Oliver Klehn
We give a generalization of an algebraic formula of Gomez-Mont for the index of a vector field with isolated zero in (C^n,0) and tangent to an isolated hypersurface singularity. We only assume that the vector field has an isolated zero on the singularity.
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The purpose of this paper is to investigate the geometric properties of real hypersurfaces of D'Angelo infinite type in ${\mathbb C}^n$. In order to understand the situation of flatness of these hypersurfaces, it is natural to ask whether there exists a nonconstant holomorphic curve tangent to a given hypersurface to infinite order. A sufficient condition for this existence is given by using Newton polyhedra,which is an important concept in singularity theory. More precisely,equivalence conditions are given in the case of some model hypersurfaces.
We compute explicitly the limits of tangents of a quasi-ordinary singularity in terms of its special monomials. We show that the set of limits of tangents of Y is essentially a topological invariant of Y .
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 …
For a hypersurface in a projective space, we consider the set of pairs of a point and a line in the projective space such that the line intersects the hypersurface at the point with a fixed multiplicity. We prove that this set of pairs forms a smooth variety for a general hypersurface.
Mather and Yau showed that an isolated complex hypersurface singularity is completely determined by its moduli algebra. It is shown, for the simple elliptic singularities, how to construct continuous invariants from the moduli algebras and, hence, associate invariants to the singularities themselves.
We characterize embedded $\C^1$ hypersurfaces of $\R^n$ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most $m<3/2$. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is $\C^1$. In the real analytic case the same conclusion holds under the weakened hypothesis that each tangent cone be a hypersurface. In particular, any convex real analytic hypersurface $X\subset\R^n$ is $\C^1$. Furthermore, if $X$ is real algebraic, strictly convex, and …
We study spaces of lines that meet a smooth hypersurface X in P^n to high order. As an application, we give a polynomial upper bound on the number of planes contained in a smooth degree d hypersurface in P^5 and provide a proof of a result of Landsberg without using moving frames.
Unirationality of hypersurfaces via highly tangent lines
本文提出基于高切线的射影空间低次完全交单有理性构造，应用于超曲面时得到更优界：当n≥2^((d-1)2^(d-5))时，射影n空间中一般d≥6次超曲面单有理，改进经典结果。
Quadratic counts of highly tangent lines to hypersurfaces
针对单点处与超曲面高次相切的直线的局部类型，本文给出基于Wronski映射和超曲面基本形式的两种几何解释，明确其为依赖定向选取的二次型值欧拉数的局部贡献。
Restricted tangent bundle of rational curves on projective hypersurfaces
本文完全分类了一般射影超曲面包含具有平衡受限切丛的有理曲线的所有次数参数，并给出了有理正规曲线情形下的显式构造方法。
The Aluffi algebra is algebraic definition of characteristic cycles of a hypersurface in intersection theory. In this paper we focus on the Aluffi algebra of quasi-homogeneous and locally Eulerian hypersurface with isolated singularities. We prove that the Jacobian ideal of an affine hypersurfac with isolated singularities is of linear type if and only if it is locally Eulerian. We show that the gradient ideal of a projective hypersurface is of linear type if and only if the corresponding affine curve in the affine chart associated to singular points is locally Eulerian. We prove that the grad…
We show that the possible drop in multiplicity in an analytic family $F(z,t)$ of complex analytic hypersurface singularities with constant Milnor number is controlled by the powers of $t$. We prove equimultiplicity of $μ$-constant families of the form $f + tg + t^2h$ if the singular set of the tangent cone of $\{f = 0}$ is not contained in the tangent cone of $\{h = 0}$.
Using the semicontinuity of the Milnor number and the Łojasiewicz exponent, we give a simple proof of Yau's characterization of isolated homogenous hypersurface singularities.
Singularity of cubic hypersurfaces and hyperplane sections of projectivized tangent bundle of projective space
研究立方超曲面的奇点性质，并通过线性代数方法描述射影空间射影化切丛的超平面截面，扩展了Mazouni-Nagaraj的高维结果。
Invariants of Non-Isolated Singularities of Hypersurfaces
本文推广了 Siersma、Pellikaan 和 de Jong 关于奇异轨迹为光滑曲线的超曲面 morsification 的结果，通过研究横截判别式及其与代数、拓扑不变量的关系，应用于 Yomdin 型孤立奇点研究。
We study the behaviour of rational curves tangent to a hypersurface under degenerations of the hypersurface. Working within the framework of logarithmic Gromov-Witten theory, we extend the degeneration formula to the logarithmically singular setting, producing a virtual class on the space of maps to the degenerate fibre. We then employ logarithmic deformation theory to express this class as an obstruction bundle integral over the moduli space of ordinary stable maps. This produces new refinements of the logarithmic Gromov-Witten invariants, encoding the degeneration behaviour of tangent curves…
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.
