Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse intersection. A problem with the multiplicity of an ideal or module is that it is only defined for modules and ideals of finite colength.
In this paper we use pairs of modules and their multiplicities as a way around this difficulty. A key tool is the multiplicity-polar theorem which applies to families of pairs of modules, linking the multiplicity at a general pair of the family to the multiplicity of the pair at the special fiber.
The specific problem we work on is:
Suppose X,0 is an equidimensional complex analytic set in C^n, and T^*_X(C^n) is the closure of the conormal vectors to the smooth part of X, s, a section of T^*(C^n), the cotangent bundle of C^n restricted to X. Suppose the intersection of the image of s, denoted im(s), with T^*_X(C^n) is isolated. Then we calculate the intersection multiplicity im(s)\cdot T^*_X(C^n) using the multiplicity of pairs. It is easy to see that if X=C^n and s(x)=df(x), where f has an isolated singularity at the origin, then this number is the Milnor number of f.
Variations of this number appear in several different situations, such as, the theory of differential forms on singular spaces, the theory of D-modules, the Milnor fiber of a function with an isolated singularity, the A_f stratification condition of Thom, and the theory of the Euler invariant. We discuss the application of our result to all of these situations.
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.
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 subm…
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 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.
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.
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.
Let $(R, \mathfrak m)$ be a Noetherian local ring and $I$ a $\mathfrak m$-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of $I$. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if $R$ has sufficiently nice singularities. We verify the inequality for regular local rings in all dimensions, for rational singularity in dimension $2$, and cDV singularities in dimension $3$. In addition, we can classify when the inequality always hold for a Cohen-Macaulay…
The famous $4n^2$-inequality is extended to generic complete intersection singularities: it is shown that the multiplicity of the self-intersection of a mobile linear system with a maximal singularity is higher than $4n^2μ$, where $μ$ is the multiplicity of the singular point.
We prove the birational rigidity of Fano complete intersections of index 1 with a singular point of high multiplicity, which can be close to the degree of the variety. In particular, the groups of birational and biregular automorphisms of these varieties are equal, and they are non-rational. The proof is based on the techniques of the method of maximal singularities, the generalized $4n^2$-inequality for complete intersection singularities and the technique of hypertangent divisors.
Soient $f,g\colon (\hbox{åC}^n,0) \to (\hbox{åC},0)$ des germes de fonctions holomorphes réduits. Nous montrons que $f$ et $g$ ont la même multiplicité en 0 si et seulement s'il existe des germes réduits $f'$ et $g'$ analytiquement équivalents à $f$ et $g$, respectivement, tels que $f'$ et $g'$ satisfassent une inégalité du type de Rouché par rapport à un `petit' cercle générique autour de~0. Comme application, nous donnons une reformulation de la question de Zariski sur la multiplicité et une réponse partielle positive à celle--ci.
We define the Segre numbers of an ideal as a generalization of the multiplicity of an ideal of finite colength. We prove generalizations of various theorems involving the multiplicity of an ideal such as a principle of specialization of integral dependence, the Rees-Böger theorem, and the formula for the multiplicity of the product of two ideals. These results are applied to the study of various equisingularity conditions, such as Verdier's condition W, and conditions $A_f$ and $W_f$.
We recover the Newton diagram (modulo a natural ambiguity) from the link for any surface hypersurface singularity with non-degenerate Newton principal part whose link is a rational homology sphere. As a corollary, we show that the link determines the embedded topological type, the Milnor fibration, and the multiplicity of such a germ. This proves (even a stronger version of) Zariski's Conjecture about the multiplicity for such a singularity.
The purpose of this paper is to show how Rees algebras can be applied in the study of singularities embedded in smooth schemes over perfect fields. In particular, we will study situations in which the multiplicity of a hypersurface is a multiple of the characteristic. As another application, here we indicate how the use of these algebras has trivialized local-global questions in resolution of singularities over fields of characteristic zero.
In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root with a prescribed multiplicity structure. More precisely, given a polynomial system f $=(f\_1, \ldots, f\_N)\in C[x\_1, \ldots, x\_n]^N$, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of $f$ such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so called inverse system that describes th…
Resolution of singularities of varieties over fields of characteristic zero can be proved by using the multiplicity as main invariant. The proof of this result leads to new questions in positive characteristic. We discuss here results which follow by induction on the dimension of the varieties. Fix a variety $X^{(d)}$ of dimension $d$ over a {\em perfect field} $k$ or, more generally, a pure dimensional scheme of finite type over $k$. Fix a closed point $x\in X^{(d)}$ of multiplicity $e>1$. Define a local simplification of the multiplicity at $x\in X^{(d)}$ as a proper birational map, say $X^{…
In 1971, Zariski proposed some questions in Theory of Singularities. One of such problems is the so-called, nowadays, Zariski's multiplicity conjecture. In this work, we consider the version of this conjecture for families. We answer positively Zariski's multiplicity conjecture for a special class of non-isolated singularities.
