Noncommutative Rational Double Points

The Dynkin Diagrams of Rational Double Points

Rational double points are the simplest surface singularities. In this essay we will be mainly concerned with the geometry of the exceptional set corresponding to the resolution of a rational double point. We will derive the classification of rational double points in terms of Dynkin diagrams.

Rational double points on supersingular K3 surfaces

We investigate configurations of rational double points with the total Milnor number 21 on supersingular K3 surfaces. The complete list of possible configurations is given. As an application, we also give the complete list of extremal (quasi-)elliptic fibrations on supersingular K3 surfaces.

Distribution of Rational Points: A Survey

Rational Points

is known to have only finitely many triples of positive integer solutions x, y, z for a given n > 2 (Faltings, 1983). In Chapter 11, special situations are described in which more precise information is accessible. For example, if x is in S, then n is bounded by a computable number C5 = Cb(pv ..., p8). From these examples, it should be clear that the book is a mine of information for workers in...

Noncommutative Resolutions and Rational Singularities

Let k be an algebraically closed field of characteristic zero. We show that the centre of a homologically homogeneous, finitely generated kalgebra has rational singularities. In particular if a finitely generated normal commutative k-algebra has a noncommutative crepant resolution, as introduced by the second author, then it has rational singularities.