Palestras e Seminários
27/09/2016
19:00
Sala 3011
Seminários de Singularidades
Palestrante: Josnei Novacoski (ICMC)
Abstract: In this lecture we will introduce the Zariski's approach to obtain a resolution of singularities for an algebraic variety $X$ over a field $k$. This method consists of two steps. The first step is to show that every valuation $\nu$ of $k(X)$ centered at $X$ admits local uniformization, i.e., that there exists a modification (a proper birational morphism) $Y\lra X$ such that the center of $\nu$ at $Y$ is regular. Then, there exists an open neighbourhood $U_\nu$ of $\nu$ (in the Zariski topology) such that the center of every valuation $\mu\in U_\nu$ at $Y$ is regular. Using the fact that the Zariski topology on the space of valuations is compact, we conclude that there exist modifications $Y_i\lra X$, $i=1,\ldots,n$, such that the center of each valuation at some $Y_i$ is regular. The second step in Zariski's approach is to glue all this solutions to obtain a resolution of singularities of $X$. We will also discuss some variations of this method to treat different types of singularities.
