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Steven Dale Cutkosky

Publications and source records attributed to Steven Dale Cutkosky.

At least 37 records · Page 2Linked to original sources
arXiv2022

Analytic spread of filtrations on two dimensional normal local rings

In this paper we prove that a classical theorem by McAdam about the analytic spread of an ideal in a Noetherian local ring continues to be true for divisorial filtrations on a two dimensional normal excellent local ring $R$, and that the Hilbert polynomial of the fiber cone of a divisorial filtration on $R$ has a Hilbert function which is the sum of a linear polynomial and a bounded function. We prove these theorems by first studying asymptotic properties of divisors on a resolution of singularities of the spectrum of $R$. The filtration of the symbolic powers of an ideal is an example of a divisorial filtration. Divisorial filtrations are often not Noetherian, giving a significant difference in the classical case of filtrations of powers of ideals and divisorial filtrations.

math.AC↗
arXiv2021

Local Uniformization of Abhyankar Valuations

We prove local uniformization of Abhyankar valuations of an algebraic function field K over a ground field k. Our result generalizes the proof of this result, with the additional assumption that the residue field of the valuation ring is separable over k, by Hagen Knaf and Franz-Viktor Kuhlmann. The proof in this paper uses different methods, being inspired by the approach of Zariski and Abhyankar.

math.AG↗
arXiv2021

Volumes of Line Bundles on Schemes

Volumes of line bundles are known to exist as limits on generically reduced projective schemes. However, it is not known if they always exist as limits on more general projective schemes. We show that they do always exist as a limit on a codimension one subscheme of a nonsingular projective variety.

math.AG↗
arXiv2021

The Minkowski Equality of Big Divisors

We give conditions characterizing equality in the Minkowski inequality for big divisors on a projective variety. Our results draw on the extensive history of research on Minkowski inequalities in algebraic geometry.

math.AG↗
arXiv2021

On the construction of valuations and generating sequences on hypersurface singularities

Suppose that (K, $ν$) is a valued field, f (z) $\in$ K[z] is a unitary and irreducible polynomial and (L, $ω$) is an extension of valued fields, where L = K[z]/(f (z)). Further suppose that A is a local domain with quotient field K such that $ν$ has nonnegative value on A and positive value on its maximal ideal, and that f (z) is in A[z]. This paper is devoted to the problem of describing the structure of the associated graded ring gr $ω$ A[z]/(f (z)) of A[z]/(f (z)) for the filtration defined by $ω$ as an extension of the associated graded ring of A for the filtration defined by $ν$. In particular we give an algorithm which in many cases produces a finite set of elements of A[z]/(f (z)) whose images in gr $ω$ A[z]/(f (z)) generate it as a gr $ν$ A-algebra as well as the relations between them. We also work out the interactions of our method of computation with phenomena which complicate the study of ramification and local uniformization in positive characteristic , such as the non tameness and the defect of an extension. For valuations of rank one in a separable extension of valued fields (K, $ν$) $\subset$ (L, $ω$) as above our algorithm produces a generating sequence in a local birational extension A1 of A dominated by $ν$ if and only if there is no defect. In this case, gr $ω$ A1[z]/(f (z)) is a finitely presented gr $ν$ A1-module. This is an improved version, thanks to a referee's remarks.

math.AG↗
arXiv2021

Multiplicities and Mixed Multiplicities of arbitrary Filtrations

We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations of $m$-primary ideals to arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities of $r$ filtrations on an analytically unramified local ring $R$ come from the coefficients of a suitable homogeneous polynomial in $r$ variables of degree equal to the dimension of the ring, analogously to the classical case of the mixed multiplicities of $m$-primary ideals in a local ring. We prove that the Minkowski inequalities hold for arbitrary filtrations. The characterization of equality in the Minkowski inequality for m-primary ideals in a local ring by Teissier, Rees and Sharp and Katz does not extend to arbitrary filtrations, but we show that they are true in a large and important subcategory of filtrations. We define divisorial and bounded filtrations. The filtration of powers of a fixed ideal is a bounded filtration, as is a divisorial filtration. We show that in an excellent local domain, the characterization of equality in the Minkowski equality is characterized by the condition that the integral closures of suitable Rees like algebras are the same, strictly generalizing the theorem of Teissier, Rees and Sharp and Katz. We also prove that a theorem of Rees characterizing the inclusion of ideals with the same multiplicity generalizes to bounded filtrations in excellent local domains. We give a number of other applications, extending classical theorems for ideals.

