arXiv subjects

Ilya Smirnov

Publications and source records attributed to Ilya Smirnov.

At least 19 recordsLinked to original sources

arXiv2026

The defect of the F-pure threshold

Introduced by Takagi and Watanabe, the F-pure threshold is an invariant defined in terms of the Frobenius homomorphism. While it finds applications in various settings, it is primarily used as a local invariant. The purpose of this note is to start filling this gap by opening the study of its behavior on a scheme. To this end, we define the defect of the F-pure threshold of a local ring $(R,\mathfrak{m})$ by setting ${\rm dfpt}(R)=\dim (R) - {\rm fpt}(\mathfrak{m})$. It turns out that this invariant defines an upper semi-continuous function on a scheme and satisfies Bertini-type theorems. We also study the behavior of the defect of the F-pure threshold under flat extensions and after blowing up the maximal ideal of a local ring.

math.AC↗

arXiv2026

Hilbert-Kunz multiplicity of quadrics via Ehrhart theory

We show that the Hilbert-Kunz multiplicity of the d-dimensional non-degenerate quadric hypersurface of characteristic p > 2 is a rational function of p composed from the Ehrhart polynomials of integer polytopes. In consequence, we prove that the Hilbert-Kunz multiplicity of quadrics of fixed characteristic is a decreasing function of dimension and recover results of Trivedi and Gessel-Monsky on the behaviour of said Hilbert-Kunz multiplicity as a function of characteristic.

math.AC↗

arXiv2026

Tight closure of products and F-rational singularities

We prove a characterization of F-rationality in terms of tight closure of products of parameter ideals. Our results are inspired by the theory of complete ideals for surfaces and, in particular, the fundamental results of Lipman-Teissier and Cutkosky characterizing rational surface singularities in terms of products of complete ideals, but are valid also in higher dimensions.

math.AC↗

arXiv2025

Lech-Mumford constant and stability of local rings

We study further Mumford's notion of local semistability and, in particular, show that semistable singularities are log canonical under mild assumptions. We provide many new examples of semistable and unstable singularities. More generally, we develop the theory of the Lech-Mumford constant, an invariant defined as an optimal constant in the Lech inequality.

math.AG↗

arXiv2025

F-singularities of polynomials with square-free support

We show that the intersection of the irreducible components of a hypersurface defined by a polynomial with square-free support has F-rational singularities in characteristic $p>0$. As a consequence, we obtain that hypersurfaces defined by irreducible polynomials with square-free support have F-rational singularities, positively answering a question of Bath, Mustaţă, and Walther.

math.AC↗

arXiv2024

Colength, multiplicity, and ideal closure operations II

Let $(R, \mathfrak{m})$ be a Noetherian local ring. This paper concerns several extremal invariants arising from the study of the relation between colength and (Hilbert--Samuel or Hilbert--Kunz) multiplicity of an $\mathfrak{m}$-primary ideal. We introduce versions of these invariants by restricting to various closures and ``cross-pollinate'' the two multiplicity theories by asking for analogues invariants already established in one of the theories. On the Hilbert--Samuel side, we prove that the analog of the Stückrad--Vogel invariant (that is, the infimum of the ratio between the multiplicity and colength) for integrally closed $\mathfrak{m}$-primary ideals is often $1$ under mild assumptions. We also compute the supremum and infimum of the relative drops of multiplicity for (integrally closed) $\mathfrak{m}$-primary ideals. On the Hilbert--Kunz side, we study several analogs of the Lech--Mumford and Stückrad--Vogel invariants.

math.AC↗

arXiv2024

Effective generic freeness and applications to local cohomology

Let $A$ be a Noetherian domain and $R$ be a finitely generated $A$-algebra. We study several features regarding the generic freeness over $A$ of an $R$-module. For an ideal $I \subset R$, we show that the local cohomology modules ${\rm H}_I^i(R)$ are generically free over $A$ under certain settings where $R$ is a smooth $A$-algebra. By utilizing the theory of Gröbner bases over arbitrary Noetherian rings, we provide an effective method to make explicit the generic freeness over $A$ of a finitely generated $R$-module.

math.AC↗

arXiv2023

An invitation to equimultiplicity of F-invariants

This note grew from the lectures I delivered at ICTP during the Summer School in honor of Hochster and Huneke. Its purpose is to provide an introduction to the notion of equimultiplicity (of numerical invariants of singularities/local rings) and survey past work on equimultiplicity of Hilbert-Kunz multiplicity and F-signature. The treatment of F-signature is from joint work with Thomas Polstra and the section on Hilbert-Kunz multiplicity updates my previous paper on the topic and gives novel applications.

