This paper applies the multiplicity polar theorem to the study of hypersurfaces with non-isolated singularities. The multiplicity polar theorem controls the multiplicity of a pair of modules in a family by relating the multiplicity at the special fiber to the multiplicity of the pair at the general fiber. It is as important to the study of multiplicities of modules as the basic theorem in ideal theory which relates the multiplicity of an ideal to the local degree of the map formed from the generators of a minimal reduction. In fact, as a corollary of the theorem, we show here that for M a submodule of finite length of a free module F over the local ring of an equidimensional complex analytic germ, that the number of points at which a generic perturbation of a minimal reduction of M is not equal to F, is the multiplicity of M.
In this paper we apply this result to the study of stratification conditions on families of hypersurfaces, obtaining the first set of invariants giving necessary and sufficient conditions for the A_f condition for hypersurfaces with non-isolated 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 …
In this paper, we investigate the multiplicities and the log canonical thresholds of abelian quotient complete intersection singularities in term of the special datum. Moreover we give bounds of the multiplicity of abelian quotient complete intersection singularities.
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.
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.
Assume that there exists a hypersurface singularity which cannot be resolved by iterated monoidal transformations in positive characteristic. We show that in the set of defining functions of hypersurface singularities which cannot be resolved, we can find a function satisfying very strong conditions. By these conditions we may be able to deduce a contradiction under the above assumption. Besides, we introduce fundamental concepts for the study of resolution of singularities of germs such as space germs, iterated analytic monoidal transforms with a normal crossing, Weierstrass representations, …
This paper studies the singularities of jet schemes of homogeneous hypersurfaces of general type. We obtain the condition of the degree and the dimension for the singularities of the jet schemes to be of dense $F$-regular type. This provides us with examples of singular varieties whose $m$-jet schemes have rational singularities for every $m$.
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 main purpose of this article is to lay the foundations for a classification of isolated hypersurface singularities in positive characteristic. Although our article is in the spirit of Arnol'd who classified real an complex hypersurfaces in the 1970's with respect to right equivalence, several new phenomena occur in positive characteristic. Already the notion of isolated singularity is different for right resp. contact equivalence over fields of characteristic other than zero. The heart of this paper consists of the study of different notions of non-degeneracy and the associated piecewise f…
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.
We study singularities f in K[[x_1,...,x_n]] over an algebraically closed field K of arbitrary characteristic with respect to right respectively contact equivalence, and we establish that the finiteness of the Milnor respectively the Tjurina number is equivalent to finite determinacy. We give improved bounds for the degree of determinacy in positive characteristic. Moreover, we consider different non-degeneracy conditions of Kouchnirenko, Wall and Beelen-Pellikaan in positive characteristic, and we show that planar Newton non-degenerate singularities satisfy Milnor's formula mu=2 delta-r+1. Th…
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.
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).
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.
It was conjectured that multiplicity of a singularity is bi-Lipschitz invariant. We disprove this conjecture, constructing examples of bi-Lipschitz equivalent complex algebraic singularities with different values of multiplicity.
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.
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 provide combinatorial/topological formula for the multiplicity of a complex analytic normal surface singularity whenever the analytic structure on the fixed topological type is generic.
We give the multiplicity of terminal singularities on threefolds by simple calculation. Then we obtain the best inequalities for the multiplicity and the index. By using this, we can improve the boundedness number of terminal weak Q-Fano 3-folds in [KMMT, Theorem 1.2]. Furthermore, we can extended [K, Theorem 3.6] for Fujita freeness conditions to nonhypersurface terminal singularities.
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 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.
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.
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 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 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…
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.
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…
This paper is the first part of a two part paper which introduces the study of the Whitney Equisingularity of families of Symmetric determinantal singularities. This study reveals how to use the multiplicity of polar curves associated to a generic deformation of a singularity to control the Whitney equisingularity type of these curves.
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…