math.AC↗
arXiv2021

The Minkowski Equality for Filtrations

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.

math.AC↗
arXiv2019

Essential Finite Generation of Valuation Rings in Characteristic Zero Algebraic Function Fields

Let $K$ be a characteristic zero algebraic function field with a valuation $ν$. Let $L$ be a finite extension of $K$ and $ω$ be an extension of $ν$ to $L$. We establish that the valuation ring $V_ω$ of $ω$ is essentially finitely generated over the valuation ring $V_ν$ of $ν$ if and only if the initial index $ε(ω|ν)$ is equal to the ramification index $e(ω|ν)$ of the extension. This gives a positive answer, for characteristic zero algebraic function fields, to a question posed by Hagen Knaf.

math.AC↗
arXiv2019

Essentially finite generation of valuation rings in terms of classical invariants

The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field $(K,ν)$ and an extension $ω$ of $ν$ to a finite extension $L$ of $K$. Then we study when the valuation ring of $ω$ is essentially finitely generated over the valuation ring of $ν$. We present a necessary condition in terms of classic invariants of the extension by Hagen Knaf and show that in some particular cases, this condition is also sufficient. We also study when the corresponding extension of graded algebras is finitely generated. For this problem we present an equivalent condition (which is weaker than the one for the finite generation of the valuation rings).

math.AC↗
arXiv2019

Positivity of Mixed Multiplicities of Filtrations

The theory of mixed multiplicities of filtrations by $m$-primary ideals in a ring is introduced in a recent paper by Cutkosky, Sarkar and Srinivasan. In this paper, we consider the positivity of mixed multiplicities of filtrations. We show that the mixed multiplicities of filtrations must be nonnegative real numbers and give examples to show that they could be zero or even irrational. When $R$ is analytically irreducible, and $\mathcal I(1),\ldots,\mathcal I(r)$ are filtrations of $R$ by $m_R$-primary ideals, we show that all of the mixed multiplicities $e_R(\mathcal I(1)^{[d_1]},\ldots,\mathcal I(r)^{[d_r]};R)$ are positive if and only if the ordinary multiplicities $e_R(\mathcal I(i);R)$ for $1\le i\le r$ are positive. We extend this to modules and prove a simple characterization of when the mixed multiplicities are positive or zero on a finitely generated module.

math.AC↗
arXiv2019

Mixed multiplicities of Divisorial Filtrations

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 $m_R$-primary divisorial filtrations. We show that some important theorems on equalities of multiplicities and mixed multiplicities of $m_R$-primary ideals, which are not true in general for filtrations, are true for divisorial filtrations. We prove that a theorem of Rees showing that if there is an inclusion of $m_R$-primary ideals $I\subset I'$ with the same multiplicity then $I$ and $I'$ have the same integral closure also holds for divisorial filtrations. This theorem does not hold for arbitrary filtrations. We show that the Teissier Rees Sharp Katz theorem on equality in the Minkowski inequality holds for divisorial filtrations in an excellent domain of dimension two. We also show that the mixed multiplicities of divisorial filtrations are anti-positive intersection products on a suitable normal scheme $X$ birationally dominating $R$, when $R$ is an algebraic local domain.

math.AC↗
arXiv2019

Mixed Multiplicities of Filtrations

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 for filtrations, including the famous Minkowski inequalities of Teissier, and Rees and Sharp.

