Lê Dũng Tráng, David B. Massey
Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is one-dimensional, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber of $f$. This result has an interesting implication on the structure of the vanishing cycles in the category of perverse sheaves.
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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.
In this paper the detailed classification of three-dimensional exceptional canonical hypersurface singularities which don't satisfy the condition of well-formedness is given. This result completes the classification of three-dimensional exceptional log canonical hypersurface singularities started in [4] (math.AG 0201025).
Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non--degenerate hypersurface singularities.
The log canonical thresholds of irreducible quasi-ordinary hypersurface singularities are computed, using an explicit list of pole candidates for the motivic zeta function found by the last two authors.
Bounding Projective Hypersurface Singularities
本研究比较了多种用奇点霍奇理论谱约束固定次数射影超曲面孤立奇点的方法，提出基于仿射锥原点非孤立奇点谱的新方法，并将其显式公式与Varchenko界建立关联。
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).
Tjurina Number Jumps and Unimodal Hypersurface Singularities in Positive Characteristic
本文通过扩展Tjurina数研究超曲面奇点的模性，证明Tjurina数跃迁必然提升模性，并在正特征p>3下对单模孤立超曲面奇点进行完整分类。
We prove a "strong" Durfee-type inequality for isolated hypersurface surface singularities, which implies Durfee's strong conjecture for such singularities with non-negative topological Euler number of the exceptional set of the minimal resolution.
C.T.C. Wall and the first author discovered an extension of Arnold's strange duality embracing on one hand series of bimodal hypersurface singularities and on the other, isolated complete intersection singularities. In this paper, we derive this duality from the mirror symmetry and the Berglund-Hübsch transposition of invertible polynomials.
This paper studies hypersurface exceptional singularities in $\mathbb C^n$ defined by non-degenerate function. For each canonical hypersurface singularity, there exists a weighted homogeneous singularity such that the former is exceptional if and only if the latter is exceptional. So we study the weighted homogeneous case and prove that the number of weights of weighted homogeneous exceptional singularities are finite. Then we determine all exceptional singularities of the Brieskorn type of dimension 3.
This text is the write-up of a series of lectures on the asymptotic mixed Hodge theory of isolated hypersurface singularities, held at the Third Latin American school on Algebraic Geometry and its applications (ELGA 3) in Guanajuato, Mexico, in august 2017. Its focus is on the classical application of the semi-continuity of the spectrum due to Varchenko and Steenbrink to the problem of bounding the possible singularities on a projective hypersurface.
We prove a formula for the polar degree of projective hypersurfaces in terms of the Milnor data of the singularities, extending to 1-dimensional singularities the Dimca-Papadima result for isolated singularities. We discuss the semi-continuity of the polar degree in deformations, and we classify the homaloidal cubic surfaces with 1-dimensional singular locus. Some open questions are pointed out along the way.
In this paper we give a positive answer to a question of Nash concerning the arc space of a singularity, for the class of quasi-ordinary hypersurface singularities, extending to this case previous results and techniques of Shihoko Ishii.
We give a survey on some aspects of the topological investigation of isolated singularities of complex hypersurfaces by means of Picard-Lefschetz theory. We focus on the concept of distinguished bases of vanishing cycles and the concept of monodromy.
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.
Linearly projecting smooth projective varieties provides a method of obtaining hypersurfaces birational to the original varieties. We show that in low dimension, the resulting hypersurfaces only have Du Bois singularities. Moreover, we conclude that these Du Bois singularities are in fact semi log canonical. However, we demonstrate the existence of counterexamples in high dimension -- the generic linear projection of certain varieties of dimension 30 or higher is neither semi log canonical nor Du Bois.
We explore the existence of irreducible and reducible arc-sections in an irreducible hypersurface singularity germ along finite projections. In particular we provide examples of irreducible isolated hypersurface singularities for which no irreducible arc-sections exist, and show that reducible ones always exist. Moreover, we give an algorithm to check if a given projection allows irreducible arc-sections, and find them if they exist.
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。
We introduce braid monodromy for the discriminant hypersurface in versal unfoldings of hypersurface singularities. Our objective is then to compute this invariant for singularities of Brieskorn Pham type: First we consider the unfolding by linear monomials in some detail and relate it to the versal unfolding. Next we present the A_n case in our framework. The two main chapters then provide the result in case of two variables and the inductive step to all higher dimensions. We finish with a presentation of the fundamental group of the discriminant complement.
We give partial answers to a metric version of Zariski's multiplicity conjecture. In particular, we prove the multiplicity of complex analytic surface (not necessarily isolated) singularities in $\mathbb{C}^3$ is a bi-Lipschitz invariant.
