Tjurina number of a local complete intersection curve

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

In 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.

Jacobian代数的分层Betti数与平面曲线的总Tjurina数

Graded Betti numbers of the Jacobian algebra and total Tjurina numbers of plane curves

本文通过Jacobian代数的分层Betti数计算平面曲线的总Tjurina数，并提出新的上界视角及必要条件。

Jacobian syzygies and plane curves with maximal global Tjurina numbers

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 Plessis and C.T.C. Wall, for the global Tjurina number of such a curve $C$, in terms of its degree $d$ and of the minimal degree $r\leq d-1$ of a Jacobian syzygy. We give a homological characterization of the curves whose global Tjurina number equals the du Plessis-Wall upper bound, which implies in particular that for such curves the upper bound for $m$ is also …

Plane curves with three syzygies, minimal Tjurina curves curves, and nearly cuspidal curves

We start the study of reduced complex projective plane curves, whose Jacobian syzygy module has 3 generators. Among these curves one finds the nearly free curves introduced by the authors, and the plus-one generated line arrangements introduced by Takuro Abe. All the Thom-Sebastiani type plane curves, and more generally, any curve whose global Tjurina number is equal to a lower bound given by A. du Plessis and C.T.C. Wall, are 3-syzygy curves. Rational plane curves which are nearly cuspidal, i.e. which have only cusps except one singularity with two branches, are also related to this class of …

关于具有孤立奇点的某些曲线和曲面的总Tjurina数的组合界

On combinatorial bounds for the total Tjurina numbers of certain curves and surfaces with isolated singularities

本文研究仅具有普通拟齐次奇点的平面线-圆锥曲线排列，导出多重交点分布的结构不等式，从而建立自由线排列和自由圆锥曲线排列的总Tjurina数的下界，并推广到三维空间中的超曲面。

On the minimal value of global Tjurina numbers for line arrangements

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.

On the Saito's basis and the Tjurina Number for Plane Branches

We introduce the concept of good Saito's basis for a plane curve $S$ and we explore it to obtain a formula for the minimal Tjurina number in a topological class. In particular, we present a positive answer for a question of Dimca and Greuel relating the Tjurina number and the Milnor number for a singular irreducible plane curve.

Milnor and Tjurina numbers for smoothings of surface singularities

For an isolated hypersurface singularity $f=0$, the Milnor number $μ$ is greater than or equal to the Tjurina number $τ$ (the dimension of the base of the semi-universal deformation), with equality if $f$ is quasi-homogeneous. K. Saito proved the converse. The same result is true for complete intersections, but is much harder. For a Gorenstein surface singularity $(V,0)$, the difference $μ- τ$ can be defined whether or not $V$ is smoothable; it was proved in [23] that it is non-negative, and equal to 0 iff $(V,0)$ is quasi-homogeneous. We conjecture a similar result for non-Gorenstein surface …

圆锥线配置的Jacobian概型与特征概型

Jacobian schemes of conic-line arrangements and eigenschemes

该研究针对射影平面圆锥线配置，给出达到全局Tjurina数上下界的几何刻画，并应用于刻画Jacobian概型为三元张量特征概型的配置及极映射几何。

平面曲线奇点与Fitting理想

Plane curve singularities and Fitting ideals

研究非拟齐次平面曲线奇点的Tjurina理想相关的Fitting理想，发现Milnor数与Tjurina数之差不超过2时的特殊性质。

The 0-th Fitting ideal of the Jacobian module of a plane curve

We describe the 0-th Fitting ideal of the Jacobian module of a plane curve in terms of determinants involving the Jacobian syzygies of this curve. This leads to new characterizations of maximal Tjurina curves, that is of non free plane curves, whose global Tjurina number equals an upper bound given by A. du Plessis and C.T.C. Wall.

A Note on 1-Forms for Reduced Curve Singularities

We collect some classical results about holomorphic 1-forms of a reduced complex curve singularity, in particular of a complete intersection, and use them to compare the Milnor number, the Tjurina number and the dimension of the torsion part of the 1-forms.

关于全纯函数芽对的米尔诺数和蒂尤林数

On Milnor and Tjurina numbers of Pairs of Holomorphic Functions Germs

该研究受全纯函数芽对的米尔诺数分岔公式启发，通过叶状理论引入蒂尤林数，证明其为解析不变量并推导公式及性质，还用此不变量刻画半驯亚纯函数芽，同时研究了全纯函数芽对的米尔诺数相关性质。