math.AC↗
arXiv2018

Extensions of Valuations to the Henselization and Completion

We show that if $R$ is a local domain which is dominated by a valuation $ν$, then there does not always exist a regular local ring $R'$ which birationally dominates R and is dominated by $ν$ and an extension of $ν$ to the Henselization $(R')^h$ of $R'$ such that the associated graded rings of $R'$ and $(R')^h$ along the valuations are equal. We also show that there does not always exist $R'$, a prime ideal $p$ of the completion $\widehat{R'}$ such that $p\cap R'=(0)$ and an extension of $ν$ to $\widehat{R'}$ such that the associated graded rings of $R'$ and $\widehat{R'}/p$ along the valuation are equal.

math.AG↗
arXiv2018

Finite Generation of Extensions of Associated Graded Rings Along a Valuation

In this paper we consider the question of when the associated graded ring along a valuation, ${\rm gr}_{ν^*}(S)$, is a finite ${\rm gr}_{ν^*}(R)$-module, where $S$ is a normal local ring which lies over a normal local ring $R$ and $ν^*$ is a valuation of the quotient field of $S$ which dominates $S$. We obtain a very general result in equicharacteristic zero in Theorem 1.5. We also obtain general results for unramified extensions of excellent local rings in Proposition 1.7.

math.AG↗
arXiv2017

Defect and Local Uniformization

We give a simple algorithm showing that the reduction of the multiplicity of a characteristic p>0 hypersurface singularity along a valuation is possible if there is a finite linear projection which is defectless. The method begins with the algorithm of Zariski to reduce multiplicity of hypersuface singularities in characteristic 0 along a valuation. This gives a simple demonstration that the only obstruction to local uniformization in positive characteristic is from defect arising in finite projections of singularities.

math.AG↗
arXiv2017

Fibers of rational maps and Jacobian matrices

A rational map $ϕ: \mathbb{P}_k^m \dashrightarrow \mathbb{P}_k^n$ is defined by homogeneous polynomials of a common degree $d$. We establish a linear bound in terms of $d$ for the number of $(m-1)$-dimensional fibers of $ϕ$, by using ideals of minors of the Jacobian matrix. In particular, we answer affirmatively Question~11 in arXiv:1511.02933v2.

math.AC↗
arXiv2017

The role of defect and splitting in finite generation of extensions of associated graded rings along a valuation

Suppose that $R$ is a 2 dimensional excellent local domain with quotient field $K$, $K^*$ is a finite separable extension of $K$ and $S$ is a 2 dimensional local domain with quotient field $K^*$ such that $S$ dominates $R$. Suppose that $ν^*$ is a valuation of $K^*$ such that $ν^*$ dominates $S$. Let $ν$ be the restriction of $ν^*$ to $K$. The associated graded ring ${\rm gr}_ν(R)$ was introduced by Bernard Teissier. It plays an important role in local uniformization. We show that the extension $(K,ν)\rightarrow (K^*,ν^*)$ of valued fields is without defect if and only if there exist regular local rings $R_1$ and $S_1$ such that $R_1$ is a local ring of a blow up of $R$, $S_1$ is a local ring of a blowup of $S$, $ν^*$ dominates $S_1$, $S_1$ dominates $R_1$ and the associated graded ring ${\rm gr}_{ν^*}(S_1)$ is a finitely generated ${\rm gr}_ν(R_1)$-algebra. We also investigate the role of splitting of the valuation $ν$ in $K^*$ in finite generation of the extensions of associated graded rings along the valuation. We will say that $ν$ does not split in $S$ if $ν^*$ is the unique extension of $ν$ to $K^*$ which dominates $S$. We show that if $R$ and $S$ are regular local rings, $ν^*$ has rational rank 1 and is not discrete and ${\rm gr}_{ν^*}(S)$ is a finitely generated ${\rm gr}_ν(R)$-algebra, then $ν$ does not split in $S$. We give examples showing that such a strong statement is not true when $ν$ does not satisfy these assumptions. We deduce that if $ν$ has rational rank 1 and is not discrete and if $R\rightarrow R'$ is a nontrivial sequence of quadratic transforms along $ν$, then ${\rm gr}_ν(R')$ is not a finitely generated ${\rm gr}_ν(R)$-algebra.

math.AG↗