Tjurina Number Research Articles

On the Jacobian scheme of a plane curve

We study the Jacobian scheme of a plane algebraic curve at an ordinary singularity, characterizing it through a geometric property. We give an algorithm that gives the analytic type of a double point using the algebraic version of the Mather-Yau Theorem. We give a characterization of plane nodal curves with no infinitely near points through their Tjurina number.

Tjurina number of a local complete intersection curve

In contrast to the Milnor number, there was no known formula relating the Tjurina number of a reducible curve to the Tjurina numbers of its components. In this work we exhibit such a formula relating Tjurina number of a complete intersection algebroid (or analytic) curve over an algebraically closed field of characteristic zero (or over C ) to the Tjurina numbers of its components, involving the intersection indices among the components and numerical analytic invariants extracted from modules of Kähler differentials on unions of branches of the curve.

Bruce–Roberts numbers and quasihomogeneous functions on analytic varieties

Given a germ of an analytic variety X and a germ of a holomorphic function f with a stratified isolated singularity with respect to the logarithmic stratification of X, we show that under certain conditions on the singularity type of the pair (f, X), the following relative analog of the well-known K. Saito’s theorem holds true: equality of the relative Milnor and Tjurina numbers of f with respect to X (also known as Bruce–Roberts numbers) is equivalent to the relative quasihomogeneity of the pair (f, X), i.e. to the existence of a coordinate system such that both f and X are quasihomogeneous with respect to the same positive rational weights.

On free and plus-one generated curves arising from free curves by addition–deletion of a line

In a recent paper, after introducing the notion of plus-one generated hyperplane arrangements, Takuro Abe has shown that if we add (respectively, delete) a line to (respectively, from) a free line arrangement, then the resulting line arrangement is either free or plus-one generated. In this paper, we prove that the same properties hold when we replace the line arrangement by a free curve and add (respectively, delete) a line. The proof uses a new version of a key result due originally to H. Schenck, H. Terao and M. Yoshinaga, in which no quasi-homogeneity assumption is needed. Two conjectures about the Tjurina number of a union of two plane curve singularities are also stated. As a geometric application, we show that, under a mild numerical condition, the projective closure of a contractible, irreducible affine plane curve is either free or plus-one generated, using a deep result due to U. Walther.

Construction of Free Curves by Adding Lines to a Given Curve

In the present note we construct new families of free plane curves starting from a curve C and adding high order inflectional tangent lines of C, lines joining the singularities of the curve C, or lines in the tangent cone of some singularities of C. These lines L have in common that the intersection C∩L\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C \\cap L$$\\end{document} consists of a small number of points. We introduce the notion of a supersolvable plane curve and conjecture that such curves are always free, as in the known case of line arrangements. Some evidence for this conjecture is given as well, both in terms of a general result in the case of quasi homogeneous singularities and in terms of specific examples. We construct a new example of maximizing curve in degree 8 and the first and unique known example of maximizing curve in degree 9. In the final section, we use a stronger version of a result due to Schenck, Terao and Yoshinaga to construct families of free conic-line arrangements by adding lines to the conic-line arrangements of maximal Tjurina number recently classified by Beorchia and Miró-Roig in arXiv:2303.04665

The Bruce–Roberts Numbers of a Function on an ICIS

Abstract We relate a number of invariants of a function germ with isolated singularity over an isolated complete intersection singularity (ICIS), generalizing our previous results, [17, 18], as a consequence we present a new relatively elementary proof of Greuel’s theorem that for a weighted homogeneous ICIS the Milnor number is equal to Tjurina number. With this relation, we are able to prove that the relative logarithmic characteristic variety is Cohen–Macaulay at any point and the logarithmic characteristic variety is Cohen–Macaulay at any point not in $X\times \{0\}$.

On Briançon–Skoda theorem for foliations

We generalize Mattei’s result relative to the Briançon–Skoda theorem for foliations to the family of foliations of the second type. We use this generalization to establish relationships between the Milnor and Tjurina numbers of foliations of second type, inspired by the results obtained by Liu for complex hypersurfaces and we determine a lower bound for the global Tjurina number of an algebraic curve.

The Tjurina Number for Sebastiani–Thom Type Isolated Hypersurface Singularities

In this note, we provide a formula for the Tjurina number of a join of isolated hypersurface singularities in separated variables. From this, we are able to provide a characterization of isolated hypersurface singularities whose difference between the Milnor and Tjurina numbers is less or equal than two arising as the join of isolated hypersurface singularities in separated variables. Also, we are able to provide new upper bounds for the quotient of Milnor and Tjurina numbers of certain join of isolated hypersurface singularities. Finally, we deduce an upper bound for the quotient of Milnor and Tjurina numbers in terms of the singularity index of any isolated hypersurface singularity, not necessarily a join of singularities.

Maximizing Curves Viewed as Free Curves

Abstract The aim of this paper is to provide a direct link between maximizing curves that occur in the construction of smooth algebraic surfaces having the maximal possible Picard numbers and reduced free plane curves with simple singularities. We also investigate odd degree plane curves with simple singularities having maximal total Tjurina number.

