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 with the same integral closure (resp. the projections associated). We also show that our theorem, applied in two special cases, on the one hand completes a theorem by Snoussi on the limits of tangent hyperplanes, and on the other hand gives an algebraic $μ$-constant theorem in linear families of planes curves.
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…
Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms of the associated primes of select powers of $I$ and of $J$. We record two applications. First, in case the field is algebraically closed, we construct primary decompositions for powers of $I+J$ from primary decompositions for powers of $I$ and $J$. Separately, we attack the persistence problem for associated primes of powers of an ideal in case one of $I$ or…
In 1936, Krull asked if the integral closure of a primary ideal is still primary. Fifty years later, Huneke partially answered this question by giving a primary polynomial ideal whose integral closure is not primary in a regular local ring of characteristic $p=2$. We provide counterexamples to Krull's question regarding polynomial rings with any characteristics. We also find that the Jacobian ideal $J$ of the polynomial $f = x^6 + y^6 + x^4 z t + z^3$ given by Briançon and Speder in 1975 is a counterexample to Krull's question. Let $V_1$ be the hypersurface defined by $f = 0$ and $V_2$ be its …
Let $G$ be a flat finite-type group scheme over a scheme $S$, and $X$ a noetherian $S$-scheme on which $G$-acts. We define and study $G$-prime and $G$-primary $G$-ideals on $X$ and study their basic properties. In particular, we prove the existence of minimal $G$-primary decomposition and the well-definedness of $G$-associated $G$-primes. We also prove a generalization of Matijevic-Roberts type theorem. In particular, we prove Matijevic-Roberts type theorem on graded rings for $F$-regular and $F$-rational properties.
An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in a polynomial ring. We characterize primary ideals in terms of PDE, punctual Hilbert schemes, relative Weyl algebras, and the join construction. Solving the PDE described by a primary ideal amounts to computing Noetherian operators in the sense of Ehrenpreis and Palamodov. We develop new algorithms for this task, and we present efficient implementations.
In this article, we obtain two sharp equality conditions in the restriction formula on complex singularity exponents: an equality between the codimension of the zero variety of related multiplier ideal sheaves and the relative codimension of the restriction of the variety on the submanifold (in the restriction formula); an equivalence between the transversality (between the variety and the submanifold) and the regularity of the restriction of the variety. As applications, we present sharp equality conditions in the fundamental subadditivity property on complex singularity exponents.
We provide explicit combinatorial descriptions of the primary components of codimension two lattice basis ideals. As an application, we compute the set of parameters for which a bivariate Horn system of hypergeometric differential equations is holonomic.
This paper deals with the study of the behaviour of the value semigroup of a curve singularity define over a global field reduced modulo a maximal ideal. We also define a global zeta function of the curve by means of motivic integration over a suitable ring of adéles, whose reduction modulo a maximal ideal will coincide with some already known zeta functions of the singularity.
Earlier the authors considered and, in some cases, computed Poincare series of two sorts of multi-index filtrations on the ring of germs of functions on a complex (normal) surface singularity (in particular on the complex plane). A filtration from the first class was defined by a curve (with several branches) on the surface singularity. The other one (so called divisorial filtration) was defined by a set of components of the exceptional divisor of a modification of the surface singularity. Here we define a filtration corresponding to an ideal or to a set of ideals in the ring of germs of funct…
We prove that certain roots of the Bernstein-Sato polynomial (i.e. b-function) are jumping coefficients up to a sign, showing a partial converse of a theorem of L. Ein, R. Lazarsfeld, K.E. Smith, and D. Varolin. We also prove that certain roots are determined by a filtration on the Milnor cohomology, generalizing a theorem of B. Malgrange in the isolated singularity case. This implies a certain relation with the spectrum which is determined by the Hodge filtration, because the above filtration is related to the pole order filtration. For multiplier ideals we prove an explicit formula in the ca…
We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many classical examples from algebraic geometry, and it has numerous applications within and beyond pure mathematics. The ideals defining toric varieties are precisely the binomial prime ideals. Our main results concern primary decomposition: If $I$ is a binomial ideal then the radical, associated primes, and isolated primary components of $I$ are again binomial, and $I$ admits primary decompositions in terms …
Suppose that $R$ is an excellent local domain with maximal ideal $m_R$. The theory of multiplicities and mixed multiplicities of $m_R$-primary ideals extends to (possibly non Noetherian) filtrations of $R$ by $m_R$-primary ideals, and many of the classical theorems for $m_R$-primary ideals continue to hold for filtrations. The celebrated theorems involving inequalities continue to hold for filtrations, but the good conclusions that hold in the case of equality for $m_R$-primary ideals do not hold for filtrations. In this article, we consider multiplicities and mixed multiplicities of $R$ by $m…
Let $(A,\frak m)$ be an excellent normal local ring with algebraically closed residue class field. Given integrally closed $\frak m$-primary ideals $I\supset J$, we show that there is a composition series between $I$ and $J$, by integrally closed ideals only. Also we show that any given integrally closed $\fm$-primary ideal $I$, the family of integrally closed ideals $J\subset I, l_A(I/J)=1$ forms an algebraic variety with dimension $\dim A -1$.
