We explore the moduli space of stable surfaces, where the simplest of questions continue to remain open for almost all invariants. A few such questions: Of the allowable singularities, which ones actually occur on a stable surface? Which of these deform to smooth surfaces? How can we use this knowledge to find divisors in the moduli spaces? Can we develop a stratification of these moduli spaces by singularity type? Our focus will be on cyclic quotient singularities, with an emphasis on discussing concrete visual examples built from rational, K3, and elliptic surfaces.
The talk will start with a pre-seminar at 2pm:
Title: Numerics on surfaces
Abstract: Many constructions of algebraic surfaces rely on a few fundamental tools that allow us to easily perform numerical computations. In this talk, we’ll focus on the canonical divisor of a surface, a related birational invariant, and the adjunction formula. This last tool will allow us to understand why blowups at a smooth point introduce a so-called (-1)-curve on a surface. If there's time, we will discuss behavior of the canonical class under double covers and/or resolutions of canonical singularities. In the actual talk, I’ll give multiple constructions of stable surfaces using blowups and double covers, so students attending the pre-talk will be in a good position to follow these constructions.