正特征下单模参数平面曲线奇点的分类

A note on exceptional unimodal singularities and K3 surfaces

This is a short note on the relation between the graded stable derived categories of 14 exceptional unimodal singularities and the derived category of K3 surfaces obtained as compactifications of the Milnor fibers. As a corollary, we obtain a basis of the numerical Grothendieck group similar to the one given by Ebeling and Ploog (arXiv:0809.2738).

在正特征下对单模孤立完全交点奇点的分类

Classification of unimodal isolated complete intersection singularities in positive characteristic

本文在接触等价下对任意特征下的单模孤立完全交点奇点进行分类，扩展了Dimca和Gibson在复数域上的工作，并将完全横截方法推广到正特征域，适用于其他分类问题。

Lipschitz单模函数芽的分类

Classification of Lipschitz unimodal function germs

本文为孤立奇点引入Lipschitz模概念，完整分类了余秩2且4-jet非零的Lipschitz单模奇点，并证明6-jet消失时模至少为2，给出上界。

The Classification of Real Singularities Using SINGULAR. Part II: The Structure of the Equivalence Classes of the Unimodal Singularities

In the classification of real singularities by Arnold et al. (1985), normal forms, as representatives of equivalence classes under right equivalence, are not always uniquely determined. We describe the complete structure of the equivalence classes of the unimodal real singularities of corank 2. In other words, we explicitly answer the question which normal forms of different type are equivalent, and how a normal form can be transformed within the same equivalence class by changing the value of the parameter. This provides new theoretical insights into these singularities and has important cons…

一类Horikawa曲面的KSBA模空间中的单模奇点与边界除子

Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces

本文研究一类Horikawa曲面典范模型模空间的KSBA紧化，构造了八个新的不可约边界除子，并探讨其与GIT紧化及退化曲面Hodge理论的关系。

The Classification of Real Singularities Using Singular. Part III: Unimodal Singularities of Corank 2

We present a classification algorithm for isolated hypersurface singularities of corank 2 and modality 1 over the real numbers. For a singularity given by a polynomial over the rationals, the algorithm determines its right equivalence class by specifying all representatives in Arnold's list of normal forms (Arnold et al. 1985) belonging to this class, and the corresponding values of the moduli parameter. We discuss how to computationally realize the individual steps of the algorithm for all singularities in consideration, and give explicit examples. The algorithm is implemented in the Singular…

Puiseux coefficients and parametric deformation of plane curve singularities

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 space. The methods used in paper are very elementary.

Resultants and Singularities of Parametric Curves

Let ${\cal C}$ be an algebraic space curve defined parametrically by ${\cal P}(t)\in {\Bbb K}(t)^{n},\,n\geq 2$. In this paper, we introduce a polynomial, the T--function, $T(s)$, which is defined by means of a univariate resultant constructed from ${\cal P}(t)$. We show that $T(s)=\prod_{i=1}^n H_{P_i}(s)^{m_i-1}$, where $H_{P_i}(s),\,i=1,\ldots,n$ are polynomials (called the fibre functions) whose roots are the fibre of the ordinary singularities $P_i\in {\cal C}$ of multiplicity $m_i,\,i=1,\ldots,n$. Thus, a complete classification of the singularities of a given space curve, via the factor…

The Classification of Real Singularities Using SINGULAR. Part I: Splitting Lemma and Simple Singularities

We present algorithms to classify isolated hypersurface singularities over the real numbers according to the classification by V.I. Arnold (Arnold et al., 1985). This first part covers the splitting lemma and the simple singularities; a second and a third part will be devoted to the unimodal singularities up to corank 2. All algorithms are implemented in the SINGULAR library realclassify.lib (Marais and Steenpass, 2012).

Equinormalizable theory for plane curve singularities with embedded points and the theory of equisingularity

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 $δ$-invariant of a (non-reduced) curve singularity which is introduced by Brücker-Greuel (\cite{BG}). The second criterion is based on the I-equisingularity of a $k$-parametric family ($k\geq 1$) of generically reduced plane curve singularities, which is introduced by Nobile (\cite{No}) for one-parametric families. The equivalence of some kinds of equisingularities of a family of generically reduced plane curve singularities is also studi…

Equinormalizability and topologically triviality of deformations of isolated curve singularities over smooth base spaces

We give a $δ$-constant criterion for equinormalizability of deformations of isolated (not necessarily reduced) curve singularities over smooth base spaces of dimension $\geq 1$. For one-parametric families of isolated curve singularities, we show that their topologically triviality is equivalent to the admission of weak simultaneous resolutions.

双角曲线的 pedal 曲线

Pedal Curves of a Bicorn

本文研究双角曲线的 pedal 曲线，分析其奇异性，比较参数与符号表示，指出有理参数化最有效，并探讨其在 STEAM 教育中的应用。

Motivic renormalization and singularities

We consider parametric Feynman integrals and their dimensional regularization from the point of view of differential forms on hypersurface complements and the approach to mixed Hodge structures via oscillatory integrals. We consider restrictions to linear subspaces that slice the singular locus, to handle the presence of non-isolated singularities. In order to account for all possible choices of slicing, we encode this extra datum as an enrichment of the Hopf algebra of Feynman graphs. We introduce a new regularization method for parametric Feynman integrals, which is based on Leray coboundari…

关于复平面曲线奇点的拓扑分类

On the topological classification of complex plane curve singularities

本文研究复平面曲线奇点的拓扑分类，通过将曲线分解为分支并提取 Puiseux 特征与相交数，证明两条曲线拓扑等价当且仅当这些数值不变量完全相同。

Equivariant approach to weighted projective curves

We investigate group actions on the category of coherent sheaves over weighted projective lines. We show that the equivariant category with respect to certain finite group action is equivalent to the category of coherent sheaves over a weighted projective curve, where the genus of the underlying smooth projective curve and the weight data are given by explicit formulas. Moreover, we obtain a trichotomy result for all such equivariant relations. This equivariant approach provides an effective way to investigate smooth projective curves via weighted projective lines. As an application, we give a…