We introduce differential primary decompositions for ideals in a commutative ring. Ideal membership is characterized by differential conditions. The minimal number of conditions needed is the arithmetic multiplicity. Minimal differential primary decompositions are unique up to change of bases. Our results generalize the construction of Noetherian operators for primary ideals in the analytic theory of Ehrenpreis-Palamodov, and they offer a concise method for representing affine schemes. The case of modules is also addressed. We implemented an algorithm in Macaulay2 that computes the minimal dec…
A criterion is given in order that the ideals of a one branch curve singularity form at most 2-parameter families. Namely, we present a list of plane curve singularities from the Arnold's classification which are the smallest among all one branch singularities having at most 2-parameter families of ideals.
We show that the $ n $-th Nash blowup of the toric surface singularity of type $ A_3 $ is singular for any $ n > 0 $. It was known that the normalization of the $ n $-th Nash blowup of a toric variety is also a toric variety associated to the Gröbner fan of a certain ideal $ J_n $. In our case, we prove that the Gröbner fan contains a non-regular cone. We determine minimal generators of the initial ideal of $ J_n $ with respect to a certain monomial ordering, and show that the reduced Gröbner basis of $ J_n $ has polynomials of certain forms for each $ n $.
We compare three naturally occurring multi-indexed filtrations of ideals on the local ring of a Newton nondegenerate hypersurface surface singularity with rational homology sphere, which in many cases are all distinct. These are the divisorial, the order, and the image filtrations. These filtrations are indexed by the lattice associated with a toric partial resolution of the singularity, or equivalently, the free abelian group generated by the compact facets of the Newton polyhedron. We prove that there exists a top dimensional cone contained in the Lipman cone having the property that the thr…
The aim of this paper is to study mixed multiplier ideals associated to a tuple of ideals in a two-dimensional local ring with a rational singularity. We are interested in the partition of the real positive orthant given by the regions where the mixed multiplier ideals are constant. In particular we reveal which information encoded in a mixed multiplier ideal determines its corresponding jumping wall and we provide an algorithm to compute all the constancy regions, and their corresponding mixed multiplier ideals, in any desired range.
Noetherian operators are differential operators that encode primary components of a polynomial ideal. We develop a framework, as well as algorithms, for computing Noetherian operators with local dual spaces, both symbolically and numerically. For a primary ideal, such operators provide an alternative representation to one given by a set of generators. This description fits well with numerical algebraic geometry, taking a step toward the goal of numerical primary decomposition.
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.
In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, generalizing the classical theory for $m_R$-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of $R$ is less than the dimension of $R$, which holds for instance if $R$ is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of $m_R$-primary ideals hold fo…
We associate convex regions in R^n to m-primary graded sequences of subspaces, in particular m-primary graded sequences of ideals, in a large class of local algebras (including analytically irreducible local domains). These convex regions encode information about Samuel multiplicities. This is in the spirit of the theory of Grobner bases and Newton polyhedra on one hand, and the theory of Newton-Okounkov bodies for linear systems on the other hand. We use this to give a new proof, as well as a generalization of a Brunn-Minkowski inequality for multiplicities due to Teissier and Rees-Sharp.