math.AG↗

arXiv2023

The theory of F-rational signature

F-signature is an important numeric invariant of singularities in positive characteristic that can be used to detect strong F-regularity. One would like to have a variant that rather detects F-rationality, and there are two theories that aim to fill this gap: F-rational signature of Hochster and Yao and dual F-signature of Sannai. Unfortunately, several important properties of the original F-signature are unknown for these invariants. We give a modification of the Hochster-Yao definition that agrees with Sannai's dual F-signature and push further the united theory to achieve a complete generalization of F-signature.

math.AC↗

arXiv2023

Uniform Lech's inequality

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d\geq 2$. We prove that if $e(\widehat{R}_{red})>1$, then the classical Lech's inequality can be improved uniformly for all $\mathfrak{m}$-primary ideals, that is, there exists $\varepsilon>0$ such that $e(I)\leq d!(e(R)-\varepsilon)\ell(R/I)$ for all $\mathfrak{m}$-primary ideals $I\subseteq R$. We also obtain partial results towards improvements of Lech's inequality when we fix the number of generators of $I$.

math.AC↗

arXiv2023

Equimultiplicity in Hilbert-Kunz theory

This paper develops a theory of equimultiplicity for Hilbert-Kunz multiplicity and uses it to study the behavior of Hilbert-Kunz multiplicity on the Brenner-Monsky hypersurface. A number of applications follows, in particular we show that Hilbert-Kunz multiplicity attains infinitely many values and that equimultiple strata may not be locally closed.

math.AC↗

arXiv2023

Hilbert-Kunz multiplicity of the powers of an ideal

We study Hilbert-Kunz multiplicity of the powers of an ideal and establish existence of the second coefficient at the full level of generality, thus extending a recent result of Trivedi. We describe the second coefficient as the limit of the Hilbert coefficients of Frobenius powers and show that it is additive in short exact sequences and satisfies a Northcott-type inequality.

math.AC↗

arXiv2023

Colength, multiplicity, and ideal closure operations

In a formally unmixed Noetherian local ring, if the colength and multiplicity of an integrally closed ideal agree, then $R$ is regular. We deduce this using the relationship between multiplicity and various ideal closure operations.

math.AC↗

arXiv2022

Deformed graphical zonotopal algebras

We study certain filtered deformations of the external zonotopal algebra of a given graph parametrized by univariate polynomials. We establish some general properties of these algebras, compute their Hilbert series for a number of graphs using Macaulay2, and formulate several conjectures.

math.CO↗

arXiv2022

Moving Other Way: Exploring Word Mover Distance Extensions

The word mover's distance (WMD) is a popular semantic similarity metric for two texts. This position paper studies several possible extensions of WMD. We experiment with the frequency of words in the corpus as a weighting factor and the geometry of the word vector space. We validate possible extensions of WMD on six document classification datasets. Some proposed extensions show better results in terms of the k-nearest neighbor classification error than WMD.

cs.CL↗

arXiv2022

Lower bounds on Hilbert--Kunz multiplicities and maximal F-signatures

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.

math.AC↗

arXiv2020

Stability and deformation of F-singularities

We study the problem of $\mathfrak{m}$-adic stability of F-singularities, that is, whether the property that a quotient of a local ring $(R,\mathfrak{m})$ by a non-zero divisor $x \in \mathfrak{m}$ has good F-singularities is preserved in a sufficiently small $\mathfrak{m}$-adic neighborhood of $x$. We show that $\mathfrak{m}$-adic stability holds for F-rationality in full generality, and for F-injectivity, F-purity and strong F-regularity under certain assumptions. We show that strong F-regularity and F-purity are not stable in general. Moreover, we exhibit strong connections between stability and deformation phenomena, which hold in great generality.

math.AC↗

arXiv2020

Asymptotic Lech's inequality

We explore the classical Lech's inequality relating the Hilbert--Samuel multiplicity and colength of an $\mathfrak{m}$-primary ideal in a Noetherian local ring $(R,\mathfrak{m})$. We prove optimal versions of Lech's inequality for sufficiently deep ideals in characteristic $p>0$, and we conjecture that they hold in all characteristics. Our main technical result shows that if $(R,\mathfrak{m})$ has characteristic $p>0$ and $\widehat{R}$ is reduced, equidimensional, and has an isolated singularity, then for any sufficiently deep $\mathfrak{m}$-primary ideal $I$, the colength and Hilbert--Kunz multiplicity of $I$ are sufficiently close to each other. More precisely, for all $\varepsilon>0$, there exists $N\gg0$ such that for any $I\subseteq R$ with $l(R/I)>N$, we have $(1-\varepsilon)l(R/I)\leq e_{HK}(I)\leq(1+\varepsilon)l(R/I)$.

math.AC↗