On Frobenius liftability of surface singularities

We classify all generalized del Pezzo surfaces (that is, minimal desingularizations of singular del Pezzo surfaces containing only rational double points) whose universal torsors are open subsets of hypersurfaces in affine space. Equivalently, their Cox rings are polynomial rings with exactly one relation. For all 30 types with this property, we describe the Cox rings in detail. These explicit descriptions can be applied to study Manin's conjecture on the asymptotic behavior of the number of rational points of bounded height for singular del Pezzo surfaces, using the universal torsor method.

The violation of the Lipman–Zariski conjecture in positive characteristic

A cusp singularity is a surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. Looijenga proved in 1981 that if a cusp singularity is smoothable, the minimal resolution of the dual cusp is the anticanonical divisor of some smooth rational surface. In 1983, the second author and Miranda gave a criterion for smoothability of a cusp singularity, in terms of the existence of a K-trivial semistable model for the central fiber of such a smoothing. We study these "Type III degenerations" of rational surfaces with an anticanonical divisor--their deformations, birational geometry, and monodromy. Looijenga's original paper also gave a description of the rational double point configurations to which a cusp singularity deforms, but only in the case where the resolution of the dual cusp has cycle length 5 or less. We generalize this classification to an arbitrary cusp singularity, giving an explicit construction of a semistable simultaneous resolution of such an adjacency. The main tools of the proof are (1) formulas for the monodromy of a Type III degeneration, (2) a construction via surgeries on integral-affine surfaces of a degeneration with prescribed monodromy, (3) surjectivity of the period map for Type III central fibers, and (4) a theorem of Shepherd-Barron producing the simultaneous contraction to the adjacency of the cusp singularity.

Article Zero cycles on a singular surface. I. Appendix: Chow groups of rational surfaces with rational double points. was published on January 1, 1985 in the journal Journal für die reine und angewandte Mathematik (volume 1985, issue 359).

INTRODUCTION Due to the work in [Brieskorn], [Tjurina], [Artin], [Wahl 2] and [Lipman] one understands very well the notion of simultaneous resolution for rational surface singularities. Given a rational surface singularity X, there exists a smooth parameter space Res representing the functor of deformations of the (minimal) resolution. The family Y ---+ Res contracts to X ---+Res. The fibers yt---+ X 1 are the minimal resolutions of Xt. There is a finite and Galois map Res ---+ Defx (the versa! base space of X). The image is the Artin component, which represents the functor of deformations of X with simultaneous resolution after finite base change, and the covering group W is a reflection group which is also the monodromy group of the component. In the case of rational double points (RDPs) W is the Weyl group of the corresponding root system (Ak, Dk, E6, E1 orEs). In general much information about the deformation, e.g. the discriminant and adjacencies, may be read off the geometry of this covering. The reflections in W are related to certain divisors, called roots, on Y and their liftings. The contraction map is induced by contracting RDP-configurations on Y. In connection with his work on deformations of cyclic quotient singularities ( CQS), the second author observed a natural Galois covering of each component in the reduced versal base space of a CQS. In the case of the Artin component the observed covering group was the same as for Res---+ Def. (See [Christophersen 1] and [Christophersen 2].) The first author conjectured that this was the monodromy cover and we asked ourselves if there was a deformation theoretic explanation. The purpose of this paper is to answer this question. We will show that something similar to the Res ---+ Def picture actually happens for every non-embedded component of the versa! base space of a quotient singularity. (For us a quotient singularity is the singularity of C2 /G where G C GL(2, C) is a finite subgroup which we can assume to be without pseudo-reflections.) For quotient singularities, the construction of the Artin component is a special case of a procedure involving deformations of certain modifications which we call M-resolutions. The application of threefold theory to deformations of rational singularities as found in [Kollfu--Shepherd-Barron] was important both for discovering M-resolutions and for the proofs of their properties. In fact, our results are anticipated in Theorem 3.5(a) of that paper. Still the definitions and statement of the main result may be made without reference to that work. The relevant results from [Kollfu--Shepherd-Barron] are postponed to §1. Consider a smoothing of a normal surface singularity X with smooth generic fiber F. The Milnor number of this smoothing is p. = rk H2 (F) and depends upon the component of the versa! base space of X on which the smoothing appears.

Let [Formula: see text] be an imaginary quadratic number field. Consider the singular cubic surface [Formula: see text] defined by [Formula: see text] in [Formula: see text] We denote by [Formula: see text] the complement of the three lines [Formula: see text] [Formula: see text] in [Formula: see text] and denote by [Formula: see text] the set of [Formula: see text]-rational points on [Formula: see text]. For any [Formula: see text] let [Formula: see text] denote the counting function of rational points in [Formula: see text] of Weil height [Formula: see text] In this paper, we shall prove the following asymptotic formula: [Formula: see text] where [Formula: see text] is a polynomial in [Formula: see text] of degree [Formula: see text] Our method is an application of the multi-variable version of the well-known Perron’s formula.