The aim of this paper is to study jumping numbers and multiplier ideals of any ideal in a two-dimensional local ring with a rational singularity. In particular we reveal which information encoded in a multiplier ideal determines the next jumping number. This leads to an algorithm to compute sequentially the jumping numbers and the whole chain of multiplier ideals in any desired range. As a consequence of our method we develop the notion of jumping divisor that allows to describe the jump between two consecutive multiplier ideals. In particular we find a unique minimal jumping divisor that is s…
Suppose that R is an analytically irreducible or excellent local domain with maximal ideal m_R. We consider multiplicities and mixed multiplicities of R by filtrations of m_R-primary ideals. We show that the theorem of Teissier, Rees and Sharp, and Katz, characterizing equality in the Minkowski inequality for multiplicities of ideals, is true for divisorial filtrations, and for the larger category of bounded filtrations. This theorem is not true for arbitrary filtrations of m_R-primary ideals.
Let (R, m) be a Cohen-Macaulay local ring and I be an m-primary ideal. We introduce ideals of almost minimal mixed multiplicty which are analogues of ideals studied by J. Sally. The main theorem describes the Hilbert series of fiber cones of these ideals.
We show the existence (and define) the mixed multiplicities of arbitrary graded families of ideals under mild assumptions. In particular, our methods and results are valid for the case of arbitrary $\mathfrak{m}$-primary graded families. Furthermore, we provide a far-reaching "Volume = Multiplicity formula" for the mixed multiplicities of graded families of ideals.
We describe a relation between two invariants that measure the complexity of a hypersurface singularity. One is the Hodge spectrum which is related to the monodromy and the Hodge filtration on the cohomology of the Milnor fiber. The other is the multiplier ideal, having to do with log resolutions.
In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding of the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are expl…
It is shown that the log-canonical threshold of a curve with an isolated singularity is computed by the term ideal of the curve in a suitable system of local parameters at the singularity. The proof uses the Enriques diagram of the singularity and shows that the log-canonical threshold depends only on a non-degenerate path of that diagram.
Let $C$ be an isolated plane curve singularity, and $(C,l)$ be a decorated curve. In this article we compare the equisingular deformations of $C$ and the sandwiched singularity $X(C,l)$. We will prove that for $l \gg 0$ the functor of equisingular deformations of $C$ and $(C,l)$ are equivalent. From this we deduce a proof of a formula for the dimension of the equisingular stratum. Furthermore we will show how compute the equisingularity ideal of the curve singularity $C$, given the minimal (good) resolution of $C$.
The first purpose of this paper is to point out a curious result announced by Macaulay on the Hilbert function of a differential module in his famous book The Algebraic Theory of Modular Systems published in 1916. Indeed, on page 78/79 of this book, Macaulay is saying the following: " A polynomial ideal $\mathfrak{a} \subset k[χ\_1$,..., $χ\_n]=k[χ]$ is of the {\it principal class} and thus {\it unmixed} if it has rank $r$ and is generated by $r$ polynomials. Having in mind this definition, a primary ideal $\mathfrak{q}$ with associated prime ideal $\mathfrak{p} = rad(\mathfrak{q})$ is such th…
We present a new approach to the study of multiplier ideals in a local, two-dimensional setting. Our method allows us to deal with ideals, graded systems of ideals and plurisubharmonic functions in a unified way. Among the applications are a formula for the complex integrability exponent of a plurisubharmonic function in terms of Kiselman numbers, and a proof of the openness conjecture by Demailly and Kollar. Our technique also yields new proofs of two recent results: one on the structure of the set of complex singularity exponents for holomorphic functions; the other by Lipman and Watanabe on…
The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a non-trivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fixed length and whose defining ideals have a fixed number of generators. We conjecture that the generating function of Euler characteristics of refined punctual Hilbert schemes is the HOMFLY polynomial of the link. The conjecture is verified for irreducible singularities y^k = x^n, whose links are the k,n torus knots, and for the singularity y^4 = x^7 - …
This note is written for the Festschrift in honor of Professor Christer Kiselman. Multiplier ideal sheaves identify the location and the extent of the failure of crucial estimates. In this note we will discuss and explain the historic evolution of the notion of multiplier ideal sheaves, especially the interpretation from the viewpoint of destabilizing subsheaves in the context of terminating or bounding an infinite process. We will also discuss the approach of constructing rational curves in Fano manifolds by using dynamic multiplier ideal sheaves and singularity-magnifying complex Monge-Amper…