关于叶的某些指标及其应用

On some indices of foliations and applications

本文研究了叶的某些指标及其应用，推导了Tjurina数的局部公式，并提供了关于叶乘数和不变曲线Milnor数的新的全局结果证明。

On the Vanishing Topology of Isolated Cohen-Macaulay Codimension 2 Singularities

Isolated Cohen-Macaulay codimension 2 singularities share many common features with isolated complete intersection singularities, but they also exhibit some striking new behaviour. One such instance was recently observed by Damon and Pike in their study of the vanishing topology and Euler characteristic, where they took this class of singularities as examples. In this article, we explore their findings further by determining the Betti numbers explicitly and explain the new phenomena. An important tool here is the Tjurina modification relating a Cohen-Macaulay codimension 2 singularity to a fin…

关于叶状结构的第 k 个 Milnor 数与第 k 个 Tjurina 数

On the k-th Milnor and k-th Tjurina Numbers of a Foliation

本文引入并系统研究了全纯叶状结构的第 k 个 Milnor 数与第 k 个 Tjurina 数，建立显式公式与拓扑性质，解决并反驳了相关猜想，还推广了 Teissier 引理并给出拟齐次情形的闭公式。

On 1-forms on isolated complete intersection curve singularities

We collect some classical results about holomorphic 1-forms of a reduced complex curve singularity. They are used to study the pull-back of holomorphic 1-forms on an isolated complete intersection curve singularity under the normalization morphism. We wonder whether the Milnor number $μ$ and the Tjurina number $τ$ of any isolated plane curve singularity satisfy the inequality $3μ<4τ$.

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 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.

关于叶状结构的Briançon-Skoda定理

On Briançon-Skoda theorem for foliations

本文将Briançon-Skoda定理推广到第二型叶状结构，建立其Milnor数与Tjurina数的关系，并给出代数曲线全局Tjurina数的下界。

最大化曲线作为自由曲线

Maximizing curves viewed as free curves

本文建立最大化曲线与自由平面曲线的联系，并研究奇数次平面曲线最大总丘里纳数问题。

Chern Classes of Logarithmic Vector Fields

Let $X$ be a nonsingular complex variety and $D$ a reduced effective divisor in $X$. In this paper we study the conditions under which the formula $c_{SM}(1_U)=c(\textup{Der}_X(-\log D))\cap [X]$ is true. We prove that this formula is equivalent to a Riemann-Roch type of formula. As a corollary, we show that over a surface, the formula is true if and only if the Milnor number equals the Tjurina number at each singularity of $D$. We also show the Rimann-Roch type of formula is true if the Jacobian scheme of $D$ is nonsingular or a complete intersection.

On rational cuspidal plane curves, and the local cohomology of Jacobian rings

This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and nearly free curves in terms of Tjurina numbers. Finally, we suggest a stronger form of Terao's conjecture on the freeness of a line arrangement being determined by its combinatorics.

On the Analytic Invariants and Semiroots of Plane Branches

The value semigroup of a $k$-semiroot $C_k$ of a plane branch $C$ allow us to recover part of the value semigroup $Γ=\langle v_0,\ldots ,v_g\rangle$ of $C$, that is, it is related to topological invariants of $C$. In this paper we consider the set of values of differentials $Λ_k$ of $C_k$, 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 $C_k$. In particular, we show that $τ\leq μ-\frac{3n_g-2}{4}μ_{g-1}$ where $n…

Tropical lifting problem for the intersection of plane curves

Given a tropical divisor $D$ in the intersection of two tropical plane curves, we study when it can be realized as the tropicalization of the intersection of two algebraic curves, and give a sufficient condition. We show that under a certain condition involving a graph determined by these tropical curves, we can algorithmically find algebraic curves such that the tropicalization of their intersection is $D$.

On degree three curves in $C^{(2)}$ with positive self-intersection

In this paper we study degree three curves in $C^{(2)}$ with positive self-intersection defined by the action of a spherical triangular group in an auxiliary curve.

Torsion of elliptic curves and unlikely intersections

We study effective versions of unlikely intersections of images of torsion points of elliptic curves on the projective line.

Intersection numbers on Deligne-Mumford moduli spaces and quantum Airy curve

We establish the Airy curve case of a conjecture of Gukov and Sułkowski by reducing to Dijkgraaf-Verlinde-Verlinde Virasoro constraints satisfied by the intersection numbers on moduli spaces of algebraic curves.

Calculating intersection numbers on moduli spaces of pointed curves

This is a note on calculating intersection numbers on moduli spaces of curves. A codimension 3 relation among tautological classes on the moduli space of genus 4 curves is given.

$\mathbb{A}^1$-curves on affine complete intersections

We generalize the results of Clemens, Ein, and Voisin regarding rational curves and zero cycles on generic projective complete intersections to the logarithmic setup.

Deforming monomial space curves into set-theoretic complete intersection singularities

We deform monomial space curves in order to construct examples of set-theoretical complete intersection space curve singularities. As a by-product we describe an inverse to Herzog's construction of minimal generators of non-complete intersection numerical semigroups with three generators.

