Del Pezzo surfaces with log terminal singularities

We prove that there are no del Pezzo surfaces with five log terminal singularities and the Picard number 1. In the course of the proof, we make use of fibrations with general fiber ?¹.     

Resolution of singularities of threefolds in positive characteristic II

Together with [Vincent Cossart, Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin–Schreier and purely inseparable coverings, J. Algebra 320 (3) (2008) 1051–1082], this article gives a complete proof of desingularization of quasiprojective varieties of dimensional 3 on fields which are differentially finite over perfect fields.     

Simple singularities of parametrized plane curves in positive characteristic

Let K be an algebraically closed field of characteristic p>0. The aim of the article is to give a classification of simple parametrized plane curve singularities over K. The idea is to give explicitly a class of families of singularities which are not simple such that almost all singularities deform to one of those and show that remaining singularities are simple.     

On log canonical divisors that are log quasi-numerically positive

Let (X Δ) be a four-dimensional log variety that is projective over the field of complex numbers. Assume that (X, Δ) is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisorK _X+Δ is log quasi-numerically positive on (X, Δ) then it is semi-ample.     

On normal two-dimensional double point singularities

Letq be a plane curve singularity and letp be the corresponding normal two-dimensional double point singularity. Let Γ and Γ₁ be the topological types of the minimal and of the canonical resolutions ofp respectively. An algorithm is given for finding the equisingular type ofq in terms of Γ₁. An algorithm is also given for finding all Γ₁ corresponding to a given Γ. There is at most one such Γ₁ in case Γ has no 1-cycles. This research was partially supported by the National Science Foundation. The author also is an Alfred P. Sloan Research Fellow.     

Universal abelian covers of superisolated singularities

We give explicit examples of Gorenstein surface singularities with integral homology sphere link, which are not complete intersections. Their existence was shown by Luengo–Velasco, Melle–Hernández and Némethi, thereby providing counterexamples to the universal abelian covering conjecture of Neumann and Wahl (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the 1-pointed curves arising as étale covers of the affine line in positive characteristic

The aim of this paper is to study smooth projective curves in positive characteristic such that the complement of a single point arise as an étale cover of the affine line. A first elementary approach leads to the study of a particular class of differential forms on these curves. A finer analysis, based on Serre duality, allows to determine the minimal degree of such a cover. This approach is particularly adapted to explicit computations.     

On nonrational divisors over non-Gorenstein terminal singularities

Let (X, o) be a germ of a 3-dimensional terminal singularity of index m ≥ 2. If (X, o) has type cAx/4, cD/3-3, cD/2-2, or cE/2, then we assume that the standard equation of X in ℂ⁴/ℤ _m is nondegenerate with respect to its Newton diagram. Let π: Y → X be a resolution. We show that there are at most 2 nonrational divisors E _i , i = 1, 2, on Y such that π(E _i ) = o and the discrepancy a(E _i , X) is at most 1. When such divisors exist, we describe them as exceptional divisors of certain blowups of (X, o) and study their birational type. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 169–184, 2005.     

Rank-1 codimension one singularities of positive quadratic differential forms

We complete the study of first-order structural stability at singular points of positive quadratic differencial forms on two manifolds. For this, we consider the generic 1-parameter bifurcation of a D₂₃-singular point. This situation consists in having, before the bifurcation, two locally stable singular points (one of type D₂ and the other of type D₃) which collapse at the D₂₃-singular point when the bifurcation parameter is reached, and afterwards disappear. In local (x,y)-coordinates, such a point appears at the origin of a planar differential equation of the form with (b²-ac)(x,y)?0, such that

(1)

the first jet of the map (a,b,c) at the origin is T₁(a,b,c)(0,0)=(y,0,-y) and

(2)

     

On two-dimensional quasihomogeneous isolated singularities,II

Supported in part by DFG.     

On the resolution of 3-dimensional terminal singularities

We show that there is at most one nonrational exceptional divisor with discrepancy 1 over a three-dimensional terminal point of type cD. If such a divisor exists, then it is birationally isomorphic to the surface ¹ × C, where C is a hyperelliptic (for g(C) > 1) curve.     

On the resolution of 3-dimensional terminal singularities

We show that there is at most one nonrational exceptional divisor with discrepancy 1 over a three-dimensional terminal point of type cD. If such a divisor exists, then it is birationally isomorphic to the surface ¹ × C, where C is a hyperelliptic (for g(C) > 1) curve.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 127–140.Original Russian Text Copyright © 2005 by D. A. Stepanov.This revised version was published online in April 2005 with a corrected issue number.