Related papers: A note on a question of Dimca and Greuel

The Milnor formula $\mu=2\delta-r+1$ relates the Milnor number $\mu$, the double point number $\delta$ and the number $r$ of branches of a plane curve singularity. It holds over the fields of characteristic zero. Melle and Wall based on a…

We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in C^4, we…

There is an elegant relation found by Fabricius-Bjerre [Math. Scand 40 (1977) 20--24] among the double tangent lines, crossings, inflections points, and cusps of a singular curve in the plane. We give a new generalization to singular curves…

In this paper we extend the concept of Milnor fiber and Milnor number of a curve singularity allowing the ambient space to be a quotient surface singularity. A generalization of the local {\delta}-invariant is defined and described in terms…

In this work, we refine a formula for the Tjurina number of a reducible algebroid plane curve defined over $\mathbb C$ obtained in the more general case of complete intersection curves in [1]. As a byproduct, we answer the affirmative to a…

We give a characterization of nearly free plane curves in terms of their global Tjurina numbers, similar to the characterization of free curves as curves with a maximal Tjurina number, due to A. A. du Plessis and C.T.C. Wall. It is also…

In this paper we give some criteria for a family of generically reduced plane curve singularities to be equinormalizable. The first criterion is based on the $\delta$-invariant of a (non-reduced) curve singularity which is introduced by…

We construct a class of noncommutative crepant resolutions of any Kleinian singularity in the form of noncommutative algebras over its crepant partial resolutions. We argue that such resolutions are Morita equivalent to the canonical…

We discuss Weber's formula which gives the quotient of two Thetanullwerte for a plane smooth quartic in terms of the bitangents. In particular, we show how it can easily be derived from the Riemann-Jacobi formula.

Reduced power series in two variables with coefficients in a field of characteristic zero satisfy a well-known formula that relates a codimension related to the normalization of a ring and the jacobian ideal. In the general case Deligne…

The purpose of this note is to suggest an analogue for genus 2 curves of part of Gross and Zagier's work on elliptic curves. Experimentally, for genus 2 curves with CM by a quartic CM field K, it appears that primes dividing the…

Greuel, Lossen and Shustin gave a general sufficient numerical condition for the T-smoothness (smoothness and expected dimension) of equisingular families of plane curves. This condition involves a new invariant \gamma for plane curve…

Let $f=0$ be a plane algebraic curve of degree $d>1$ with an isolated singular point at the origin of the complex plane. We show that the Milnor number $\mu_0(f)$ is less than or equal to $(d-1)^2-\left[\frac{d}{2}\right]$, unless $f=0$ is…

An old conjecture of Durfee 1978 bounds the ratio of two basic invariants of complex isolated complete intersection surface singularities: the Milnor number and the singularity (or geometric) genus. We give a counterexample for the case of…

The present paper addresses a semimodule counting conjecture of Moreno-Fr\'{i}as and Rosales for numerical semigroups. Applying Pfister and Steenbrink's Theory for punctual Hilbert schemes of curve singularities, we show that this…

In this note we study a problem of A'Campo about the minimal non-zero difference between the Milnor numbers of a germ of plane curve and one of its deformation.

Let f_0 be a plane curve singularity. We study the Minor numbers of singularities in deformations of f_0. We completely describe the set of these Milnor numbers for homogeneous singularities f_0 in the case of non-degenerate deformations…

First we give a sharp upper bound for the cardinal $m$ of a minimal set of generators for the module of Jacobian syzygies of a complex projective reduced plane curve $C$. Next we discuss the sharpness of an upper bound, given by A. du…

We study deformations of plane curve singularities from an analytic point of view and obtain some new concrete results. We show some rather unexpected properties of Puiseux coefficients treated as functions on a suitably defined parameter…

We generalize two classical formulas for complete intersection curves by introducing the the complete intersection discrepancy of a curve as a correction term. The first is a well-known multiplicity formula in singularity theory, due to…