On plane algebroid curves

单项式代数曲线与代数体曲线及其在基环上的Lipschitz饱和

Monomial algebraic and algebroid curves and their Lipschitz saturation over the ground ring

本文在基环R满足温和条件时，利用数值半群Γ=⟨γ₁,…,γₙ⟩显式描述了子代数A⊆R[[t^{γ₁},…,t^{γₙ}]]在R上的Lipschitz饱和A^*_{B,R}。

平面代数曲线的导子

Derivations of plane algebroid curves

本文针对非正则不可约平面代数曲线，研究其导子的判定条件，建立了诱导导子的幂级数阶数下界，并证明了其值半群的相关结果。

双拼接代数的Cohen-Macaulay和Gorenstein性质及其在代数曲线中的应用

Cohen-Macaulay and Gorenstein Properties of Bi-Amalgamated Algebras with Applications to Algebroid Curves

本文研究双拼接代数的基本同调性质，推导了其维数和深度，并通过最大和大Cohen-Macaulay模刻画其Cohen-Macaulay性质，同时通过canonical模刻画Gorenstein性质，并应用于代数曲线奇异性的理论。

Equisingular calculations for plane curve singularities

We present an algorithm which, given a deformation with trivial section of a reduced plane curve singularity, computes equations for the equisingularity stratum (that is, the mu-constant stratum in characteristic 0) in the parameter space of the deformation. The algorithm works for any, not necessarily reduced, parameter space and for algebroid curve singularities C defined over an algebraically closed field of characteristic 0 (or of characteristic p>ord(C)). It provides at the same time an algorithm for computing the equisingularity ideal of J. Wahl. The algorithms have been implemented in t…

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 …

关于第三类复平面曲线

On type three complex plane curves

本文研究复平面曲线的第三类，重点探讨低次曲线和直线排列特性。

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.

Euclid meets Bezout: Intersecting algebraic plane curves with the Euclidean algorithm

We show how the Eulcidean algorithm for polynomials can be used to find the intersection points, with multiplicities, of two plane algebraic curves.

Normal form for space curves in a double plane

This note is an attempt to relate explicitly the geometric and algebraic properties of a space curve that is contained in some double plane. We show in particular that the minimal generators of the homogeneous ideal of such a curve can be written in a very specific form. As applications we characterize the possible Hartshorne-Rao modules of curves in a double plane and the minimal curves in their even Liaison classes.

A remark on 1-forms and plane curve singularities

We study the pull-back of regular 1-forms on a complex irreducible plane curve singularity under the normalization morphism.

Rational plane curves parameterizable by conics

We introduce the class of rational plane curves parameterizable by conics as an extension of the family of curves parameterizable by lines (also known as monoid curves). We show that they are the image of monoid curves via suitable quadratic transformations in projective plane. We also describe all the possible proper parameterizations of them, and a set of minimal generators of the Rees Algebra associated to these parameterizations, extending well-known results for curves parameterizable by lines.

Embedding dimension of a good semigroup

In this paper, we study good semigroups of \mathbb{N}^n, a class of semigroups that contains the value semigroups of algebroid curves with n branches. We give the definition of embedding dimension of a good semigroup showing that, in the case of good semigroups of \mathbb{N}^2, some of its properties agree with the analogue concepts defined for numerical semigroups.

Saturation of Jacobian ideals: some applications to nearly free curves, line arrangements and rational cuspidal plane curves

In this note we describe the minimal resolution of the ideal $I_f$, the saturation of the Jacobian ideal of a nearly free plane curve $C:f=0$. In particular, it follows that this ideal $I_f$ can be generated by at most 4 polynomials. Related general results by Hassanzadeh and Simis on the saturation of codimension 2 ideals are discussed in detail. Some applications to rational cuspidal plane curves and to line arrangements are also given.

On the equations of the moving curve ideal of a rational algebraic plane curve

Given a parametrization of a rational plane algebraic curve C, some explicit adjoint pencils on C are described in terms of determinants. Moreover, some generators of the Rees algebra associated to this parametrization are presented. The main ingredient developed in this paper is a detailed study of the elimination ideal of two homogeneous polynomials in two homogeneous variables that form a regular sequence.