3:30-4:45 pm, Lecture Hall 5
Lectures on Rational Singularities
Starting with the definition of a rational singularity by M. Artin (and D. Burns for dimension > 2) we will give several naturally occuring examples of rational singularities. Next, we will give another equivalent criterion for rationality which is often more useful for checking rationality. Some general results about normal surface singularities, including the existence of the fundamental cycle, will be discussed. Artin's purely combinatorial criterion of rationality will be proved. Combinatorics associated to the intersection theory of exceptional divisors of rational singularities for surfaces, and results of Artin, Brieskorn, Laufer, Lipman, Gurjar-Wagh, Spivakovky, Meral Tosun-Le Dung Trang, Tjurina,... will be discussed. In the end we will mention some tantalizing open problems. Familiarity with divisors and coherent cohomology will be assumed.