Program

Jacob Fowler _ Smoothings of surface singularities

A smoothing of a singularity is a way of deforming the defining equations of the variety to obtain a smooth manifold, called the Milnor fiber. The properties of this smooth space reveal interesting information about the original singularity. Of particular interest is the Milnor number of the smoothing, which is a topological invariant of the Milnor fiber. In the case where the Milnor number is 0, the Milnor fiber is a rational homology disk (QHD). Recent work has lead to a complete list of those surface singularities which admit QHD-smoothings. These smoothings have found applications to the rational blow-down procedure of Fintushel--Stern and the construction of new surfaces of general type by Lee--Park.

The goal of these lectures is to introduce the subject of smoothings of surface singularities through explicit examples. We will discuss the recent results on QHD-smoothings, including the complete list of singularities which have such smoothings. Examples will show how QHD-smoothings may be constructed in two ways: by taking quotients of hypersurface singularities, or by blowing up special configurations of curves in the plane.

Kyoung-Seog Lee _ Derived categories of coherent sheaves on algebraic surfaces

Derived categories of coherent sheaves are one of the most important and mysterious invariants of algebraic varieties. In these lecture series I will review the theory of derived categories of algebraic surfaces and introduce the recent developments of the study of derived categories of surfaces of general type with $p_g=q=0$.

Lecture 1. Introduction to derived category

We will briefly recollcet some necessary tools from homological algebra and discuss their applications to algebraic geometry.

Lecture 2. Derived categories of algebraic surfaces

We will recall the notions of semiorthogonal decomposition and exceptional sequence which are one of the most important tools to study derived categories of algebraic varieties. We will also reveiw several important theorems about derived categories of algebraic surfaces.

Lecture 3. Exceptional sequences on some algebraic surfaces

We will review the recent developments of the study of derived categories of surfaces of general type with $p_g=q=0$. To be more precise, we will discuss how to construct exceptional sequences of maximal length on some surfaces with $p_g=q=0$.

Giancarlo Urzúa _ Exploring KSBA boundary

These lectures will be about singular surfaces of general type appearing as "natural" degenerations of smooth ones, and tools to work with these degenerations. Following the choice of Kollár--Shepherd-Barron--Alexeev (KSBA) based on Mori's theory, these singular surfaces compactify the moduli space of surfaces of general type. We will focus on a certain type of singularities and smoothings, which will be introduced in this Winter school by Fowler, and on explicit computations around the necessary Mori's theory to explore KSBA boundary.

Through concrete examples, the aim is also to discuss on some open problems related to KSBA moduli spaces and the corresponding Mori's theory.

The content will be mainly based on arXiv:1310.1580 (joint with Paul Hacking and Jenia Tevelev), arXiv:1310.4353, arXiv:1311.4844, and arXiv:1409.4985 (joint with Arié Stern), plus some work in progress joint with Jenia Tevelev.

(1) Basics on singularities / Lee-Park construction / KSBA moduli

(2) Running MMP & Flipping surfaces / extremal P-resolutions/ Identification

(3) W-surfaces in general / Non-normal limits / Flips and surfaces via examples