Research
I study moduli spaces of algebraic varieties, with current emphasis on compactifications and universal families.
My recent work focuses on KSBA compactifications of marked surface pairs, especially marked cubic surfaces and related del Pezzo surfaces. I am interested in how geometric properties of a family appear globally on its moduli space and, conversely, how geometric structures on the moduli space constrain the universal family.
I am particularly interested in:
- Moduli spaces of algebraic varieties: compactifications, stability conditions, and modular interpretations.
- Universal families: relative projective geometry and geometric structures on families over compactified moduli spaces.
- KSBA moduli of surfaces: compactifications of marked surface pairs, especially marked del Pezzo surfaces.
- Intersection theory: divisor classes and tautological classes on compactified moduli spaces.
Current Project
Universal Families of Weighted KSBA-Stable Marked Cubic Surfaces
In preparation
My current project studies the universal family over the minimally weighted KSBA compactification of the moduli space of marked cubic surfaces.
A smooth cubic surface admits an anticanonical embedding as a cubic hypersurface in projective space. This project studies how that construction globalizes over the universal family. The main result realizes the universal family as a relative cubic hypersurface inside the projectivization of a rank four vector bundle obtained from the relative anticanonical sheaf.
This realization provides a geometric description of the universal family and allows one to study divisor classes and intersection theory directly from the relative projective geometry.
As applications, the project studies:
- the geometry of the total space of the universal family;
- the divisor class of the relative cubic hypersurface;
- the relationship between the universal marked divisor and the relative anticanonical class; and
- weighted tautological classes in the minimal chamber.
Additional Directions of Interest
I am interested in extending these questions to related moduli problems, including:
- explicit wall-crossing transformations for weighted KSBA-stable marked cubic surfaces;
- weighted KSBA compactifications of lower-degree del Pezzo surfaces;
- moduli spaces associated with real cubic surfaces; and
- modular compactifications related to Coble surfaces.
At present these are directions of interest rather than active projects.
