The Dynkin diagrams of rational double points

We treat nine of fourteen triangle singularities in Arnold's classification list of singularities. We consider what kind of combinations of rational double points can appear on their small deformation fibers. We show their combinations are described by a simple priciple using Dynkin graphs.

A general strategy is given for the classification of graphs of rational surface singularities. For each maximal rational double point configuration we investigate the possible multiplicities in the fundamental cycle. We classify completely certain types of graphs. This allows to extend the classification of rational singularities to multiplicity 8. We also discuss the complexity of rational resolution graphs.

Fourteen kinds of triangle singularities with modality one in Arnold's classification list are discussed. We consider which kinds of combinations of rational double points can appear on small deformation fibers of the singularities. We show that possible combinations of rational double points can be described by a unique principle from the view point of Dynkin graphs.

We establish a one-to-one correspondence between the singularity categories of rational double points and the simply-laced Dynkin graphs in arbitrary characteristic. This correspondence is well-known in characteristic zero since the rational double points are quotient singularities in characteristic zero whereas not necessarily in positive characteristic. Considering some rational double points are not taut in characteristic less than seven, we can see there exist two rationa...

We classify normal supersingular K3 surfaces Y with total Milnor number 20 in characteristic p, where p is an odd prime that does not divide the discriminant of the Dynkin type of the rational double points on Y.

Locally analytically, any isolated double point occurs as a double covering of a smooth surface. It can be desingularized via the canonical resolution, as it is well-known. In this paper we explicitly compute the fundamental cycle of both the canonical and minimal resolution of a double point singularity and we classify those for which the fundamental cycle differs from the fiber cycle. Finally we compute the conditions that a double point imposes to pluricanonical systems.

We classify the singularities of a surface ruled by conics: they are rational double points of type $A_n$ or $D_n$. This is proved by showing that they arise from a precise series of blow-ups of a suitable surface geometrically ruled by conics. We determine also the family of such surfaces which are birational models of a given surface ruled by conics and obtained in a "minimal way" from it.

We classify, up to some lattice-theoretic equivalence, all possible configurations of rational double points that can appear on a surface whose minimal resolution is a complex Enriques surface.

Let $k$ be an algebraically closed field, $S$ a variety over $k$ and m a nonnegative integer. There is a space $S_m$ over $S$ , called the jet scheme of $X$ of order $m$, parameterizing $m$-th jets on $S$. The fiber over the singular locus of $S$ is called the singular fiber. In this paper, we consider the singular fibers of the jet schemes of 2-dimensional rational double points over a field $k$ of characteristic $2$ whose resolution graph is of type $D_4$. There are two typ...

This is an expository paper which presents the holomorphic classification of rational complex surfaces from a simple and intuitive point of view, which is not found in the literature. Our approach is to compare this classification with the topological classification of real surfaces.