For a weighted quasihomogeneous two dimensional hypersurface singularity, we define a smoothing with unipotent monodromy and an isolated graded normal singularity. We study the natural weighted blow up of both the smoothing and the surface. In particular, we describe our construction for the quasihomogeneous singularities of type I, the 14 unimodal exceptional singularities and we relate it to their stable replacement.
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.
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.
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。
We introduce a Lie algebra of initial terms of logarithmic vector fields along a hypersurface singularity. Extending the formal structure theorem in [GS06, Thm. 5.4], we show that the completely reducible part of its linear projection lifts formally to a linear Lie algebra of logarithmic vector fields. For quasihomogeneous singularities, we prove convergence of this linearization. We relate our construction to the work of Hauser and M"uller [M"ul86, HM89] on Levi subgroups of automorphism groups of singularities, which proves convergence even for algebraic singularities. Based on the initial L…
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。
Examples of Hirzebruch-Milnor classes of projective hypersurfaces detecting higher du Bois or rational singularities
本文表明经典 Hirzebruch-Milnor 类仅在特殊情形下能检测射影超曲面的更高 du Bois 或有理奇点，并澄清了作者早期论文中的若干要点。
We introduce two generalizations of Newton-non-degenerate (Nnd) singularities of hypersurfaces. Roughly speaking, an isolated hypersurface singularity is called topologically Newton-non-degenerate (tNnd) if the local embedded topological singularity type can be restored from a collection of Newton diagrams (for some coordinate choices). A singularity that is not tNnd is called essentially Newton-degenerate. For plane curves we give an explicit characterization of tNnd singularities; for hypersurfaces we provide several examples. Next, we treat the question: whether Nnd or tNnd is a property of…
Higher Du Bois and higher rational singularities of hypersurfaces
本研究针对全局定义的超曲面，利用消失循环混合霍奇模的Hodge滤过给出高阶Du Bois与高阶有理奇点的等价刻画，并提出基于谱Hirzebruch-Milnor特征类的同调检测准则。
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…
Birationally rigid hypersurfaces with quadratic singularities of low rank
本文证明了满足特定一般位置条件、具有低秩二次奇点的$M$次超曲面是双有理超刚性Fano簇，并给出了其补集的余维数下界。
From the degree zero part of logarithmic vector fields along an algebraic hypersurface singularity we indentify the maximal multihomogeneity of a defining equation in form of a maximal algebraic torus in the embedded automorphism group. We show that all such maximal tori are conjugate and in one-to-one correspondence to maxmimal tori in the degree zero jet of the embedded automorphism group. The result is motivated by Kyoji Saito's characterization of quasihomogeneity for isolated hypersurface singularities and extends its formal version and a result of Hauser and Mueller.
Stability and singularities of relative hypersurfaces
本文研究曲线上的相对超曲面，证明纤维的不稳定性条件，给出对数规范阈值上界，并与中山Zariski分解信息比较。
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.
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。
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.
Kähler's paper "Über die Verzweigung einer algebraischen Funktion zweier Veränderlichen in der Umgebung einer singulären Stelle" offered a more perceptual view of the link of a complex plane curve singularity than that provided shortly before by Brauner. Kähler's innovation of using a "square sphere" became standard in the toolkit of later researchers on singularities. We describe his contribution and survey developments since then, including a brief discussion of the topology of isolated hypersurface singularities in higher dimension.
The paper is motivated on the open problem of resolution of singularities in positive characteristic. The aim is to present a form of induction which is different from that used by Hironaka. In characteristic zero induction is formulated by restriction to smooth hypersurfaces (hypersurfaces of maximal contact). Our alternative approach, introduced here, replaces restrictions to smooth sub-schemes by generic projections on smooth schemes of smaller dimension. We also introduce a generalization of the discriminant, and our result makes use of the elimination theory. In the case of fields of char…
We determine conditions on q for the nonexistence of deep holes of the standard Reed-Solomon code of dimension k over F_q generated by polynomials of degree k+d. Our conditions rely on the existence of q-rational points with nonzero, pairwise-distinct coordinates of a certain family of hypersurfaces defined over F_q. We show that the hypersurfaces under consideration are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of these hypersurfaces, from which the existence of q-rational p…
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…
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 $V$.
In the present paper MAPLE subroutines computing Milnor and Tyurina numbers of an isolated algebraic hypersurface singularity are presented and described in detail. They represents examples, and perhaps the first ones, of a MAPLE implementation of local monomial ordering. As an application, the last section is devoted to writing down equations of algebraic stratifications of Kuranishi spaces of simple Arnol'd singularities: they geometrically represents, by means of inclusions of algebraic subsets, the partial ordering on classes of simple singularities induced by the adjacency relation.
