Boundary divisors in the moduli space of stable quintic surfaces

Abstract

BOUNDARY DIVISORS IN THE MODULI SPACE OF STABLE QUINTIC SURFACES FEBRUARY 2014 JULIE RANA, B.S., MARLBORO COLLEGE M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Jenia Tevelev My research incorporates several central themes in algebraic geometry, including moduli spaces and their compactifications, singular spaces, and deformation theory. I am especially interested in Gieseker’s moduli space MK2,χ of minimal surfaces of general type with fixed numerical invariants, and its Kollar–Shepherd-Barron, Alexeev compactification MK2,χ. Some of the questions I am interested in include describing which singularities might appear on a stable surface with given invariants, finding concrete models for singular surfaces, and describing the structure of MK2,χ along the boundary, especially in the presence of obstructions to Q-Gorenstein deformations of stable surfaces. In this thesis, I give a bound on which singularities may appear on stable surfaces for a wide range of topological invariants, and use this result to describe all stable numerical quintic surfaces, i.e. stable surfaces with K2 = χ = 5, whose unique non Du Val singularity is a Wahl singularity. Quintic surfaces are the simplest examples of surfaces of general type and the question of describing their moduli is a long-standing question in algebraic geometry. I then extend the deformation theory of Horikawa in [Hor75] to the log setting in order to describe the boundary divisor of the moduli space M5,5 corresponding to