On the self-intersection number of the nonsingular models of rational cuspidal plane curves

In this paper, we consider rational cuspidal plane curves having at least three cusps. We give an upper bound of the self-intersection number of the proper transforms of such curves via the minimal embedded resolution of the cusps. For a curve having exactly three cusps, we show that the self-intersection number is equal to the bound if and only if the curve coincides with the quartic curve having three cusps.

On the intersection of ACM curves in $\PP^3$

Bezout's theorem gives us the degree of intersection of two properly intersecting projective varieties. As two curves in P^3 never intersect properly, Bezout's theorem cannot be directly used to bound the number of intersection points of such curves. In this work, we bound the maximum number of intersection points of two integral ACM curves in P^3. The bound that we give is in many cases optimal as a function of only the degrees and the initial degrees of the curves.

The genus of projective curves on complete intersection surfaces

We bound the genus of a projective curve lying on a complete intersection surface in terms of its degree and the degrees of the defining equations of the surface on which it lies.

Very free curves on Fano Complete Intersections

In this paper, we show that general Fano complete intersections over an algebraically closed field of arbitrary characteristics are separably rationally connected. Our proof also implies that general log Fano complete intersections with smooth tame boundary divisors admit very free $A^1$-curves.

Complete intersections primitive structures on space curves

A multiple (loc. Cohen Macaulay) structure, X, on a space curve C in P3 is said to be primitive if X is locally contained in a smooth surface. We give numerical conditions for C to be a "primitive" set theoretic complete intersection (i.e. to have a primitive multiple structure which is a complete intersection).

曲线和稳定交点的热带结式

Tropical Resultants for Curves and Stable Intersection

在热带几何的框架下，引入了两条平面曲线的结式概念。证明了代数结式的热带化可用于计算两条热带平面曲线的稳定交线。它表明，对于两个一般的原像的曲线的代数框架，他们的交集正好投影到稳定的交叉的曲线。并利用曲线定义方程的代数系数的残差系数给出了这种普遍性的充分条件。

Intersections of psi-classes on moduli spaces of m-stable curves

We explain how to compute top-dimensional intersections of psi-classes on moduli spaces of m-stable curves. On the moduli spaces of Deligne-Mumford stable pointed curves of genus one, these intersection numbers are determined by two recursions, namely the string equation and the dilaton equation. We establish, for each integer m>0, an analogous pair of recursions which determine these intersection numbers on the moduli spaces of m-stable pointed curves of genus one.

Smooth tropical complete intersection curves of genus $3$ in $\mathbb{R}^3$

We develop a method for describing the tropical complete intersection of a tropical hypersurface and a tropical plane in $\mathbb{R}^3$. This involves a method for determining the topological type of the intersection of a tropical plane curve and $\mathbb{R}_{\leq 0}^2$ by using a polyhedral complex. As an application, we study smooth tropical complete intersection curves of genus $3$ in $\mathbb{R}^3$. In particular, we show that there are no smooth tropical complete intersection curves in $\mathbb{R}^3$ whose skeletons are the lollipop graph of genus $3$. This gives a partial answer to a pro…

费马曲面上的负曲线自交数

Self-intersection Number of Negative Curves on Fermat Surfaces

本文给出了费马曲面上负曲线自交数的显式公式，为验证负数界限猜想提供了新思路。

曲线模空间的相交理论导论

An introduction to the intersection theory of the moduli space of curves

介绍曲线模空间的相交理论及其环状结构，综述开放问题并概述证明周环由环状类生成与否的技术。

曲线上的超商方案交理论

Intersection Theory of Hyperquot Schemes on curves

本文研究曲线上的超商方案虚拟交理论，推广了Vafa-Intriligator公式，并为曲线到部分旗流形的映射提供闭合计数公式。

Rational curves on complete intersections in positive characteristic

We study properties of rational curves on complete intersections in positive characteristic. It has long been known that in characteristic 0, smooth Calabi-Yau and general type varieties are not uniruled. In positive characteristic, however, there are well-known counterexamples to this statement. We will show that nevertheless, a \emph{general} Calabi-Yau or general type complete intersection in projective space is not uniruled. We will also show that the space of complete intersections of degree $(d_1, \cdots, d_k)$ containing a rational curve has codimension at least $\sum_{i=1}^k d_i - 2n +…

Geometry of intersections of some secant varieties to algebraic curves

For a smooth projective curve, the cycles of subordinate or, more generally, secant divisors to a given linear series are among some of the most studied objects in classical enumerative geometry. We consider the intersection of two such cycles corresponding to secant divisors of two different linear series on the same curve and investigate the validity of the enumerative formulas counting the number of divisors in the intersection. We study some interesting cases, with unexpected transversality properties, and establish a general method to verify when this intersection is empty.

