Research

Research interests

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.

Lower bounds on Hilbert-Kunz multiplicities and maximal F-signatures, joint with Jack Jeffries, Yusuke Nakajima, Kei-ichi Watanabe, and Ken-ichi Yoshida.: 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.
Arxiv
Toward the theory of F-rational signature, joint with Kevin Tucker.: F-rational signature was defined by Hochster and Yao to detect F-rational singularities in the same way F-signature detects strong F-regularity. We propose a modification of their definition that has a better behavior. In particular, we are able to show upper semicontinuity of the new invariant, altough after having to extend it again! Our big conjecture is that the last extension was not necessary and that the new invariant coincides with Sannai's dual F-signature that also detects F-rational singularities.
Arxiv
A transformation rule for natural multiplicities, joint with Jack Jeffries.: 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.
Arxiv. To appear in IMRN
Equimultiplicity theory of strongly F-regular rings, joint with Thomas Polstra.: Originally, I wanted to extend my earlier Hilbert-Kunz paper by building an equimultiplicity theory for F-signature, but with Thomas's incredible 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.
Arxiv Michigan Math. J., accepted.
On semicontinuity of multiplicities in families.: 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.
Arxiv, Doc. Math., 25, 381-399, 2020
Hilbert-Kunz multiplicity of the powers of an ideal.: Extending and reconceptualizing a recent 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.
Arxiv, Proc. Amer. Math. Soc., 147(8), 3331-3338, 2019.
Hilbert-Kunz multiplicities and F-thresholds, joint with Luis Núñez-Betancourt.: We sharpen the inequality between Hilbert-Kunz multiplicity of a ring and its quotient by an element using F-thresholds and conjecture a generalization.
Bol. Soc. Mat. Mex., 26(1), 15-25, 2020.
Continuity of Hilbert-Kunz multiplicity and F-signature, joint with Thomas Polstra.: While both of us were still in Virginia, Luis Núñez-Betancourt asked another 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: our claim that F-signature is continuous is incorrect as stated and requires the ring to be Gorenstein to hold.
Arxiv, Nagoya Math. J., 239, 322-345, 2020.
Upper semi-continuity of the Hilbert-Kunz multiplicity.: 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.
Arxiv, Compos. Math., 152(3), 477–488, 2016.
Equimultiplicity in Hilbert-Kunz theory.: 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.
Arxiv, Math.Z., 291(1-2), 245-278, 2019.
On generalized Hilbert-Kunz function and multiplicity, joint with Hailong Dao.: 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.
Arxiv, Israel J. Math., 237(1), 155–184, 2020.

Asymptotic Lech's inequality, joint with Craig Huneke, Linquan Ma, and Pham Hung Quy.: 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 we can get arbitrarily close to omitting e(R) from Lech's inequality for sufficiently deep ideals. As a consequence, we show that for singular rings Lech's inequality is not sharp even asymptotically -- we can always replace e(R) by something smaller.
Arxiv. To appear in Advances Math.
Colength, multiplicity, and ideal closure operations, joint with Linquan Ma and Pham Hung Quy.: 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.
Arxiv, Comm. Algebra, 48(4), 1601-1607, 2020.
Lech's inequality, the Stückrad-Vögel conjecture, and uniform behavior of Koszul homology, joint with Patricia Klein, Linquan Ma, Pham Hung Quy, and Yongwei Yao.: 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.
Arxiv, Adv. Math., 347, 442-472, 2019.
A generalization of an inequality of Lech relating multiplicity and colength, joint with Craig Huneke and Javid Validashti.: 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.
Arxiv, Comm. Algebra., 47(6), 2436-2449, 2019.
The multiplicity and the number of generators of an integrally closed ideal, joint with Hailong Dao.: 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.
Arxiv, J. Singularities, 19, 61–75, 2019.

Stability of singularities.

Many deep theorems concern relations between analytic and algebraic. An example of such theorem and relation is finite determinacy of hypersurface isolated singularities of Samuel: a hypersurface defined by powers series can be truncated to give an equivalent singularity. In fact, he showed that we always get an equivalent singularity if we perturb the equation by adding terms of sufficiently high degree. Unfortunately, these results do not hold without assuming isolated singularity, and we instead must study what properties and invariants are stable.

Stability and deformation of F-singularities, joint with Alessandro De Stefani.: We start 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.
Arxiv
Filter regular sequence under small perturbations, joint with Linquan Ma and Pham Hung Quy.: This paper was pretty much entirely done in a week when Quy and I visited Linquan in Purdue! 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.
Arxiv, Math. Ann., 378(1-2), 243-254, 2020.
Continuity of Hilbert-Kunz multiplicity and F-signature, joint with Thomas Polstra.: While both of us were still in Virginia, Luis Núñez-Betancourt asked another 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: our claim that F-signature is continuous is incorrect as stated and requires the ring to be Gorenstein to hold.
Arxiv, Nagoya Math. J., 239, 322-345, 2020.

Decomposition of graded local cohomology tables, joint with Alessandro De Stefani.: 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.
Arxiv, Math. Z., 297(1), 1-24, 2021.
D-module and F-module length of some local cohomology modules, joint with Mordechai Katzman, Linquan Ma and Wenliang Zhang.: 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.
Arxiv Trans. Amer. Math. Soc., 370(12), 8551-8580, 2018.
Prime filtrations of the powers of an ideal, joint with Craig Huneke.: 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.
Arxiv Bull. Lond. Math. Soc., 47(4), 585–592, 2015.

Research interests

Hilbert-Kunz theory and related invariants

Multiplicity and colength

Stability of singularities.

Uncategorized projects