Plática dada por Karl Schwede (The University of Utah) en el Segundo Congreso Nacional de Geometría Algebraica el lunes 26 de febrero del 2018 Casa Matemática Oaxaca, en Oaxaca de Juárez, México
In characteristic zero, Elkik proved that if R is a local domain and that f in R is nonzero such that R/fR has rational singularities, then R has rational singularities as well. In characteristic p, there is an analog of rational singularities called F(robenius)-rational singularities and the analogous result to Elkik's also holds in that context. Additionally K. Smith showed that if R in characteristic zero has F-rational singularities after reduction to p 0, then R has rational singularities in characteristic zero.
Suppose now that R is a local ring of mixed characteristic (0, p) and that R/pR has F-rational singularities. We show that this implies that R has a form of rational singularities which we call BCM-rational. This implies that R has rational singularities in the usual sense and hence the localization to characteristic zero also has rational singularities. In particular, if R in characteristic zero has rational singularities after reduction to a single characteristic p (even a small one), then R has rational singularities in characteristic zero. This also gives a way for a computer to show that a ring in characteristic zero has rational singularities.
If time permits, I will also discuss generalizations to other ways to measure singularities such as multiplier ideals. This is all joint work with Linquan Ma.