Methods of graph theory are used to obtain rational projective surfaces with only rational double points as singularities and with rational cohomology rings isomorphic to that of the complex projective plane. Uniqueness results for such cohomologyCP2's and for rational and integral homologyCP2's are given in terms of the typesAk,Dk, orEk of singularities allowed by the construction.

Unirationality of RDP Del Pezzo surfaces of degree 2

Manin's conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the Poisson formula. An alternative approach to Manin's conjecture via universal torsors was used so far mainly over the field Q of rational numbers. In this note, we give a proof of Manin's conjecture over the Gaussian rational numbers Q(i) and over other imaginary quadratic number fields with class number 1 for the singular toric cubic surface defined by t^3=xyz.

Complex surface singularities from the combinatorial point of view

It was conjectured by Mordell [6] that the Hasse principle holds for cubic surfaces in 3-dimensional projective space other than cones†: i.e., that such a surface defined over the rational field 0 has a rational point whenever it has points defined over every p -adic field Q p . This conjecture was verified for singular cubic surfaces by Skolem [11” and for surfaces with by Selmer [9]: but it was disproved for cubic surfaces in general by Swinnerton-Dyer [12] (see also Mordell [7]). It therefore becomes of interest to specify fairly wide classes of cubic surfaces for which the Hasse principle does hold. It was shown independently by F. Châtєlet and by Swinnerton-Dyer (both, apparently, unpublished) that this is the case when it contains a set of either 3 or 6 mutually skew lines which are rational as a whole (and trivially true when there is a rational pair of lines, since then there are always rational points). Selmer [9] conjectures on the basis of numerical evidence that the Hasse principle is also true for all surfaces of the type (1). It is the object of this note to disprove this by showing that the Hasse principle fails for

By the famous ADE classification rational double points are simple. Rational triple points are also simple. We conjecture that the simple normal surface singularities are exactly those rational singularities, whose resolution graph can be obtained from the graph of a rational double point or rational triple point by making (some) vertex weights more negative. For rational singularities we show one direction in general, and the other direction (simpleness) within the special classes of rational quadruple points and of sandwiched singularities.

We determine all configurations of rational double points that occur on RDP del Pezzo surfaces of arbitrary degree and Picard rank over an algebraically closed field $k$ of arbitrary characteristic ${\rm char}(k)=p \geq 0$, generalizing classical work of Du Val to positive characteristic. Moreover, we give simplified equations for all RDP del Pezzo surfaces of degree $1$ containing non-taut rational double points.Comment: 27 pages, final version

Let (SpecR,m) be a rational double point defined over an algebraically closed field k of characteristic p≥0. We evaluate further the dimensions of the local cohomology groups, which were treated by Wahl in 1975 as vanishing theorem C (resp., D) under the assumption that p is a very good prime (resp., good prime) with respect to (SpecR,m). We use Artin’s classification of rational double points and completely determine the dimensions dimkHE1(SX) and dimkHE1(SX⊗OX(E)), supplementing Wahl’s theorems. In the proof, we concretely construct derivations that do not lift to the minimal resolution X→SpecR and an equisingular family that injects into a versal deformation of the rational double point (SpecR,m).

In this paper we introduce the term to refer to those graphs which characterize resolutions of certain isolated singular points of complex surfaces. Using techniques for graphical evaluation of determinants, we reduce questions about perfect graphs to problems involving partial fraction representations of positive integers; the solutions to those Diophantine problems thus have interesting geometric interpretations. 1. Introduction and statement of results. In (5) Brieskorn gave the first examples of isolated singularities of complex n- varieties, n> 3, that are topologically non-singular (locally homeomorphic to the 2/2-ball) but analytically singular. Earlier Mumford (16) had shown that this is impossible in dimension 2. In this paper we pursue the natural analogue of the Brieskorn singularities for complex surfaces, namely those singular points xeX which are homologically non-singular in the sense of being locally homeomorphic to the cone on a homology 3-sphere. (The rational double point E8 is the most familiar example.) This condition is equivalent to the requirement that the local fundamental group of x in X be a perfect group (cf., for example, (16), (17), and (19), where the topic of classifying isolated two-dimensional singularities by the group-theoretic properties of the local fundamental group is introduced and developed). Let x be an isolated singularity of a normal complex surface X, and let p: X^>X be the minimal resolution of singularities . We will assume that the exceptional curve C=p~1(x)=\Jni = 1Ci is contractible, that each component Cf is non-singular rational, and that the components meet transversally with no triple intersections. In this case the topology of the singularity is completely determined by the weighted dual intersection graph Gp of the exceptional curve. In particular, the local fundamental group πx(x) can be computed directly from Gp in terms of generators and relations, by the technique of Mumford (16). Using this method it can be shown that πx(x) is perfect exactly when the intersection matrix { — Ci'C^ has determinant 1. Indeed, the following are necessary and sufficient conditions for a weighted graph G to be the dual graph of the minimal resolution of a normal complex surface singularity whose minimal resolution is normal (good) and whose local fundamental group is perfect: (a) G is a tree (a connected graph with no circuits). (b) Each weight wt is an integer >2. (c) The associated intersection matrix is positive definite with determinant 1. (Section 1 of (4) gives an elementary expository review of the geometry of complex surface