Discriminant of a generic projection of a minimal normal surface singularity

A discriminant criterion of irreducibility

In this paper we give a criterion of irreducibility for a complex power series in two variables, using the notion of jacobian Newton diagrams, defined with respect to any direction. Moreover we study the singularity at infinity of a plane affine curve with one point at infinity for which the global counterpart of our main result holds.

Newton polyhedra of discriminants of projections

For a system of polynomial equations, whose coefficients depend on parameters, the Newton polyhedron of its discriminant is computed in terms of the Newton polyhedra of the coefficients. This leads to an explicit formula (involving mixed fiber polyhedra and Euler obstructions of toric varieties) in the unmixed case, suggests certain open questions in general, and generalizes a number of similar known results. The argument is based on a formula for the support function of a mixed fiber body, which also suggests a new proof for existence of mixed fiber bodies.

Using approximate roots for irreducibility and equi-singularity issues in K[[x]][y]

We provide an irreducibility test in the ring K[[x]][y] whose complexity is quasi-linear with respect to the valuation of the discriminant, assuming the input polynomial F square-free and K a perfect field of characteristic zero or greater than deg(F). The algorithm uses the theory of approximate roots and may be seen as a generalization of Abhyankhar's irreducibility criterion to the case of non algebraically closed residue fields. More generally, we show that we can test within the same complexity if a polynomial is pseudo-irreducible, a larger class of polynomials containing irreducible one…

代数曲面的Whitney折叠与尖点，以及判别式的奇点

Whitney fold and cusp for algebraic surfaces, and singularities of discriminants

本文研究代数曲面投影的Whitney型奇点稳定性，通过分析判别式奇点轨迹和Vandermonde/Schur矩阵退化，在温和组合条件下枚举所有稳定的多奇点类型。

局部判别式到全局障碍的扭转轨迹

Torsion Trajectories from Local Discriminants to Global Obstructions

研究通过局部奇点数据到全局障碍理论的轨迹，揭示有限判别 torsion 的二维现象，提出 Enriques 2-torsion 的全球性问题。

The Entropic Discriminant

The entropic discriminant is a non-negative polynomial associated to a matrix. It arises in contexts ranging from statistics and linear programming to singularity theory and algebraic geometry. It describes the complex branch locus of the polar map of a real hyperplane arrangement, and it vanishes when the equations defining the analytic center of a linear program have a complex double root. We study the geometry of the entropic discriminant, and we express its degree in terms of the characteristic polynomial of the underlying matroid. Singularities of reciprocal linear spaces play a key role.…

野循环商奇点的判别群

Discriminant groups of wild cyclic quotient singularities

本文显式描述二维野循环商奇点的消解，证明奇素数下 p 的任意幂可为相交矩阵行列式，并支持野作用仅在原点分歧的猜想。

关于正规曲面上曲线经平面投影的像

On the image of a curve in a normal surface by a plane projection

本文研究了复解析正规曲面到平面的有限解析态射下约化曲线像的拓扑结构，通过迭代线性系方法描述了像的拓扑，并推广了已有光滑情形的结论，同时给出了判别式曲线的拓扑型。

矩阵分解与曲线奇点的通用平面投影

Matrix factorization and the generic plane projection of curve a singularity

本文研究了$\mathbb{C}^3$中曲线奇点通用平面投影定义的矩阵分解，并反向探讨了给定平面曲线奇点$Y$对应的空间曲线奇点$X$所定义的矩阵分解族。

On the exceptional locus of the birational projections of normal surface singularity into a plane

Given a normal surface singularity $(X, Q)$ and a birational morphism to a non- singular surface $π: X \to S$, we investigate the local geometry of the exceptional divisor $L$ of $π$. We prove that the dimension of the tangent space to $L$ at $Q$ equals the number of exceptional components meeting at $Q$. Consequences relative to the existence of such birational projections contracting a prescribed number of irreducible curves are deduced. A new characterization of minimal singularities is obtained in these terms.

Rationality of real conic bundles with quartic discriminant curve

We study real double covers of $\mathbb P^1\times\mathbb P^2$ branched over a $(2,2)$-divisor, which have the structure of a conic bundle threefold with smooth quartic discriminant curve via the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space of the conic bundle is rational. For five of the six isotopy classes we construct $\mathbb C$-rational examples that have obstructions to rationality over $\mathbb R$, and for the sixth class, we show that the models we consider are all rational. Moreover, for three of the five classes with ir…

超曲面的判别式的赋值

The valuation of the discriminant of a hypersurface

本文利用 Zariski 主定理和退化论证，刻画了超曲面判别式赋值为 1 的充要条件，并给出了奇点情形下的下界。

Fine polar invariants of minimal singularities of surface

We consider the polar curves $\PSO$ arising from generic projections of a germ $(S,0)$ of complex surface singularity onto $\C^2$. Taking $(S,0)$ to be a minimal singularity of normal surface (i.e. a rational singularity with reduced tangent cone), we give the $δ$-invariant of these polar curves, as well as the equisingularity-type of their generic plane projections, which are also the discriminants of generic projections of $(S,0)$. These two (equisingularity)-data for $\PSO$ are described in term, on the one side of the geometry of the tangent cone of $(S,0)$ and on the other side of the lim…