We study the multiplicity of the jumping numbers of an $\mathfrak m$-primary ideal $\mathfrak a$ in a two-dimensional local ring with a rational singularity. The formula we provide for the multiplicities leads to a very simple and efficient method to detect whether a given rational number is a jumping number. We also give an explicit description of the Poincaré series of multiplier ideals associated to $\mathfrak a$ proving, in particular, that it is a rational function.
Given a family of pairs of modules parametrised by a smooth space Y, the Multiplicity-Polar Theorem relates the multiplicity of the pair of modules at a special point of the parameter to the multiplicity of the pair at a generic point. This theorem is proved in this paper (Corollary 1.4). The applications of the Multiplicity-Polar theorem are of two types. In the first we construct a deformation so that we can understand the significance of the multiplicity of the pair, or show we can use the multiplicity of the pair to describe some geometric behavior. Applications of this type are the geomet…
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 $2(n-2)$.
We give a new differential proof of our result on the maximal rank of generic unions of points of multiplicity two in projective space in degrees greater than five. This simplifies somewhat our proof of the Waring conjecture.
Let $d,m_1,...,m_r$ be ($r+1$) positive integers, and $P_1,...,P_r$ be $r$ general points in the projective plane ; let $m$ be a positive integer. We prove that there exists a bound $d_0(m)$ such that : If $m_i < m$ ($0<i<r+1$), and $d > d_0(m)$ then the linear system $L$ of plane curves of degree $d$ having a multiplicity at least $m_i$ at each point $P_i$ has the expected dimension ; moreover, if $L$ is not empty, there exists an irreducible plane curve of degree $d$, smooth away from the $r$ points $P_i$, and having an ordinary singularity of the prescribed multiplicity $m_i$ at each point …
Hilbert-Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptions. A natural question is for what singular rings these invariants are closest to one. For Hilbert--Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for F-signature and revisit lower bounds on Hilbert--Kunz multiplicity.
Let (R,m) be an n-dimensional regular local ring, essentially of finite type over a field of characteristic zero. In this paper we study the relationship between the singularities of the scheme defined by an m-primary ideal I of R and the multiplier ideals J(I^c), with c varying among the positive rational numbers. In particular, we prove that, for every nonnegative integer k, the Samuel multiplicity of I is greater or equal to (n+k)^n/c^n whenever J(I^c) is contained in the (k+1)-th power of the maximal ideal m. This formula generalizes an inequality on log canonical thresholds previously obt…
We introduce an upper semi-continuous function that stratifies the highest multiplicity locus of a hypersurface in arbitrary characteristic (over a perfect field). The blow-up along the maximum stratum defined by this function leads to a form of simplification of the singularities, also known as a reduction to the monomial case.
The objective of this paper is to discuss invariants of singularities of algebraic schemes over fields of positive characteristic, and to show how they yield the simplification of singularities. We focus here on invariants which arise in an inductive manner, namely by successive elimination of variables. When applied to hypersurface singularities they lead us to a refinement of the notion of multiplicity. The main theorem proves that, under some numerical conditions expressed by these invariants, singularities can be simplified by blowups at centers prescribed by this refinement.
We show that the Euler number of the compactified Jacobian of a rational curve $C$ with locally planar singularities is equal to the multiplicity of the $δ$-constant stratum in the base of a semi-universal deformation of $C$. In particular, the multiplicity assigned by Yau, Zaslow and Beauville to a rational curve on a K3 surface $S$ coincides with the multiplicity of the normalisation map in the moduli space of stable maps to $S$.
In \cite{NNM} the author with A. Némethi computed the multiplicity of generic surface singularities, the formula is purely topological computable from the resolution graph of the surface singularity. In the present paper we extend the results partly to the relative case, when there is a pair of resolution graphs $\mathcal{T}_1 \subset \mathcal{T}$, a fixed singularity $\tX_1$ with resolution graph $\mathcal{T}_1$, a relatively generic singularity $\tX$ corresponding to the subsingularity $\tX_1$ with resolution graph $\mathcal{T}$. We determine the base points of the natural line bundles (unde…
The main goal of the paper is to find an effective estimation for the minimal number of generic points in $\mathbb K^2$ for which the basis for Hermite interpolation consists of the first $\ell$ terms (with respect to total degree ordering). As a result we prove that the space of plane curves of degree $d$ having generic singularities of multiplicity $\leq m$ has the expected dimension if the number of low order singularities (of multiplicity $k\leq12$) is greater then some $r(m,k)$. Additionally, the upper bounds for $r(m,k)$ are given.
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.
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.
We give several constructions of bicuspidal rational complex projective plane curves, and list the Newton pairs and the multiplicity sequences of the singularities on the resulting curves. Although the existence of some of the listed cusp configurations is already proved in the literature, the advantage of our approach is that on one hand we explicitly state the Newton pairs and the multiplicity sequences of the singularities on the curves, on the other hand, we also give birational transformations transforming a conic or a bicuspidal curve with simple equation to these rational bicuspidal cur…
We study the equisingularity of a family of function germs $\{f_t\colon(X_t,0)\to (\mathbb{C},0)\}$, where $(X_t,0)$ are $d$-dimensional isolated determinantal singularities. We define the $(d-1)$th polar multiplicity of the fibers $X_t\cap f_t^{-1}(0)$ and we show how the constancy of the polar multiplicities is related to the constancy of the Milnor number of $f_t$ and the Whitney equisingularity of the family.
