Log canonical foliation singularities on surfaces

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. As an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert–Riemenschneider–type vanishing theorem for foliated surfaces with special log canonical foliation singularities.     

Limits of Puiseux Expansions and Parametric Deformations of Plane Curve Singularities

We consider a family of parametrizations of unibranched plane curve singularities. Each member of the family can be expressed via the Puiseux expansion. Studying the limits of suitably renormalized coefficients of the Puiseux expansions, we obtain in some cases obstructions for a possible adjacency of unibranched plane curve singularities.

Furthermore, we show that the coefficients of the Puiseux expansion are not generic (in a precisely defined sense) as functions of the coefficients of parametrization of the singularity.     

Algebraic surfaces with log canonical singularities and the fundamental groups of their smooth parts

Let be a log surface with at worst log canonical singularities and reduced boundary such that is nef and big. We shall prove that either has finite fundamental group or is affine-ruled. Moreover, and the structure of are determined in some sense when .

     

Regular canonical covers

We construct three sequences of regular surfaces of general type with unbounded numerical invariants whose canonical map is 2–to–1 onto a canonically embedded surface. Only sporadic examples of surfaces with these properties were previously known.     

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)     

Isolated singularities of polyharmonic inequalities

We study nonnegative classical solutions u of the polyharmonic inequality

     

Bi-Lipschitz geometry of complex surface singularities

We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface with particular emphasis on when it is metrically conical.      

Extended canonical algebras and Fuchsian singularities

The authors introduce a new class of finite dimensional algebras called extended canonical, and determine the shape of their derived categories. Extended canonical algebras arise from a canonical algebra ?? by onepoint extension or coextension by an indecomposable projective module. Our main results concern the case of negative Euler characteristic of the corresponding weighted projective line ${\mathbb{X}}$ ; more specifically we establish, for a base field of arbitrary characteristic, a link to the Fuchsian singularity R of ${\mathbb{X}}$ which for the base field of complex numbers is isomorphic to an algebra of automorphic forms. By means of a recent result of Orlov we show that the triangulated category of the graded singularities of R (in the sense of Buchweitz and Orlov) admits a tilting object whose endomorphism ring is the corresponding extended canonical algebra. Of particular interest are those cases where the attached Coxeter transformation has spectral radius one. A K-theoretic analysis then shows that this happens exactly for 38 cases including Arnold??s 14 exceptional unimodal singularities. The paper is related to recent independent work by Kajiura, Saito and Takahashi.     

Existence of curves with prescribed topological singularities

Throughout this paper we study the existence of irreducible curves on smooth projective surfaces with singular points of prescribed topological types . There are necessary conditions for the existence of the type for some fixed divisor on and suitable coefficients , and , and the main sufficient condition that we find is of the same type, saying it is asymptotically proper. Ten years ago general results of this quality were not known even for the case . An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up of the form , deduced from the Kawamata-Vieweg Vanishing Theorem. Its proof covers the first part of the paper, while the middle part is devoted to the existence theorems. In the last part we investigate our conditions on ruled surfaces, products of elliptic curves, surfaces in , and K3-surfaces.

     

The geometric genus of splice-quotient singularities

We prove a formula for the geometric genus of splice-quotient singularities (in the sense of Neumann and Wahl). This formula enables us to compute the invariant from the resolution graph; in fact, it reduces the computation to that for splice-quotient singularities with smaller resolution graphs. We also discuss the dimension of the first cohomology groups of certain invertible sheaves on a resolution of a splice-quotient singularity.

     

The canonical sheaf of Du Bois singularities

We prove that a Cohen-Macaulay normal variety X has Du Bois singularities if and only if π_∗ωX^′(G)?ωX for a log resolution π:X^′→X, where G is the reduced exceptional divisor of π. Many basic theorems about Du Bois singularities become transparent using this characterization (including the fact that Cohen-Macaulay log canonical singularities are Du Bois). We also give a straightforward and self-contained proof that (generalizations of) semi-log-canonical singularities are Du Bois, in the Cohen-Macaulay case. It also follows that the Kodaira vanishing theorem holds for semi-log-canonical varieties and that Cohen-Macaulay semi-log-canonical singularities are cohomologically insignificant in the sense of Dolgachev.     

Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity

We are concerned on the possibility of finite time singularity in a partially viscous magnetohydrodynamic equations in Rn, n=2,3, namely the MHD with positive viscosity and zero resistivity. In the special case of zero magnetic field the system reduces to the Navier-Stokes equations in Rn. In this paper we exclude the scenario of finite time singularity in the form of self-similarity, under suitable integrability conditions on the velocity and the magnetic field. We also prove the nonexistence of asymptotically self-similar singularity. This provides us information on the behavior of solutions near possible singularity of general type as described in Corollary 1.1.     

The canonical decomposition of bivariate distributions

The ordinary notion of a bivariate distribution has a natural generalisation. For this generalisation it is shown that a bivariate distribution can be characterised by a Hilbert space and a family _p, 0 ≤ p ≤ 1, of subspaces of . specifies the marginal distributions whilst _p is a summary of the dependence structure. This characterisation extends existing ideas on canonical correlation.     

Del Pezzo foliations with log canonical singularities

In this paper, we classify del Pezzo foliations of rank at least 3 on projective manifolds and with log canonical singularities in the sense of McQuillan.     

The canonical model of a singular curve

We give refined statements and modern proofs of Rosenlicht’s results about the canonical model C′ of an arbitrary complete integral curve C. Notably, we prove that C and C′ are birationally equivalent if and only if C is nonhyperelliptic, and that, if C is nonhyperelliptic, then C′ is equal to the blowup of C with respect to the canonical sheaf ω. We also prove some new results: we determine just when C′ is rational normal, arithmetically normal, projectively normal, and linearly normal.