Related papers: A note on a question of Dimca and Greuel
The purpose of this paper is to study low degree points on plane curves. We prove results analogous to those of Debarre and Klassen for singular plane curves with a finite number $\delta$ of ordinary nodes/cusps, where $\delta$ is bounded…
We present an intersection-theoretical approach to the invariants of plane curve singularities $\mu$, $\delta$, $r$ related by the Milnor formula $2\delta=\mu+r-1$. Using Newton transformations we give formulae for $\mu$, $\delta$, $r$…
This paper develops a theory of isolated hypersurface singularities in mixed characteristic $(0,p)$, focusing on quotient rings over a Discrete Valuation Ring (DVR). We introduce and study analogues of the classical Tjurina and Milnor…
We compute Bernstein-Sato polynomials of some pairs of topologically equivalent plane curve singularities. Some pairs have the same Tjurina number but distinct Bernstein-Sato polynomials, which implies that they are not analytically…
We study the partial resolutions of singularities related to Hilbert schemes of points on an affine space. Consider a quotient of a vector space $V$ by an action of a finite group $G$ of linear transforms. Under some additional assumptions,…
The quotient of a finite-dimensional vector space by the action of a finite subgroup of automorphisms is usually a singular variety. Under appropriate assumptions, the McKay correspondence relates the geometry of nice resolutions of…
The jump of the Milnor number of an isolated singularity $f_0$ is the minimal non-zero difference between the Milnor numbers of $f_0$ and one of its deformations $f_s$. We determinate the jump of quasihomogeneous singularities in the class…
We show that a general lower bound for the global Tjurina number of a reduced complex projective plane curve, given by A. A. du Plessis and C.T.C. Wall, can be improved when the curve is a line arrangement.
In this paper we characterize the value set $\Delta$ of the $R$-modules of the form $R+zR$ for the local ring $R$ associated to a germ $\xi$ of an irreducible plane curve singularity with one Puiseux pair. In the particular case of the…
This is a short addendum to a note of Beauville on the subject of the title. We prove an inequality that takes into account the constant part of the Jacobian.
Partial duality is a duality of ribbon graphs relative to a subset of their edges generalizing the classical Euler-Poincare duality. This operation often changes the genus. Recently J.L.Gross, T.Mansour, and T.W.Tucker formulated a…
Generalizing recent work of Brundan and Kleshchev, we introduce grading on Dipper-James' $q$-Schur algebra, and prove a graded analogue of the Leclerc and Thibon's conjecture on the decomposition numbers of the $q$-Schur algebra when…
We give a survey on some aspects of deformations of isolated singularities. In addition to the presentation of the general theory, we report on the question of the smoothability of a singularity and on relations between different…
Given an algebroid plane curve $f=0$ over an algebraically closed field of characteristic $p\geq 0$ we consider the Milnor number $\mu(f)$, the delta invariant $\delta(f)$ and the number $r(f)$ of its irreducible components. Put $\bar…
The present paper describes a relation between the quotient of the fundamental group of a smooth quasi-projective variety by its second commutator and the existence of maps to orbifold curves. It extends previously studied cases when the…
Differently from the Milnor number, no formula relating the Tjurina number of a reducible algebroid curve to invariants of its branches was known. The aim of this work is to provide such a formula for a complete intersection algebroid curve…
We investigate combinatorial bounds for the total Tjurina numbers of plane curve arrangements. Focusing on arrangements of lines and conics in $\mathbb{P}^2$ that admit only ordinary quasi-homogeneous singularities, we derive new structural…
Let $C$ be a projective plane curve of degree $d$ whose singularities are all isolated. Suppose $C$ is not concurrent lines. P{\l}oski proved that the Milnor number of an isolated singlar point of $C$ is less than or equal to…
The aim of this note is to announce some results on the GIT problem for the Hilbert and Chow scheme of curves of degree d and genus g in P^{d-g}, whose full details will appear in a subsequent paper. In particular, we extend the previous…
In this article we study quotients of deformations of simple singularities, and attempt to characterize them in terms of subsystems of simple root systems. The quotient of a semiuniversal deformation of a simple singularity of inhomogeneous…