On the cotangent cohomology of rational surface singularities with almost reduced fundamental cycle

We prove dimension formulas for the cotangent spaces T ¹ and T ² for a class of rational surface singularities by calculating a correction term in the general dimension formulas. We get that it is zero if the dual graph of the rational surface singularity X does not contain a particular type of configurations, and this generalizes a result of Theo de Jong stating that the correction term c (X ) is zero for rational determinantal surface singularities. In particular our result implies that c (X ) is zero for Riemenschneiders quasi‐determinantal rational surface singularities, and this also generalizes results for quotient singularities. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the number of generators of the module of derivations and multiplicity of certain rings

The present paper studies the module of derivations of certain rings and the multiplicity of certain rational surface singularities. We also consider some relation between these two for 2-dimensional quotient singularities. We conjecture a relation between the multiplicity of a rational surface singularity and the order of the divisor class group of the singularity and verify the same for several cases.     

Rational blowdowns and smoothings of surface singularities

In this paper, we give a necessary combinatorial condition fora negative-definite plumbing tree to be suitable for rationalblowdown, or to be the graph of a complex surface singularitywhich admits a rational homology disk smoothing. New examplesof surface singularities with rational homology disk smoothingsare also presented; these include singularities with resolutiongraph having valency 4 nodes. Received July 25, 2007.     

On Auslander modules of normal surface singularities

Abstract We study thefundamental sequences of normal surface singularities. Our main result asserts that for rational singularities (with a technical side-condition) and for minimally elliptic singularities the middle termA, theAuslander module, is isomorphic to the module of Zariski differentials if and only if the singularity is quasihomogeneous.     

Determinantal Rational Surface Singularities

In this paper we give explicit equations for determinantal rational surface singularities and prove dimension formulas for the T ¹ and T ² for those singularities.     

P~3中3次曲面与平面4次曲线间的关系

本文通过二重覆盖建立了P3中仅有二重的孤立奇点的3次曲面与平面带孤立奇点的4次曲线的对应关系，这种关系使我们能用平面4次曲线的奇点分类及几何性质研究P3中3次曲面的奇点类型及几何性质     

The Hilbert scheme of points for supersingular abelian surfaces

We study the geometry of Hilbert schemes of points on abelian surfaces and Beauville’s generalized Kummer varieties in positive characteristics. The main result is that, in characteristic two, the addition map from the Hilbert scheme of two points to the abelian surface is a quasifibration such that all fibers are nonsmooth. In particular, the corresponding generalized Kummer surface is nonsmooth, and minimally elliptic singularities occur in the supersingular case. We unravel the structure of the singularities in dependence of p-rank and a-number of the abelian surface. To do so, we establish a McKay Correspondence for Artin’s wild involutions on surfaces. Along the line, we find examples of canonical singularities that are not rational singularities.     

On weakly rational singularities in complex analytic geometry

Summary We study particular singularities of complex analytic spaces that we call weakly rational and that contain rational singularities. In fact, a weakly rational singularity is rational if and only if it is Cohen-Macauley. Invariance under morphisms and deformations of weakly rational singularities is also studied.Partially supported by C.N.R.     

Jet schemes and minimal toric embedded resolutions of rational double point singularities

Using the structure of the jet schemes of rational double point singularities, we construct “minimal embedded toric resolutions” of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every “minimal embedded toric resolution”. We prove that this correspondence is bijective except for the E₈ singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities.     

Cotangent cohomology of rational surface singularities

In this paper we show that the number of generators of the cotangent cohomology groups T _Y ⁿ , n≥2, is the same for all rational surface singularities Y of fixed multiplicity. For a large class of rational surface singularities, including quotient singularities, this number is also the dimension. For them we obtain an explicit formula for the Poincaré series P _Y (t)=∑dim T ⁿ _Y ·t ⁿ . In the special case of the cone over the rational normal curve we give the multigraded Poincaré series. Oblatum: 18-XI-1998 & 25-III-1999 / Published online: 6 July 1999     

Singularities of rational functions and minimal factorizations: The noncommutative and the commutative setting

We show that the singularities of a matrix-valued noncommutative rational function which is regular at zero coincide with the singularities of the resolvent in its minimal state space realization. The proof uses a new notion of noncommutative backward shifts. As an application, we establish the commutative counterpart of the singularities theorem: the singularities of a matrix-valued commutative rational function which is regular at zero coincide with the singularities of the resolvent in any of its Fornasini-Marchesini realizations with the minimal possible state space dimension. The singularities results imply the absence of zero-pole cancellations in a minimal factorization, both in the noncommutative and in the commutative setting.     

