Florian Enescu,
Singularities of F-Pure Rings of Dimension Two
Nowadays, in commutative algebra, Noetherian rings of characteristic p
defined by various properties of the Frobenius homomorphism play an
important role. Together with the tight closure theory, they proved to be
useful in studying a great number of problems such as Briancon - Skoda
type theorems, the direct summand conjecture, big Cohen-Macaulay algebras,
Hilbert-Kunz multiplicity, the uniform Artin-Rees type theorems, to name a
few. Along the same lines, this methods give a great deal of information
in equal characteristic zero, by reduction to characteristic p.
In this talk, we will discuss F-pure rings and F-rational rings and talk
about their relation to singularities that arise from birational geometry.
In dimension two, the singularities of F-pure normal rings were classified
independently by V.B. Mehta and V. Srinivas, and K.-I. Watanabe (in the
Gorenstein case). We will sketch the proof of Watanabe's result asserting
that a normal F-pure ring of dimension two is either a rational double
point, a cusp or a simple elliptic singularity.