arXiv subjects

Emily E. Witt

Publications and source records attributed to Emily E. Witt.

18 recordsLinked to original sources

arXiv2022

Lower Bounds on the F-pure Threshold and Extremal Singularities

We prove that if $f$ is a reduced homogenous polynomial of degree $d$, then its $F$-pure threshold at the unique homogeneous maximal ideal is at least $\frac{1}{d-1}$. We show, furthermore, that its $F$-pure threshold equals $\frac{1}{d-1}$ if and only if $f\in \mathfrak m^{[q]}$ and $d=q+1$, where $q$ is a power of $p$. Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such "extremal singularities," and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are "extremal," for example, in terms of the configurations of lines they can contain.

math.AC↗

arXiv2022

Classification of Frobenius Forms in five variables

We classify Frobenius forms, a special class of homogeneous polynomials in characteristic $p>0$, in up to five variables over an algebraically closed field. We also point out some of the similarities with quadratic forms.

math.AC↗

arXiv2022

Cubic Surfaces of Characteristic Two

Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that, the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and that up to projective equivalence, there are finitely many non-Frobenius split cubic surfaces. We explicitly describe defining equations for each and characterize them as extremal in terms of configurations of lines on them. In particular, a (possibly singular) cubic surface in characteristic two fails to be Frobenius split if and only if no three lines on it form a "triangle".

math.AC↗

arXiv2022

Bernstein's inequality and holonomicity for certain singular rings

In this manuscript we prove the Bernstein inequality and develop the theory of holonomic D-modules for rings of invariants of finite groups in characteristic zero, and for strongly F-regular finitely generated graded algebras with FFRT in prime characteristic. In each of these cases, the ring itself, its localizations, and its local cohomology modules are holonomic. We also show that holonomic D-modules, in this context, have finite length. We obtain these results using a more general version of Bernstein filtrations.

math.AC↗

arXiv2021

Bernstein-Sato functional equations, $V$-filtrations, and multiplier ideals of direct summands

This paper investigates the existence and properties of a Bernstein-Sato functional equation in nonregular settings. In particular, we construct $D$-modules in which such formal equations can be studied. The existence of the Bernstein-Sato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of $V$-filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals and Hodge ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular direct summands of polynomial rings.

math.AG↗

arXiv2019

Frobenius powers

This article extends the notion of a Frobenius power of an ideal in prime characteristic to allow arbitrary nonnegative real exponents. These generalized Frobenius powers are closely related to test ideals in prime characteristic, and multiplier ideals over fields of characteristic zero. For instance, like these well-known families of ideals, Frobenius powers also give rise to jumping exponents that we call critical Frobenius exponents. In fact, the Frobenius powers of a principal ideal coincides with its test ideals, but appear to be a more refined measure of singularities in general. Herein, we develop the theory of Frobenius powers in regular domains, and apply it to study singularities, especially those of generic hypersurfaces. These applications illustrate one way in which multiplier ideals behave more like Frobenius powers than like test ideals.

math.AC↗

arXiv2019

The TestIdeals package for Macaulay2

This note describes a \emph{Macaulay2} package for computations in prime characteristic commutative algebra. This includes Frobenius powers and roots, $p^{-e}$-linear and $p^{e}$-linear maps, singularities defined in terms of these maps, different types of test ideals and modules, and ideals compatible with a given $p^{-e}$-linear map.

math.AC↗

arXiv2019

The FrobeniusThresholds package for Macaulay2

This article describes the \emph{Macaulay2} package \emph{FrobeniusThresholds}, designed to estimate and calculate $F$-pure thresholds, more general $F$-thresholds, and related numerical invariants arising in the study of singularities in prime characteristic commutative algebra.

math.AC↗

arXiv2018

Frobenius powers of some monomial ideals

In this paper, we characterize the (generalized) Frobenius powers and critical exponents of two classes of monomial ideals of a polynomial ring in positive characteristic: powers of the homogeneous maximal ideal, and ideals generated by positive powers of the variables. In doing so, we effectively characterize the test ideals and $F$-jumping exponents of sufficiently general homogeneous polynomials, and of all diagonal polynomials. Our characterizations make these invariants computable, and show that they vary uniformly with the congruence class of the characteristic modulo a fixed integer. Moreover, we confirm that for a diagonal polynomial over a field of characteristic zero, the test ideals of its reduction modulo a prime agree with the reductions of its multiplier ideals for infinitely many primes.

math.AC↗

arXiv2018

Local m-adic constancy of F-pure thresholds and test ideals

In this note, we consider a corollary of the ACC conjecture for F-pure thresholds. Specifically, we show that the F-pure threshold (and more generally, the test ideals) associated to a polynomial with an isolated singularity are locally constant in the m-adic topology of the corresponding local ring. As a by-product of our methods, we also describe a simple algorithm for computing all of the F-jumping numbers and test ideals associated to an arbitrary polynomial over an F-finite field.