一类半稳定曲线族的高度与交线

Height and Intersection for a family of semi-stable curves

本文考虑一类半稳定曲线f：X \to B的算术Hodge指标定理，其中f_{\bf C}是光滑的，推广了算术曲面的Faltings-Hriljac的算术Hodge指标定理.

完全交上的曲线与无理度的度量

Curves on complete intersections and measures of irrationality

研究完全交上曲线的最小次数和亏格，证明曲线次数由完全交次数从下方界定，并以此回答了关于完全交无理度的一个问题。

On isolated smooth curves of low genera in Calabi-Yau complete intersection threefolds

Building on results of Clemens and Kley, we find criteria for a continuous family of curves in a nodal $K$-trivial threefold $Y_0$ to deform to a scheme of finitely many smooth isolated curves in a general deformation $Y_t$ of $Y_0$. As an application, we show the existence of smooth isolated curves of bounded genera and unbounded degrees in Calabi-Yau complete intersections threefolds.

The gonality of complete intersection curves

The purpose of this paper is to show that for a complete intersection curve $C$ in projective space (other than a few stated exceptions), any morphism $f: C \to \mathbb{P}^r$ satisfying $\text{deg}\, f^*\mathcal{O}_{\mathbb{P}^r}(1) <\text{deg}\, C$ is obtained by projection from a linear space. In particular, we obtain bounds on the gonality of such curves and compute the gonality of general complete intersection curves. We also prove a special case of one of the well-known Cayley-Bacharach conjectures posed by Eisenbud, Green, and Harris.

Complete intersections and movable curves on the moduli space of six-pointed rational curves

A curve on a projective variety is called movable if it belongs to an algebraic family of curves covering the variety. We consider when the cone of movable curves can be characterized without existence statements of covering families by studying the complete intersection cone on a family of blow-ups of complex projective space, including the moduli space of stable six-pointed rational curves, $\bar{M}_{0,6}$, and the permutohedral or Losev-Manin moduli space of four-pointed rational curves. Our main result is that the movable and complete intersection cones coincide for the toric members of th…

On smooth and isolated curves in general complete intersection Calabi-Yau threefolds

Recently Knutsen found criteria for the curves in a complete linear system $|\mathcal{L}|$ on a smooth surface $X$ in a nodal K-trivial threefold $Y_0$ to deform to a scheme of finitely many smooth isolated curves in a general deformation $Y_t$ of $Y_0$. In this article we develop new methods to check whether the set of nodes of $Y_0$ imposes independent conditions on $|\mathcal{L}|$. As an application, we find new smooth isolated curves in complete intersection Calabi-Yau threefolds.

节曲线II上的几何：循环映射和交演算

Geometry on nodal curves II: cycle map and intersection calculus

我们研究一族节点（或光滑）曲线的相对Hilbert方案，通过它的（双有理）循环映射，去相对对称产品。我们证明了循环图是判别轨迹的爆破，它由具有重点的循环组成。本文在相对Hilbert概型上导出重言丛的Chern类的交演算，它在计数几何中有应用。

On intersections of symmetric determinantal varieties and theta characteristics of canonical curves

From a block-diagonal $(n+1) \times (m+1) \times (m+1)$ tensor symmetric in the last two entries one obtains two varieties: an intersection of symmetric determinantal hypersurfaces $X$ in $n$-dimensional projective space, and an intersection of quadrics $\mathfrak{C}$ in $m$-dimensional projective space. Under mild technical assumptions, we characterize the accidental singularities of $X$ in terms of $\mathfrak{C}$. We apply our methods to algebraic curves and show how to construct theta characteristics of certain canonical curves of genera 3, 4, and 5, generalizing a classical construction of…

Intersection theory on mixed curves

We consider two mixed curve $C,C'\subset {\Bbb C}^2$ which are defined by mixed functions of two variables $\bf z=(z_1,z_2)$. We have shown in \cite{MC}, that they have canonical orientations. If $C$ and $C'$ are smooth and intersect transversely at $P$, the intersection number $I_{top}(C,C';P)$ is topologically defined. We will generalize this definition to the case when the intersection is not necessarily transversal or either $C$ or $C'$ may be singular at $P$ using the defining mixed polynomials.

Extended Klein model and a bound on curves with negative self-intersection

Let $S$ be an irreducible smooth projective surface and $\mathcal{F}$ a collection of curves with negative self-intersection on $S$ such that no positive combination $aC_1 + bC_2$ is connected nef. In this paper, we provide an alternate proof that $\mathcal{F}$ is bounded by an exponential function of the Picard number $ρ(S)$ of $S$ using an extended version of the Klein disc model for hyperbolic space.

平面曲线奇点的极不变量：相交理论方法

Polar Invariants of Plane Curve Singularities: Intersection Theoretical Approach

这篇文章是根据2008年6月在Castro Urdiales的Pierrettefest上的一位作者的演讲，概述了平面曲线奇点的极不变量的一些最新结果。