I study singularities using algebraic methods. More generally, I am interested in many aspects of local algebra, including connections with homological algebra and combinatorics. I especially enjoy asymptotical methods in a broad sense: integral and tight closures, local cohomology, Hilbert functions, Hilbert-Kunz multiplicity, and more. Below I will describe my projects in more details.

• The defect of the F-pure threshold 

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  Alessandro De Stefani and Luis and Smirnov Núñez-Betancourt

  Adv. Math., 488, pp. Paper No. 110792, 50, 2026.

  We define the defect of the F-pure threshold of an F-pure local ring as the difference between its dimension and the F-pure threshold of the maximal ideal. We provide a characterization of this invariant using differential operations and show that it defines an upper semicontinuous invariant on a variety, further stratifying the F-pure locus. Notably, the function is also constructive in the Q-Gorenstein case, the first such result for positive characteristic invariant.
• An invitation to equimultiplicity of F-invariants 

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  Ilya Smirnov

  Lectures on tight closure and its applications, 831, pp. 179-202, 2025.

  These notes were written after lecture series at ICTP to serve as a brief introduction to the topic. It also slightly improves a result from my paper "Equimultiplicity in Hilbert-Kunz theory" and presents a new application.
• Stability and deformation of F-singularities 

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  Alessandro De Stefani and Ilya Smirnov

  Israel J. Math., 264(1), pp. 1-35, 2024.

  We started developing a theory of m-adic stability of F-singulaties. By proving a continuity result for F-rational signature, we show that F-rational singularities are stable. Stability is usually a stronger property than deformation, and we use the example of Singh to show that F-pure and strongly F-regular singularities are not generally stable. This also shows a mistake in my earlier paper with Thomas Polstra: even an individual splitting number is not continuous. We also present a few cases where F-injective singularities are stable.
• The theory of F-rational signature 

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  Ilya Smirnov and Kevin Tucker

  J. Reine Angew. Math., 812, pp. 1-58, 2024.

  F-rational signature was defined by Hochster and Yao to detect F-rational singularities in the same way F-signature detects strong F-regularity. Later, Sannai defined dual F-signature, an invariant of different spirit but same purpose. We show that the dual F-signature is equal to a modified version of the Hochster-Yao invariant and use this to show a number of desirable properties pushing the state of knowledge almost to that of F-signature.
• Effective generic freeness and applications to local cohomology 

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  Yairon Cid-Ruiz and Ilya Smirnov

  J. Lond. Math. Soc. (2), 110(4), pp. Paper No. e12995, 31, 2024.

  Motivated by base change properties that Yairon previously worked on, we prove a generic freeness result for local cohomology when the algebra is smooth over the base and contains a field. We also prove some effective generic freeness results by utilizing Groebner bases. Interestingly, the above generic freeness result does not hold for smooth Z-algebras, does the generic flatness hold?
• Lower bounds on Hilbert-Kunz multiplicities and maximal F-signatures 

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  Jack Jeffries, Yusuke Nakajima, Ilya Smirnov, Kei-ichi Watanabe, and Ken-Ichi Yoshida

  Math. Proc. Cambridge Philos. Soc., 174(2), pp. 247-271, 2023.

  Watanabe and Yoshida asked if the simple A₁ singularity has the smallest Hilbert-Kunz multiplicity, or the mildest Hilbert-Kunz singularity, among singular rings. This question was answered affirmatively in many cases, but is still open. We consider an analogous conjecture for F-signature and are able to prove it in the toric case. Furthermore, it is known that non-Gorenstein rings cannot have F-signature larger than 1/2, and we are able to classify toric rings for which the equality happens.
• Uniform Lech's inequality 

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  Linquan Ma and Ilya Smirnov

  Proc. Amer. Math. Soc., 151(6), pp. 2387-2397, 2023.

  We found a trick to settle the mixed characteristic case of the uniform improvement of Lech's inequality left in the previous paper with Huneke and Quy. Yet, more question remain in mixed characteristic since we don't know how to improve Lech's inequality for ideals with a small number of generators in this case.
• A transformation rule for natural multiplicities 

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  Jack Jeffries and Ilya Smirnov

  Int. Math. Res. Not. IMRN, 2022(2), pp. 999-1015, 2022.

  We conceptualize the transformation rule for F-signature obtained by Carvajal-Rojas, Schwede, and Tucker in a more abstract setup. We apply the framework to get a transformation rule for differential signature, a characteristic-free invariant due to Brenner, Jeffries, and Núñez-Betancourt. Similarly to F-signature the consequence is a bound on the order of the local étale fundamental group.
• Equimultiplicity Theory of Strongly F-regular rings 

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  Thomas Polstra and Ilya Smirnov

  Michigan Math. J., 70(4), pp. 837-856, 2021.

