arXiv subjects

Nicolás Vilches

Publications and source records attributed to Nicolás Vilches.

5 recordsLinked to original sources

arXiv2026

Weak stability conditions on coherent systems of genus four curves

The derived category of coherent systems is an interesting triangulated category associated with a smooth, projective curve $C$. These categories admit Bridgeland stability conditions, as recently shown by Feyzbakhsh and Novik. Their construction depends explicitly on the higher rank Brill-Noether theory of $C$. In this short note, we study the Feyzbakhsh--Novik stability conditions for a general curve of genus four. We show that these stability conditions degenerate to a stability condition on the Kuznetsov component of the corresponding nodal cubic threefold, using a result of Alexeev-Kuznetsov.

math.AG↗

arXiv2025

Categorical resolutions of curves and Bridgeland stability

Categorical resolutions of singularities are a replacement of resolution of singularities within the realm of triangulated categories. They allow the study of the derived category of a singular variety $X$ via a triangulated category that behaves like the derived category of a smooth variety. We follow these ideas to study the bounded derived category of a singular, reduced curve $C$ (with arbitrary singularities and number of components). We start by describing an explicit categorical resolution of singularities, specializing a general construction of Kuznetsov and Lunts. We prove the existence of Bridgeland stability conditions on these categories. As a consequence, we get the existence of proper, good moduli spaces of semistable objects. If the curve $C$ is irreducible, then we relate these moduli spaces to the moduli of slope-semistable torsion-free sheaves on $C$, and to the moduli of slope-semistable vector bundles on the (geometric) resolution $\tilde{C}$. This extends classical constructions by Oda and Seshadri, Bhosle and many others. Finally, we use these results to give explicit descriptions of the moduli of torsion-free sheaves on a curve with a single node, cusp, or tacnode.

math.AG↗

arXiv2025

Some examples of moduli spaces on surfaces via wall-crossing

We describe new explicit examples of moduli spaces of Bridgeland semistable objects on surfaces, parametrizing objects whose numerical class agrees with the class of a point. This follows ideas of Tramel and Xia, using stability conditions constructed in our previous work. Our main technical tools are a careful analysis of the wall-crossing from the geometric chamber, and explicit models for the differential graded Lie algebra governing the local structure of the moduli spaces.

math.AG↗

arXiv2025

Stability conditions on surfaces and contractions of curves

We study Bridgeland stability conditions on smooth surfaces arising from birational morphisms $S \to T$ to a singular surface. Assuming that $T$ has only ADE singularities or certain cyclic quotient singularities, we produce pre-stability conditions on $S$ whose central charges depend on the pullback of an ample divisor on $T$. Moreover, we prove the support property for the cyclic quotient case (including the $A_n$ singularities). These stability conditions arise as limits of the Arcara-Bertram stability conditions on $S$. In a complementary direction, we study birational maps $S \to T$ for which a stability condition $S$ can be obtained using the pullback of an ample class on $T$. We prove that the morphism cannot contract smooth curves of positive genus.

math.AG↗

arXiv2021

On wormholes in the moduli space of surfaces

We study a certain wormholing phenomenon that takes place in the Kollár--Shepherd-Barron--Alexeev (KSBA) compactification of the moduli space of surfaces of general type. It occurs because of the appearance of particular extremal P-resolutions in surfaces on the KSBA boundary. We state a general wormhole conjecture, and we prove it for a wide range of cases. At the end, we discuss some topological properties and open questions.

math.AG↗