du Val singularity

Abstract: We give a bound on which singularities may appear on Koll\'ar--Shepherd-Barron--Alexeev stable surfaces for a wide range of topological invariants and use this result to describe all stable numerical quintic surfaces (KSBA-stable surfaces with $K^2=\chi=5$) whose unique non Du Val singularity is a Wahl singularity. We then extend the deformation theory of Horikawa to the log setting in order to describe the boundary divisor of the moduli space $\overline{\mathcal{M}}_{5,5}$ corresponding to these surfaces. 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.

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

Abstract: We study terminal 3-fold divisorial extractions σ: (E Y ) → (C X) which extract a prime divisor E from a singular curve C centred at a point P in a smooth 3-fold X. Given a presentation of the equations defining C, we give a method for calculating the graded ring of Y explicitly by serial unprojection. We compute some important examples and classify such extractions when the general hyperplane section SX containing C has a Du Val singularity at (P ∈ SX) of type A1, A2, D2k, E6, E7 or E8.

Abstract: We show that the partition function on a generalised conifold $C_{m,n}$ with ${m+n \choose m}$ crepant resolutions can be equivalently computed on the compound du Val singularity $A_{m+n-1}\times \mathbb C$ with a unique crepant resolution.