Growth properties of Nevanlinna matrices for rational moment problems

We consider rational moment problems on the real line with their associated orthogonal rational functions. There exists a Nevanlinna-type parameterization relating to the problem, with associated Nevanlinna matrices of functions having singularities in the closure of the set of poles of the rational functions belonging to the problem. We prove results related to the growth at the singularities of the functions in a Nevanlinna matrix, and in particular provide bounds on the growth analogous to the corresponding result in the classical polynomial case, when the number of singularities is finite.     

Universal Abelian Covers of Rational Surface Singularities

The paper gives fundamental results on the universal abeliancovers of rational surface singularities. Let (X,o) be a normalcomplex surface singularity germ with a rational homology spherelink. Then (X,o) has the universal abelian cover (Y,o) - (X,o).It is shown that if (X,o) is rational or minimally elliptic,and if it has a star-shaped resolution graph, then (Y,o) isa complete intersection (a partial answer to the conjectureof Neumann and Wahl). A way is given to compute the multiplicityand the embedding dimension of (Y,o) from the resolution graphof (X,o) in the case when (X,o) is rational.     

KSBA surfaces with elliptic quotient singularities, π1 = 1, pg = 0, and K2 = 1, 2

Among log canonical surface singularities, those which have a rational homology disk smoothing are the cyclic quotient singularities \(\frac{1}{{{n^2}}}\left( {1,na - 1} \right)\) with gcd(a, n) = 1, and three distinguished elliptic quotient singularities. We show the existence of smoothable KSBA normal surfaces with π₁ = 1, p_g = 0, and K² = 1, 2 for each of these three singularities. We also give a list of new (and old) normal surface singularities in smoothable KSBA surfaces for invariants π₁ = 1, p_g = 0, and K² = 1, 2, 3, 4.     

The l-adic Dualizing Complex on an Excellent Surface with Rational Singularities

In this article,we show that if X is an excellent surface with rational singularities,the constant sheaf Qe is a dualizing complex.In coefficient Ze,we also prove that the obstruction for Ze to become a dualizing complex,lies on the divisor class groups at singular points.As applications,we study the perverse sheaves and the weights of-adic cohomology groups on such surfaces.     

On normal surface singularities and a problem of enriques

We construct families of normal surface singularities with the following property: given any fiat projective connected family V →B of smooth, irreducible, minimal algebraic surfaces, the general singularity in one of our families cannot occur, analytically, on any algebraic surfaces which is Irrationally equivalent to a surface in V→B. In particular this holds for V→B consisting of a single rational surface, thus answering negatively to a long standing problem posed by F. Enriques. In order to prove the above mentioned results, wo develop a general, though elementary, method, based on the consideration of suitable correspondences, for comparing a given family of minimal surfaces with a family of surface singularities. Specifically the method in question gives us the possibility of comparing the parameters on which the two families depend, thus leading to the aforementioned results.     

On Mordell–Weil groups of isotrivial abelian varieties over function fields

We show that the Mordell–Weil rank of an isotrivial abelian variety with cyclic holonomy depends only on the fundamental group of the complement to the discriminant, provided the discriminant has singularities in CM class introduced here. This class of singularities includes all unibranched plane curves singularities. As a corollary, we describe a family of simple Jacobians over the field of rational functions in two variables for which the Mordell–Weil rank is arbitrarily large.     

Graph theoretic techniques in algebraic geometry II: construction of singular complex surfaces of the rational cohomology type of CP2

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 cohomologyCP ²'s and for rational and integral homologyCP ²'s are given in terms of the typesA _k,D _k, orE _k of singularities allowed by the construction. Supported in part by National Science Foundation grant no. MCS 77-03540.     

A note on irregularity of two dimensional simple hyperbolic singularities

Deformation theory is an important aspect of the study about isolated singularities. The invariant called irregularity is very useful in the study on the deformation of isolated singularities. In this note we give an optimal upper bound for a class of surface singularities by the computation of cohomology. Moreover a sufficient condition is given for the positivity of irregularity of some simple hyperbolic surface singularities. Therefore a class of surface singularities with non-rigid deformation is constructed.     

Differential Forms on Quasihomogeneous Noncomplete Intersections

In this article, we discuss a few simple methods for computing the Poincaré series of modules of differential forms given on quasihomogeneous noncomplete intersections of various types. Among them are curves associated with a semigroup, bouquets of such curves, affine cones over rational or elliptic curves, and normal determinantal and toric varieties, including some types of quotient singularities, as well as cones over the Veronese embedding of projective spaces or over the Segre embedding of products of projective spaces, rigid singularities, fans, etc. In many cases, correct formulas can be derived without resorting to analysis of complicated resolvents or using computer systems of algebraic calculations. The obtained results allow us to compute the basic invariants of singularities in an explicit form by means of elementary operations on rational functions.