Bounds on the F-Pure Threshold of Isolated Hypersurface Singularities
本文在正特征代数闭域上，利用Milnor数和Tjurina数等经典奇点不变量，给出了孤立超曲面奇点F-纯阈值的界，并针对曲线奇点得到了基于值半群生成元的显式计算，推广了Igusa公式。
We prove that a smooth surface, non of general type, in projective four-space, which lies on a quartic hypersurface with isolated singularities has degree at most 27 (in fact we prove a slightly more general result).
Deformations of the tangent bundle of a projective hypersurface
本文研究了射影超曲面切丛的无穷小形变，证明了其形变空间的维数公式及无阻碍性，并通过构造显式双有理模型证明包含切丛的稳定丛模空间不可约分支是有理簇。
We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a projective hypersurface $V$ with isolated singularities and the Torelli properties of $V$ (in the sense of Dolgachev-Kapranov). We show in particular that hypersurfaces with a small Tjurina numbers are Torelli in this sense. When $V$ is a plane curve, or more interestingly, a surface in $P^3$, we discuss the stability of the reflexive sheaf of logarithmic vector fields along $V$. A new lower bound for the minimal degree of a syzygy associated to a 1-dimensional complete intersection is also g…
The Schwartz index and the residue of logarithmic foliations along a hypersurface with isolated singularities
本文证明了对数叶状结构沿孤立奇点超曲面的Baum-Bott型公式，用Schwartz指数表达留数，并推广了Poincaré问题的障碍结果至任意偶数维射影空间。
It is proved that a general Fano hypersurface of index 1 (in the projective space) with isolated singularities of general position is birationally rigid. Therefore it cannot be fibered into uniruled varieties of a smaller dimension by a rational map. The group of birational self-maps is either trivial or cyclic of order two.
We explicitly fix a mistake in a preliminary statement of our previous paper on the conductor at a multiplanar singularity. The correction is not immediate and, though the mistake does not affect correctness of the subsequent results, the wrong statement could easily be misleading.
Minkowski Inequalities and non-isolated hypersurface singularities
本文研究非孤立超曲面奇点，通过带不等式的 Lê-Iomdine 公式并结合 Teissier 的截面 Milnor 数 Minkowski 不等式，建立了关于 Lê 数的 Minkowski 不等式。
In the study of equisingularity of isolated singularities we have the classical theorem of Briancon, Speder and Teissier which states that a family of isolated hypersurface singularities is Whitney equisingular if and only if the mu^*-sequence for a hypersurface is constant in the family. In this paper we generalize to non-isolated hypersurface singularities. By assuming non-contractibility of strata of a Whitney stratification of the non-isolated singularities outside the origin we show that Whitney equisingularity of a family is equivalent to constancy of a certain selection of invariants fr…
We study the Milnor monodromies of non-isolated hypersurface singularities and show that the reduced cohomology groups of the Milnor fibers are concentrated in the middle degree for some eigenvalues of the monodromies. As an application of this result, we give an explicit formula for some parts of their Jordan normal forms.
In this note we give a simple proof of the following relative analog of the well known Milnor-Palamodov theorem: the Bruce-Roberts number of a function relative to an isolated hypersurface singularity is equal to its topological Milnor number (the rank of a certain relative (co)homology group) if and only if the hypersurface singularity is quasihomogeneous. The proof relies on an interpretation of the Bruce-Roberts number in terms of differential forms and the Lê-Greuel formula.
We study the Brieskorn modules associated to a germ of holomorphic function with non-isolated singularities, and show that the Brieskorn module has naturally a structure of a module over the ring of microdifferential operators of nonpositive degree, and that the kernel of the morphism to the Gauss-Manin system coincides with the torsion part for the action of $t$ and also with that for the action of the inverse of the Gauss-Manin connection. This torsion part is not finitely generated in general, and we give a sufficient condition for the finiteness. We also prove a Thom-Sebastiani type theore…
We show that the dimension of the set of limits of tangent spaces along a subvariety at an irreducible isolated singularity is one less than the dimension of the subvariety itself.
Nakai conjectures for isolated weighted homogeneous hypersurface singularities
本文验证了Nakai猜想在孤立齐次超曲面奇点情况下的成立，探讨微分算子与奇点检测的关系。
Let V be a projective hypersurface of fixed degree and dimension which has only isolated singular points. We show that, if the sum of the Milnor numbers at the singular points of V is large, then V cannot have a point of large multiplicity, unless V is a cone. As an application, we give an affirmative answer to a conjecture of Dimca and Papadima.
