Research
My research studies KSBA moduli spaces of surfaces, with particular emphasis on del Pezzo surfaces and their weighted marked variants. I am computing \(\kappa\)-classes on these moduli spaces using explicit intersection-theoretic techniques. These del Pezzo examples are guided by the combinatorics of surface degenerations, including their relationships with Weyl groups and associated graphs, as well as serve as concrete settings in which to study the interaction between intersection theory, combinatorics, and birational geometry.
I am particularly interested in:
- KSBA moduli of surfaces: KSBA compactifications of surface pairs, especially marked del Pezzo surfaces.
- Tautological classes on moduli spaces: in particular the \(\kappa\)-classes, their intersection theory, and their role in the structure of the Chow ring.
- Del Pezzo surfaces and their degenerations: analyzing how del Pezzo surfaces appear in KSBA compactifications and how their degenerations are reflected in boundary strata.
Some guiding questions for my work include:
- How does the birational geometry of KSBA moduli spaces of surfaces reflect the combinatorics of surface degenerations?
- To what extent do tautological classes such as the \(\kappa\)-classes satisfy structural properties on these moduli spaces, for instance generating the Chow ring or behaving functorially under natural morphisms?
- When weights are introduced on boundary divisors, how do the \(\kappa\)-classes transform under wall-crossings between chambers, in analogy with Hassett’s moduli spaces of weighted pointed curves?
Current Project
Kappa Classes on KSBA Moduli of Stable Marked Degrees 3 and 4 del Pezzo Surfaces
In preparation (expected 2026)
This project studies the \(\kappa\)-classes on the KSBA moduli space of stable marked degrees \(3\) and \(4\) del Pezzo surfaces. The main goals are:
- to compute the \(\kappa\)-classes on this moduli space and establish analogues of known properties of \(\kappa\)-classes on moduli spaces of stable rational curves, and
- to determine what these \(\kappa\)-classes generate in the Chow ring of the moduli space.
I further investigate the intersection theory of these \(\kappa\)-classes by computing their intersections inside the Chow ring. A related direction considers weighted stable marked degree \(3\) and \(4\) del Pezzo surfaces, where rational weights are placed on boundary components. In this setting, I study how the \(\kappa\)-classes vary across walls in the space of weights, mirroring wall-crossing phenomena for Hassett’s moduli spaces of weighted pointed curves.
Related Interests
While my current focus is on the degrees \(3\) and \(4\) cases, I am interested in extending these questions to:
- del Pezzo surfaces of lower degrees,
- other classes of surfaces admitting KSBA compactifications,
- comparisons between \(\kappa\)-classes on moduli spaces of surfaces and those on moduli spaces of curves, and
- the formulation of surface-level analogues of \(\psi\)-classes on moduli of marked curves.