This article, based on the talk given by one of the authors at the Pierrettefest in Castro Urdiales in June 2008, is an overview of a number of recent results on the polar invariants of plane curve singularities.
We study classical invariants for plane curve singularities $f\in K[[x,y]]$, $K$ an algebraically closed field of characteristic $p\geq 0$: Milnor number, delta invariant, kappa invariant and multiplicity. It is known, in characteristic zero, that $μ(f)=2δ(f)-r(f)+1$ and that $κ(f)=2δ(f)-r(f)+\mathrm{mt}(f)$. For arbitrary characteristic, Deligne prove that there is always the inequality $μ(f)\geq 2δ(f)-r(f)+1$ by showing that $μ(f)-\left( 2δ(f)-r(f)+1\right)$ measures the wild vanishing cycles. By introducing new invariants $γ,\tildeγ$, we prove in this note that $κ(f)\geq γ(f)+\mathrm{mt}(f)…
We give a sharp upper bound on the multiplicity of a fake weighted projective space with at worst canonical singularities. This is equivalent to giving a sharp upper bound on the index of the sublattice generated by the vertices of a lattice simplex containing only the origin as an interior lattice point. We also completely characterise when equality occurs and discuss related questions and conjectures.
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…
In this paper, we study polar quotients and Łojasiewicz exponents of plane curve singularities, which are {\em not necessarily reduced}. We first show that the polar quotients is a topological invariant. We next prove that the Łojasiewicz gradient exponent can be computed in terms of the polar quotients, and so it is also a topological invariant. As an application, we give effective estimates of the Łojasiewicz exponents in the gradient and classical inequalities of polynomials in two (real or complex) variables.
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 study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in C^4, we obtain a Lê-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the 1- form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last se…
We study lattice polarizations of five exceptional pairs of families of K3 surfaces obtained via compactifications of strange dual pairs of bimodal singularities. We show that the polarizations induced by embedding these paired families in paired toric varieties obtained from polar dual reflexive polytopes cannot be mirror lattices and identify mirror sublattices.
Burban-Drozd showed that the degenerate cusp singularities have tame Cohen-Macaulay representation type, and classified all indecomposable Cohen-Macaulay modules over them. One of their main example is the non-isolated singularity $W=xyz$. On the other hand, Abouzaid-Auroux-Efimov-Katzarkov-Orlov showed that $W=xyz$ is mirror to a pair of pants. In this paper, we investigate homological mirror symmetry of these indecomposable Cohen-Macaulay modules for $xyz$. Namely, we show that closed geodesics (with a flat $\mathbb{C}$-bundle) of a hyperbolic pair of pants have a one-to-one correspondence w…
This paper addresses a very classical topic that goes back at least to Plücker: how to understand a plane curve singularity using its polar curves. Here, we explicitly construct the singular points of a plane curve singularity directly from the weighted cluster of base points of its polars. In particular, we determine the equisingularity class (or topological equivalence class) of a germ of plane curve from the equisingularity class of generic polars and combinatorial data about the non-singular points shared by them.
The problem of resolution of singularities in positive characteristic can be reformulated as follows: Fix a hypersurface $X$, embedded in a smooth scheme, with points of multiplicity at most $n$. Let an $n$-sequence of transformations of $X$ be a finite composition of monoidal transformations with centers included in the $n$-fold points of $X$, and of its successive strict transforms. The open problem (in positive characteristic) is to prove that there is an $n$-sequence such that the final strict transform of $X$ has no points of multiplicity $n$ (no $n$-fold points). In characteristic zero, …
In the present article we work out a relative setup of generic structures on surface singularities. We fix an analytic type on a subgraph of a rational homology sphere resolution graph $\mathcal{T}$ and we choose a relatively generic normal surface singularity $\tX$ with resolution graph $\mathcal{T}$. We provide formulae for the geometric genus and the analytical Poincaré series of $\tX$. We determine the base point structure of natural line bundles on $\tX$ and give a lower bound on the multiplicity of $\tX$ which is expected to be sharp. We prove similar results about cohomology numbers of …
A quasi-ordinary polynomial is a monic polynomial with coefficients in the power series ring such that its discriminant equals a monomial up to unit. In this paper we study higher derivatives of quasi-ordinary polynomials, also called higher order polars. We find factorizations of these polars. Our research in this paper goes in two directions. We generalize the results of Casas-Alvero and our previous results on higher order polars in the plane to irreducible quasi-ordinary polynomials. We also generalize the factorization of the first polar of a quasi-ordinary polynomial (not necessary irred…