In this series of lectures, I will discuss results for complex hypersurfaces with non-isolated singularities. In Lecture 1, I will review basic definitions and results on complex hypersurfaces, and then present classical material on the Milnor fiber and fibration. In Lecture 2, I will present basic results from Morse theory, and use them to prove some results about complex hypersurfaces, including a proof of Lê's attaching result for Milnor fibers of non-isolated hypersurface singularities. This will include defining the relative polar curve. Lecture 3 will begin with a discussion of intersect…
Resolutions of differential operators of low order for an isolated hypersurface singularity
本文提出新方法研究孤立奇点分次超曲面环的微分算子，构造二阶和三阶微分算子模的极小生成集及自由解消，推广了已有结果，并利用奇点范畴中的矩阵分解实现构造。
Given a nonconstant polynomial map over the reals having an isolated critical point in the origin and with zero locus of positive dimension, we establish a formula for the singular homology groups of a Milnor fibre relative to its boundary.
Geometry of Hypersurfaces with Isolated Singularities
本文研究具有孤立奇点的超曲面中直线的Fano簇，证明过奇点的直线簇的不可约性，并研究多奇点三次超曲面的直线Fano簇。
Zariski's multiplicity conjecture for quasihomogeneous hypersurfaces with non-isolated singularities
本文针对2≤n≤4维拟齐次非孤立奇点超曲面，利用拟齐次正规形推导出由权重和次数决定的重数显式公式，证明重数在拓扑等价下不变，从而证实Zariski重数猜想。
Isolated Hypersurface Singularities May Be Stably Degenerate
证明稳定退化超曲面奇点的存在性，证明要素包括奇点接触轨迹上的混合霍奇结构以及大维度极化阿贝尔簇的模空间是一般型的。
The Nakai Conjecture for isolated hypersurface singularities of modality $\le 2$
本文将Nakai猜想推广到局部整环，并针对模态≤2的孤立超曲面奇点验证了该猜想，推广了已有结果。
We establish birational superrigidity for a large class of singular projective Fano hypersurfaces of index one. In the special case of isolated singularities, our result applies for instance to: (1) hypersurfaces with semi-homogeneous singularities of multiplicity asymptotically bounded by twice the square root of the dimension of the hypersurface, (2) hypersurfaces with isolated singularities whose Tyurina numbers satisfy a similar bound, and (3) hypersurfaces with isolated singularities whose dual variety is a hypersurface of degree sufficiently close to the expected degree.
We recall first the relations between the syzygies of the Jacobian ideal of the defining equation for a projective hypersurface $V$ with isolated singularities and the versality properties of $V$, as studied by du Plessis and Wall. Then we show how the bounds on the global Tjurina number of $V$ obtained by du Plessis and Wall lead to substantial improvements of our previous results on the stability of the reflexive sheaf $T\langle V\rangle$ of logarithmic vector fields along $V$, and on the Torelli property in the sense of Dolgachev-Kapranov of $V$.
Let $(X,0)$ be an isolated hypersurface singularity defined by $ϕ\colon(\mathbb C^n,0)\to(\mathbb C,0)$ and $f\colon(\mathbb C^n,0)\to\mathbb C$ such that the Bruce-Roberts number $μ_{BR}(f,X)$ is finite. We first prove that $μ_{BR}(f,X)=μ(f)+μ(ϕ,f)+μ(X,0)-τ(X,0)$, where $μ$ and $τ$ 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 $(X,0)$ …
We prove stability of logarithmic tangent sheaves of singular hypersurfaces D of the projective space with constraints on the dimension and degree of the singularities of D. As main application, we prove that determinants and symmetric determinants have stable logarithmic tangent sheaves and we describe an open dense piece of the associated moduli space.
We study the boundary $L_t$ of the Milnor fiber for the reduced holomorphic germs $f:(\Bbb C^3,0) \rightarrow (\Bbb C,0)$ having a non-isolated singularity at $0$. We prove that $L_t$ is a graph manifold by using a new technique of carrousel depending on one parameter varying on a circle. Our results enable one to compare the topology of $L_t$ and of the link of the normalization of $f^{-1}(0)$.