Higher singularities for hypersurfaces
本文研究了复超曲面的高阶奇异性，证明了在特定条件下，若某个微分模同构，则所有低阶微分模也同构，并讨论了正特征下的类似结论。
Pencils of projective hypersurfaces, Griffiths heights and geometric invariant theory. II Hypersurfaces with semihomogeneous singularities
本文确立了具有半齐次奇点的射影超曲面束的中间维上同调的稳定 Griffiths 高度公式，通过构造有限覆盖以获得半稳定光滑模型，成功规避了确定奇点纤维初等指数的难题。
All varieties, extremal contractions, singularities are divided on exceptional and non-exceptional ones. Roughly speaking, there are the infinite families of non-exceptional varieties, extremal contractions or singularities and only the finite number of the types of exceptional ones. This subdivision is well demonstrated by the example of Du Val singularities. There are two infinite series of non-exceptional singularities: $A_n$ and $D_n$ and only three types of exceptional ones: $E_6$, $E_7$ and $E_8$. Also the importance of exceptionality phenomenon follows from the next observation: A). If …
We show that if a homogeneous polynomial $f$ in $n$ variables has Waring rank $n+1$, then the corresponding projective hypersurface $f=0$ has at most isolated singularities, and the type of these singularities is completely determined by the combinatorics of a hyperplane arrangement naturally associated with the Waring decomposition of $f$. We also discuss the relation between the Waring rank and the type of singularities on a plane curve, when this curve is defined by the suspension of a binary form, or when the Waring rank is 5.
We show the existence of ($1+\frac{w_2}{w_3}$)-fast loops on semi-weighted homogeneous hypersurface singularities with weights $w_1\geq w_2>w_3$. In particular we show that semi-weighted homogeneous hypersurface singularities have metrical conical structure only if its two low weights are equal.
We give an upper bound for the minimal discrepancies of hypersurface singularities. As an application, we show that Shokurov's conjecture is true for log-terminal threefolds.
We give a full exceptional collection in the triangulated category of singularities in the sense of Orlov for a hypersurface singularity of Fermat type, and discuss its relation with homological mirror symmetry for simple elliptic hypersurface singularities.
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 …
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.
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。
Singularities of totally invariant hypersurfaces of endomorphisms
本文证明复射影流形极化自同态的全不变超曲面的奇点轨迹各分支余维数为一，并由此推出$\mathbb P^4$上全不变除子的线性性。
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…
The Quotient of Milnor Number by Tjurina Number of Hypersurface Singularities in Arbitrary Characteristic
本文通过Hilbert-Samuel乘数等工具，建立了任意特征下孤立hypersurface奇异性的广义Milnor数与Tjurina数商的精确上界，并推导出特征零下的上界，证明了孤立表面奇点的μ/τ≤3/2，回应了Almirón猜想并验证了Durfee猜想的弱形式。
Suppose that $f$ defines a singular, complex affine hypersurface. If the critical locus of $f$ is one-dimensional at the origin, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber, $F_{f, \mathbf 0}$, of $f$ at the origin, with either integral or $\mathbb Z/p\mathbb Z$ coefficients. If the critical locus of $f$ has arbitrary dimension, we show that the smallest possibly non-zero reduced Betti number of $F_{f, \mathbf 0}$ completely determines if $f$ defines a family of isolated singularities, over a smooth base, with constant Milnor number. This result has a n…
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。
Let $\mathbb{K}$ be an algebraically closed field of characteristic zero, and $V$ a hypersurface defined by an irreducible polynomial $f$ with coefficients in $\mathbb{K}$ . In this article we prove that an Embedded Deformation of $V$ which admits a Simultaneous Embedded Resolution induces, under certain mild conditions, a deformation of the reduced scheme associated to the space of $m$-jets $V_m$, $m\geq 0$. An example of an Embedded Deformation of $V$ which admits a Simultaneous Embedded Resolution is a $Γ(f)$-deformation of $V$, where $V$ has at most one isolated singularity, and $f$ is non…
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 $f^{-1}(0)$.
We show the existence of surfaces of degree $d$ in $\dP^3(\dC)$ with approximately ${3j+2\over 6j(j+1)} d^3$ singularities of type $A_j, 2\le j\le d-1$. The result is based on Chmutov's construction of nodal surfaces. For the proof we use plane trees related to the theory of Dessins d'Enfants. Our examples improve the previously known lower bounds for the maximum number $μ_{A_j}(d)$ of $A_j$-singularities on a surface of degree $d$ in most cases. We also give a generalization to higher dimensions which leads to new lower bounds even in the case of nodal hypersurfaces in $\dP^n, n\ge5$. To conc…
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