math.AC↗

arXiv2016

Cohomological dimension, Lyubeznik numbers, and connectedness properties in mixed characteristic

We establish a "second vanishing theorem" for local cohomology modules over regular rings of unramified mixed characteristic, which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology. Applying this, and new results on the mixed characteristic Lyubeznik numbers, we further study connectedness properties of the spectra of a certain class of rings.

math.AC↗

arXiv2015

Lyubeznik numbers and injective dimension in mixed characteristic

We investigate the Lyubeznik numbers, and the injective dimension of local cohomology modules, of finitely generated $\mathbb{Z}$-algebras. We prove that the mixed characteristic Lyubeznik numbers and the standard ones agree locally for almost all reductions to positive characteristic. Additionally, we address an open question of Lyubeznik that asks whether the injective dimension of a local cohomology module over a regular ring is bounded above by the dimension of its support. Although we show that the answer is affirmative for several families of $\mathbb{Z}$-algebras, we also exhibit an example where this bound fails to hold. This example settles Lyubeznik's question, and illustrates one way that the behavior of local cohomology modules of regular rings of equal characteristic and of mixed characteristic can differ.

math.AC↗

arXiv2014

A survey on the Lyubeznik numbers

The Lyubeznik numbers are invariants of a local ring containing a field that capture ring-theoretic properties, but also have numerous connections to geometry and topology. We discuss basic properties of these integer-valued invariants, as well as describe some significant results and recent developments (including certain generalizations) in the area.

math.AC↗

arXiv2014

F-pure thresholds of homogeneous polynomials

In this article, we investigate F-pure thresholds of polynomials that are homogeneous under some N-grading, and have an isolated singularity at the origin. We characterize these invariants in terms of the base p expansion of the corresponding log canonical threshold. As an application, we are able to make precise some bounds on the difference between F-pure and log canonical thresholds established by Mustaţă and the fourth author. We also examine the set of primes for which the F-pure and log canonical threshold of a polynomial must differ. Moreover, we obtain results in special cases on the ACC conjecture for F-pure thresholds, and on the upper semi-continuity property for the F-pure threshold function.

math.AC↗

arXiv2013

Local cohomology with support in ideals of maximal minors and sub-maximal Pfaffians

We compute the GL-equivariant description of the local cohomology modules with support in the ideal of maximal minors of a generic matrix, as well as of those with support in the ideal of 2n x 2n Pfaffians of a (2n+1)x(2n+1) generic skew-symmetric matrix. As an application, we characterize the Cohen-Macaulay modules of covariants for the action of the special linear group SL(G) on G^m. The main tool we develop is a method for computing certain Ext modules based on the geometric technique for computing syzygies and on Matlis duality.

math.AC↗

arXiv2012

Generalized Lyubeznik numbers

Given a local ring containing a field, we define and investigate a family of invariants that includes the Lyubeznik numbers, but that captures finer information. These "generalized Lyubeznik numbers" are defined as lengths of certain iterated local cohomology modules in a category of D-modules, and in order to define them, we develop the theory of a functor Lyubeznik utilized in proving that his original invariants are well defined. In particular, this functor gives an equivalence of categories with a category of D-modules. These new invariants are indicators of F-regularity and F-rationality in characteristic p>0, and have close connections with characteristic cycle multiplicities in characteristic zero. We compute the generalized Lyubeznik numbers associated to monomial ideals using interpretations as lengths in a category of straight modules, as well as provide examples of these invariants associated to certain determinantal ideals.

math.AC↗

arXiv2012

Lyubeznik numbers in mixed characteristic

This manuscript defines a new family of invariants, analogous to the Lyubeznik numbers, associated to any local ring whose residue field has prime characteristic. In particular, as their nomenclature suggests, these "Lyubeznik numbers in mixed characteristic" are defined for all local rings of mixed characteristic. Some properties similar to those in equal characteristic hold for these new invariants. Notably, the "highest" Lyubeznik number in mixed characteristic is a well-defined notion. Although the Lyubeznik numbers in mixed characteristic and their equal-characteristic counterparts are the same for certain local rings of equal characteristic p>0, we also provide an example where they differ.

math.AC↗

arXiv2011

Local cohomology with support in ideals of maximal minors

Suppose that k is a field of characteristic zero, X is an r by s matrix of indeterminates, where r \leq s, and R = k[X] is the polynomial ring over k in the entries of X. We study the local cohomology modules H^i_I(R), where I is the ideal of R generated by the maximal minors of X. We identify the indices i for which these modules vanish, compute H^i_I(R) at the highest nonvanishing index, i = r(s-r)+1, and characterize all nonzero ones as submodules of certain indecomposable injective modules. These results are consequences of more general theorems regarding linearly reductive groups acting on local cohomology modules of polynomial rings.

math.AC↗