Quasi-homogeneous singularities of projective hypersurfaces and Jacobian syzygies
该研究发现射影超曲面Jacobian合冲与奇点性质的一般关系，提出识别拟齐次孤立奇点的新方法，推广了此前关于自由与殆自由平面曲线的研究成果。
On symplectic aspects of $μ$-equivalence of a class of isolated hypersurface singularities
本文证明了孤立超曲面奇点在$\mu$-常数形变下保持Milnor纤维微分同胚类的辛版本，适用于加权齐次多项式奇点。
We investigate the problem of the number of flats simultaneously tangent to several convex hypersurfaces in real projective space from a random point of view. More precisely, we say that smooth convex hypersurfaces $X_1, \ldots, X_{d_{k,n}}\subset \mathbb{R}\textrm{P}^n$, where $d_{k,n}=(k+1)(n-k)$, are in random position if each one of them is randomly translated by elements $g_1, \ldots, g_{d_{k,n}}$ sampled independently and uniformly from the Orthogonal group; we denote by $τ_k(X_1, \ldots, X_{d_{k,n}})$ the average number of $k$-dimensional projective subspaces (flats) which are simultane…
In the paper "Algebraic classification of rational CR structures on topological 5-sphere with transversal holomorphic S^1-action in C^4" (Yau and Yu, Math. Nachrichten 246-247(2002), 207-233), we give algebraic classification of rational CR structures on the topological 5-sphere with transversal holomorphic S^1-action in C^4. Here, algebraic classification of compact strongly pseudoconvex CR manifolds X means classification up to algebraic equivalence, i.e. roughly up to isomorphism of the normalization of the complex analytic variety V which has X as boundary. The problem is intimately relate…
We prove that if $\{f_t\}$ is a family of line singularities with constant Lê numbers and such that $f_0$ is a homogeneous polynomial, then $\{f_t\}$ is equimultiple. This extends to line singularities a well known theorem of A. M. Gabrièlov and A. G. Kušnirenko concerning isolated singularities. As an application, we show that if $\{f_t\}$ is a topologically $\mathscr{V}$-equisingular family of line singularities, with $f_0$ homogeneous, then $\{f_t\}$ is equimultiple. This provides a new partial positive answer to the famous Zariski multiplicity conjecture for a special class of non-isolated…
Let $X$ be a hypersurface in $\mathbb{P}^{4}$ of degree $d$ that has at most isolated ordinary double points. We prove that $X$ is factorial in the case when $X$ has at most $(d-1)^{2}-1$ singular points.
Generic vector fields on isolated complex hypersurface germs
本文研究孤立超曲面奇点上的全纯向量场，推导出全纯向量场存在的全局障碍。通过GSV指数刻画向量场的最小可能指数，并证明其与Tjurina-Greuel数的关系。进一步证明当向量场可延拓到C^{n+1}时，指数相等，且此类延拓构成开密集集。
We describe a multivariable polynomial invariant for certain class of non isolated hypersurface singularities generalizing the characteristic polynomial on monodromy. The starting point is an extension of a theorem due to Le Dung Trang and K.Saito on commutativity of the local fundamental groups of certain hypersurfaces. The description of multivariable polynomial invariants is given in terms of the ideals and polytopes of quasiadjunction generalizing corresponding data used in the study of the homotopy groups of the complements to projective hypersurfaces and Alexander invariants of plane red…
Criteria of isolated weighted homogeneous hypersurface singularities using Logarithmic vector fields
本文证明了da Silva Machado和Seade的猜想：通过存在横截于链环的对数向量场来刻画加权齐次孤立超曲面奇点，并给出了等价形式。
We introduce Hodge ideal spectrum for isolated hypersurface singularities to see the difference between the Hodge ideals and the microlocal $V$-filtration modulo the Jacobian ideal. Via the Tjurina subspectrum, we can compare the Hodge ideal spectrum with the Steenbrink spectrum which can be defined by the microlocal $V$-filtration. As a consequence of a formula of Mustata and Popa, these two spectra coincide in the weighted homogeneous case. We prove sufficient conditions for their coincidence and non-coincidence in some non-weighted-homogeneous cases where the defining function is semi-weigh…
Spectrum of cones of projective hypersurfaces with singularities isolated
本文研究具有孤立奇点的射影超曲面锥的谱，利用奇点谱数与全局次数给出描述，并推广至非约化平面曲线及普通奇点情形。
Strong monodromy conjecture for defining polynomials of projective hypersurfaces having only weighted homogeneous isolated singularities
研究显示，在特定条件下，定义多项式的强monodromy猜想可通过已知结果和Denef-Loeser公式推导，且在某些情况下可能不存在反例。
Isolated hypersurface singularities, spectral invariants, and quantum cohomology
本文利用 Floer 理论的谱不变量研究孤立超曲面奇点与量子上同调环的关系，在半单假设下证明了奇点必为 $A_m$ 型，并揭示了 $A_m$-配置对 Hofer 几何的影响及 Dehn 扭曲减小谱不变量的结论。
The moduli space of generalized deformations of a Calabi-Yau hypersurface is computed in terms of the Jacobian ring of the defining polynomial. The fibers of the tangent bundle to this moduli space carry algebra structures, which are identified using subalgebras of a deformed Jacobian ring.
We prove the non-rationality of a double cover of $\mathbb{P}^{n}$ branched over a hypersurface $F\subset\mathbb{P}^{n}$ of degree $2n$ having isolated singularities such that $n\ge 4$ and every singular points of the hypersurface $F$ is ordinary, i.e. the projectivization of its tangent cone is smooth, whose multiplicity does not exceed $2(n-2)$.
This article presents the theory of focal locus applied to the hypersurfaces in the projective space which are (finitely) covered by linear spaces and such that the tangent space is constant along these spaces.
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