On a Sharp Lower Bound for the Tjurina Number of Zero-Dimensional Complete Intersections

On a Sharp Lower Bound for the Tjurina Number of Zero-Dimensional Complete Intersections

K-th Milnor numbers and k-th Tjurina numbers of weighted homogeneous singularities

k-th Milnor numbers and k-th Tjurina numbers of weighted homogeneous singularities

On the quotient of Milnor and Tjurina numbers for two‐dimensional isolated hypersurface singularities

AbstractIn this paper we give a complete answer to a question posed by Dimca and Greuel about the quotient of the Milnor and Tjurina numbers of a plane curve singularity. We put this question into a general framework of the study of the difference of Milnor and Tjurina numbers for isolated complete intersection singularities showing its connection with other problems in singularity theory.

On the Milnor and Tjurina Numbers of Zero-Dimensional Singularities

On the Milnor and Tjurina Numbers of Zero-Dimensional Singularities

On the analytic invariants and semiroots of plane branches

The value semigroup of a k-semiroot Ck of a plane branch C allows us to recover part of the value semigroup Γ=〈v0,…,vg〉 of C, that is, it is related to topological invariants of C. In this paper we consider the set of values of differentials Λk of Ck, that is an analytical invariant, and we show how it determine part of the set of values of differentials Λ of C. As a consequence, in a fixed topological class, we relate the Tjurina number τ of C with the Tjurina number of Ck. In particular, we show that τ≤μ−3ng−24μg−1 where ng=gcd(v0,…,vg−1), μ and μg−1 denote the Milnor number of C and Cg−1 respectively. If ng=2, we have that τ=μ−μg−1 for any curve in the topological class determined by Γ that is a generalization of a result obtained by Luengo and Pfister.

Topological type of discriminants of some special families

We will describe the topological type of the discriminant curve of the morphism (ell , f), where ell is a smooth curve and f is an irreducible curve (branch) of multiplicity less than five or a branch such that the difference between its Milnor number and Tjurina number is less than 3. We prove that for a branch of these families, the topological type of the discriminant curve is determined by the semigroup, the Zariski invariant and at most two other analytical invariants of the branch.

The relative Bruce–Roberts number of a function on a hypersurface

AbstractWe consider the relative Bruce–Roberts number $\mu _{\textrm {BR}}^{-}(f,\,X)$ of a function on an isolated hypersurface singularity $(X,\,0)$. We show that $\mu _{\textrm {BR}}^{-}(f,\,X)$ is equal to the sum of the Milnor number of the fibre $\mu (f^{-1}(0)\cap X,\,0)$ plus the difference $\mu (X,\,0)-\tau (X,\,0)$ between the Milnor and the Tjurina numbers of $(X,\,0)$. As an application, we show that the usual Bruce–Roberts number $\mu _{\textrm {BR}}(f,\,X)$ is equal to $\mu (f)+\mu _{\textrm {BR}}^{-}(f,\,X)$. We also deduce that the relative logarithmic characteristic variety $LC(X)^{-}$, obtained from the logarithmic characteristic variety $LC(X)$ by eliminating the component corresponding to the complement of $X$ in the ambient space, is Cohen–Macaulay.

Jacobian syzygies, Fitting ideals, and plane curves with maximal global Tjurina numbers

Jacobian syzygies, Fitting ideals, and plane curves with maximal global Tjurina numbers

The minimal Tjurina number of irreducible germs of plane curve singularities

In this paper we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and the Tjurina numbers for any irreducible germ of plane curve singularity. This result is based on a closed formula for the minimal Tjurina number of an equisingularity class in terms of the sequence of multiplicities of the strict transform along a resolution. The key points for the proof are previous results by Genzmer, Wall and Mattei.

The Bruce–Roberts Number of A Function on A Hypersurface with Isolated Singularity

AbstractLet $(X,0)$ be an isolated hypersurface singularity defined by $\phi \colon ({\mathbb{C}}^n,0)\to ({\mathbb{C}},0)$ and $f\colon ({\mathbb{C}}^n,0)\to{\mathbb{C}}$ such that the Bruce–Roberts number $\mu _{BR}(f,X)$ is finite. We first prove that $\mu _{BR}(f,X)=\mu (f)+\mu (\phi ,f)+\mu (X,0)-\tau (X,0)$, where $\mu $ and $\tau $ are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. Second, we show that the logarithmic characteristic variety $LC(X,0)$ is Cohen–Macaulay. Both theorems generalize the results of a previous paper by some of the authors, in which the hypersurface $(X,0)$ was assumed to be weighted homogeneous.

Hypersurfaces with defect

A projective hypersurface X⊆Pn has defect if hi(X)≠hi(Pn) for some i∈{n,…,2n−2} in a suitable cohomology theory. This occurs for example when X⊆P4 is not Q-factorial. We show that hypersurfaces with defect tend to be very singular: In characteristic 0, we present a lower bound on the Tjurina number, where X is allowed to have arbitrary isolated singularities. For X with mild singularities, we prove a similar result in positive characteristic. As an application, we obtain an estimate on the asymptotic density of hypersurfaces without defect over a finite field.