda–Seshadri, Bhosle and many others, by relating the objects in the categorical resolution to torsion-free sheaves on the singular curve and its normalization. For surfaces, we exhibit new interesting examples of wall-crossing, including moduli spaces with arbitrarily many irreducible components, and explicit local descriptions of the singularities using differential graded Lie algebras.