On the Nash modification of a germ of complex analytic singularity

For a germ $(X,0) \subset (\mathbb{C}^n,0)$ of reduced, equidimensional complex analytic singularity its Nash modification can be constructed as an analytic subvariety $ Z \subset \mathbb{C}^n \times G(k,n)$. We give a characterization of the subvarieties of $\mathbb{C}^n \times G(k,n)$ that are the Nash modification of its image under the projection to $\mathbb{C}^n$. This result generalizes the characterization of conormal varieties as Legendrian subvarieties of $\mathbb{C}^n \times \check{\mathbb{P}}^{n-1}$ with its canonical contact structure. As a by-product we define the $d$-conormal spa…

节点曲面的簇、编码理论与三次超曲面的判别式。第一部分：一般理论与节点K3曲面。第二部分：三次超曲面及其关联判别式。第三部分：节点五次曲面。第四部分：节点六次曲面

Varieties of Nodal surfaces, coding theory and Discriminants of cubic hypersurfaces. Part 1: Generalities and nodal K3 surfaces. Part 2: Cubic Hypersurfaces, associated discriminants. Part 3: Nodal quintics. Part 4: Nodal sextics

本文通过为射影节点曲面附加二元码，系统研究了P^3中节点曲面的Severi簇的不可约分量及其关联层级，完整分类了节点K3曲面，并揭示了Togliatti五次曲面及最大节点数六次曲面与三次超曲面判别式之间的深刻联系。

General elements of an m-primary ideal on a normal surface singularity

In this paper, we show how to apply a theorem by Lê D.T. and the author about linear families of curves on normal surface singularities to get new results in this area. The main concept used is a specific definition of {\em general elements} of an ideal in the local ring of the surface. We make explicit the connection between this notion and the elementary notion of general element of a linear pencil, through the use of {\em reduction}. This allows us to prove the invariance of the generic Milnor number (resp. of the multiplicity of the discriminant), between two pencils generating two ideals …

A polyhedral characterization of quasi-ordinary singularities

Given an irreducible hypersurface singularity of dimension $d$ (defined by a polynomial $f\in K[[ {\bf x} ]][z]$) and the projection to the affine space defined by $K[[ {\bf x} ]]$, we construct an invariant which detects whether the singularity is quasi-ordinary with respect to the projection. The construction uses a weighted version of Hironaka's characteristic polyhedron and successive embeddings of the singularity in affine spaces of higher dimensions. When $ f $ is quasi-ordinary, our invariant determines the semigroup of the singularity and hence it encodes the embedded topology of the s…

曲面上秩二丛的冒泡

Bubbling of rank two bundles over surfaces

本文研究曲面上秩二全纯向量丛族的奇点形成，定义并刻画带泡可育族，利用初等变换证明其存在性，并以泡树和反例回应相关问题。

Toric embedded resolutions of quasi-ordinary hypersurface singularities

We build two embedded resolution procedures of a quasi-ordinary singularity of complex analytic hypersurface, by using toric morphisms which depend only on the characteristic monomials associated to a quasi-ordinary projection of the singularity. This result answers an open problem of Lipman in Equisingularity and simultaneous resolution of singularities, Resolution of Singularities, Progress in Mathematics No. 181, 2000, 485-503. In the first procedure the singularity is embedded as hypersurface. In the second procedure, which is inspired by a work of Goldin and Teissier for plane curves (see…

Distance to the discriminant

We will study algebraic hyper-surfaces on the real unit sphere $\mathcal S^{n-1}$ given by an homogeneous polynomial of degree d in n variables with the view point, rarely exploited, of Euclidian geometry using Bombieri's scalar product and norm. This view point is mostly present in works about the topology of random hyper-surfaces \cite{ShubSmale93, GayetWelschinger14}. Our first result (lemma \ref{distgen} page \pageref{distgen}) is a formula for the distance $\dist(P,Δ)$ of a polynomial to the {\em real discriminant} $Δ$, i.e. the set of polynomials with a real singularity on the sphere. Th…

On Lipschitz Normally Embedded complex surface germs

We undertake a systematic study of Lipschitz Normally Embedded normal complex surface germs. We prove in particular that the topological type of such a germ determines the combinatorics of its minimal resolution which factors through the blowup of its maximal ideal and through its Nash transform, as well as the polar curve and the discriminant curve of a generic plane projection, thus generalizing results of Spivakovsky and Bondil that were known for minimal surface singularities. In an appendix, we give a new example of a Lipschitz Normally Embedded surface singularity.

通过局部不变量研究 $\mathbb C^3$ 中的曲面奇点的扎里斯基等奇异性和

Zariski equisingularity of surface singularities in $\mathbb C^3$ by a local invariant

本文通过定义一个新的局部不变量，研究了 $\mathbb C^3$ 中曲面奇点的扎里斯基等奇异性和性质。

Equi-singularity of real families and Lipschitz Killing curvature densities at infinity

Fix an o-minimal structure expanding the ordered field of real numbers. Let $(W_y)_{y\in\mathbb{R}^s}$ be a definable family of closed subsets of $\mathbb{R}^n$ whose total space $W = \cup_y W_y\times y$ is a closed connected $C^2$ definable sub-manifold of $\mathbb{R}^n\times\mathbb{R}^s$. Let $φ:W \to\mathbb{R}^s$ be the restriction of the projection to the second factor. After defining $K(φ)$, the set of generalized critical values of $φ$, showing that they are closed and definable of positive codimension in $\mathbb{R}^s$, contain the bifurcation values of $φ$ and are stable under generic …

正常表面奇点的积分 perverse 障碍：解析确定性和monodromy

Integral Perverse Obstructions for Normal Surface Singularities: Resolution Determinants and Monodromy

研究正常复分析表面奇点的积分pervers障碍，通过解析确定性和monodromy理论揭示其局部不变性。