LEAPS-MPS: Describing Compactifications of Moduli Spaces of Varieties and Pairs.

Algebraic geometry is an area of mathematics that explores the geometry associated with the solution sets of polynomials. As the coefficients of these polynomials often vary, this, in turn, changes the geometry of their solution sets. Moreover, these geometric shapes carry linear data, known as their Hodge structures, which typically identify them. The PI will study the behavior of such geometric and linear data when these coefficients shift toward infinity. Particular focus is given to geometric cases that arise from applications to other mathematical areas like combinatorics and mathematical physics. Additionally, the researcher aims to foster an inclusive and diverse learning environment, providing students with training in geometric and computational tools, thereby empowering them to become scholars. This focus is particularly essential for first-generation students and underrepresented minorities, for whom the path to academia is often more uncertain. This project will study GIT, Hodge theoretic, and KSBA compactifications for the moduli space of surfaces of a general type and log pairs, such as the ones associated with configurations of plane conics and log del Pezzo surfaces with their anticanonical divisors. In particular, the PI will describe well-behaved degenerations from a Hodge theory perspective and utilize techniques such as eigenspace decompositions for studying the Limiting Mixed Hodge structures of the varieties parametrized by the relevant compact moduli spaces. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.