  Originally, I wanted to extend my earlier Hilbert-Kunz paper by building an equimultiplicity theory for F-signature, but with Thomas' sharp insight the paper turned into much more! The gist of the paper is rigidity of many Frobenius-induced asymptotic invariants in strongly F-regular rings: if the values of the limit at a point and its specialization are equal, then necessarily entire functions are equal. This gives nice proofs of some classic results.
• Decomposition of graded local cohomology tables 

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  Alessandro De Stefani and Ilya Smirnov

  Math. Z, 297(1), pp. 1-24, 2021.

  Daniel Erman asked whether there is a Boij-Söderberg theory for local cohomology tables similar to the theory developed for vector bundles and coherent sheaves. We are able to answer this question in dimension two. We provide explicit decomposition and equations of facets.
• Hilbert–Kunz multiplicities and F-thresholds 

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  Luis Núñez-Betancourt and Ilya Smirnov

  Bol. Soc. Mat. Mex., 26(1), pp. 15-25, 2020.

  We sharpen the inequality between Hilbert-Kunz multiplicity of a ring and its quotient by an element using F-thresholds and conjecture a generalization.
• On the generalized Hilbert-Kunz function and multiplicity 

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  Hailong Dao and Ilya Smirnov

  Israel J. Math., 237(1), pp. 155-184, 2020.

  In this paper, we set up a variant of Hilbert-Kunz multiplicity for arbitrary ideals, being inspired by the epsilon multiplicity of Ulrich and Validashti. We give some existence results and also found some interesting homological consequences over complete intersections. It would be nice to understand these results better.
• Colength, multiplicity, and ideal closure operations 

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  Linquan Ma, Pham Hung Quy, and Ilya Smirnov

  Comm. Algebra, 48(4), pp. 1601-1607, 2020.

  In this short note, we give a unified treatment on many results connecting colength of some closure of an ideal and its multiplicity. Most of the results are already known and our new contribution is showing under very mild assumptions that in a singular ring the colength of an integrally closed ideals is always strictly less than its multiplicity.
• On semicontinuity of multiplicities in families 

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  Ilya Smirnov

  Doc. Math., 25, pp. 381-399, 2020.

  A family depending on a parameter is perhaps the most natural notion of deformation. This paper gives a somewhat new proof of semicontinuity of Hilbert-Samuel multiplicity in families and shows semicontinuity of Hilbert-Kunz multiplicity in families of finite type. The most surprising aspect is that the bound happens to be characteristic-free, so the result can be applied for families over the integers, giving a partial answer to a question of Brenner, Li, and Miller.
• Continuity of Hilbert–Kunz multiplicity and F-signature 

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  Thomas Polstra and Ilya Smirnov

  Nagoya Math. J., 239, pp. 322-345, 2020.

  Luis Núñez-Betancourt once asked me a deformation question: how does Hilbert-Kunz multiplicity change if we add a term of a very high order to the equation? In this note we provide the expected continuity property, so the answer is: slightly.
  Caution: The published version has an incorrect claim that F-signature is continuous, as discussed on arXiv and in the erratum. This claim is true in the Q-Gorenstein case, see my paper with De Stefani.
• Filter regular sequence under small perturbations 

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  Linquan Ma, Pham Hung Quy, and Ilya Smirnov

  Math. Ann., 378(1-2), pp. 243-254, 2020.

  Srinivas and Trivedi investigated what happens to the Hilbert-Samuel function when we perturb generators of an ideal by adding terms of very high order. After showing a number of nice results, they asked whether a sufficiently small perturbation of a filter-regular sequence (a generalization of a regular sequence) does not change the Hilbert-Samuel function. We answer this question affirmatively, giving the greatest possible level of generality where such a result may hold.
• Asymptotic Lech's inequality 

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  Craig Huneke, Linquan Ma, Pham Hung Quy, and Ilya Smirnov

  Adv. Math., 372, pp. 107296, 33, 2020.

  Lech's inequality asserts that e(I) ≤ d! e(R) col(I) and is not sharp if the dimension of R is at least 2. We show that for an isolated singularity of positive charactristic we can get arbitrarily close to omitting e(R) from Lech's inequality for sufficiently deep ideals. It follows (see Ma-Smirnov for mixed characteristic) that Lech's inequality is not sharp uniformly – under very mild assumption we can replace e(R) by something smaller.
• Equimultiplicity in Hilbert-Kunz theory 

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  Ilya Smirnov

  Math. Z, 291(1-2), pp. 245-278, 2019.

  This is the first part of my thesis. I study when the Hilbert-Kunz multiplicity of the localization at a prime ideal is equal to the Hilbert-Kunz multiplicity of the original ring. As it often happens, the behavior is similar but different from the classical situation. As an application, I show that the Hilbert-Kunz multiplicity may attain infinitely many values on Spec and that the equimultiple strata need not to be locally closed, a contrast with the classical theory.
• A generalization of an inequality of Lech relating multiplicity and colength 

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  Craig Huneke, Ilya Smirnov, and Javid Validashti

  Comm. Algebra, 47, pp. 2436-2449, 2019.

  Lech's inequality is never sharp in dimension at least three. We propose a way to fix this and verify the formula in dimension three. We also prove a weaker generalization of Lech's formula and propose an extension of the Lech-type inequality on the number of generators.
• Hilbert-Kunz multiplicity of the powers of an ideal 

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  Ilya Smirnov

  Proc. Amer. Math. Soc., 147(8), pp. 3331-3338, 2019.

  Extending and reconceptualizing a result of Trivedi, I show that Hilbert-Kunz multiplicity of Iⁿ has a n^d-1 term that behaves as if we could freely switch between Frobenius and ordinary powers. Naturally, I conjectured that the entire function should be the limit of Hilbert-Samuel polynomials of the Frobenius powers of I.
• Lech's inequality, the Stückrad–Vogel conjecture, and uniform behavior of Koszul homology 

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  Patricia Klein, Linquan Ma, Pham Hung Quy, Ilya Smirnov, and Yongwei Yao

  Adv. Math., 347, pp. 442-472, 2019.

  We prove Lech's inequality for modules and also prove its opposite, conjectured by Stückrad and Vögel. Roughly, the two inequalities say that multiplicity and colength are quantities of the same order: ce(I) ≤ col(I) ≤ Ce(I) for some constant c and C independent of I.
• The multiplicity and the number of generators of an integrally closed ideal 

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  Hailong Dao and Ilya Smirnov

  J. Singularities, 19, pp. 61-75, 2019.

  We prove a Lech-type of inequality that bounds the multiplicity of an ideal by a multiple of the number of generators. Similar to the classical Lech inequality, we study how it will improve if the singularity is mild. In dimension two we are able to get a very nice characterization: a ring satisfies same inequality as a regular ring if and only if it has minimal multiplicity and the powers of the maximal ideal are integrally closed.
• D-module and F-module length of local cohomology modules 

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  Mordechai Katzman, Linquan Ma, Ilya Smirnov, and Wenliang Zhang

  Trans. Amer. Math. Soc., 370(12), pp. 8551-8580, 2018.

  We study how the natural D-module structure of local cohomology modules interacts with the natural Frobenius action. Besides proving some general bounds on lengths, we continue Blickle's work and show that the two lengths mentioned in the title could be different.
• Upper semi-continuity of the Hilbert-Kunz multiplicity 

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  Ilya Smirnov

  Compos. Math., 152(3), pp. 477-488, 2016.

  This is the second part of my thesis, I prove that Hilbert-Kunz multiplicity is upper semicontinuous on Spec, as in the classical theory. Among other things, upper semicontinuity is thought to be a necessary requirement for a measure of singularity in a resolution algorithm.
• Prime filtrations of the powers of an ideal 

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  Craig Huneke and Ilya Smirnov

  Bull. Lond. Math. Soc., 47(4), pp. 585-592, 2015.

  In a short but fun paper, we prove an overlooked result: for an ideal I there exist prime filtrations of R/Iⁿ such that the set of prime factors is finite as n varies.
• Differential Krull dimension in differential polynomial extensions 

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  Ilya Smirnov

  J. Algebra, 344, pp. 354-372, 2011.

  In this paper written as my undergraduate thesis, I studied the differential-algebraic analog of the classical result: if R is a commutative ring of Krull dimension d then the Krull dimension of the polynomial ring R[x] is between d+1 and 2d+1.
• Colength, multiplicity, and ideal closure operations II 

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  Linquan Ma, Pham Hung Quy, and Ilya Smirnov

  Michigan Math. J..

  Despite the title, the paper is not a sequal to the first part, but rather a standalone paper on the same topics. This paper raises questions in Hilbert-Samuel theory inspired by results in the Hilbert-Kunz theory and vice-versa. Many of these questions come by asking to restrict previous studied infima and suprema to integrally closed or tightly closed ideals. In particular, we answer a question of Watanabe by showing that the relative Hilbert-Kunz multiplicity of integrally closed ideals is 1 under mild assumptions.