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+34 946 567 842
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+34 946 567 842
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ismirnov@bcamath.org
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Ikerbasque profileI am a Ramón y Cajal and Ikerbasque Research Fellow at BCAM since 2022. Personal webpage.
My principal research area is local algebra; loosely speaking this means that I study singularities, as local rings, using techniques of commutative algebra. A particular way to do so is via numerical invariants, numbers that are supposed to represent an aspect of the complexity.
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Colength, Multiplicity, and Ideal Closure Operations II
(2025)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 ...
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Hilbert--Kunz multiplicity of quadrics via Ehrhart theory
(2025)We show that the Hilbert--Kunz multiplicity $h_{p, d}$ 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 ...
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Lech-Mumford constant and stability of local rings
(2025)This work systematically develops a theory of the Lech–Mumford constant, an invariant defined as an optimal constant in the classical Lech’s inequality and underlined Mumford’s notion of local semistability. We establish ...
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Stability and deformation of F-singularities
(2024-12-01)We study the problem of m-adic stability of F-singularities, that is, whether the property that a quotient of a local ring (R,m) by a non-zero divisor x∈m has good F-singularities is preserved in a sufficiently small m-adic ...
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Effective generic freeness and applications to local cohomology
(2024-10-01)Let (Formula presented.) be a Noetherian domain and (Formula presented.) be a finitely generated (Formula presented.) -algebra. We study several features regarding the generic freeness over (Formula presented.) of an ...
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The theory of F-rational signature
(2024-07-01)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 ...
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COLENGTH, MULTIPLICITY, AND IDEAL CLOSURE OPERATIONS II
(2024)Let (R,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 m-primary ideal. We ...
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Lower bounds on Hilbert-Kunz multiplicities and maximal F-signature
(2022)ABSTRACT. 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 ...
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Uniform Lech's inequality
(2022)Let (R,m) be a Noetherian local ring, and let M be a finitely generated R-module of dimension d. We prove that the set [Formula presented] is bounded below by 1/d!e(R‾) where R‾=R/Ann(M). Moreover, when Mˆ is equidimensional, ...
School on Commutative Algebra and Algebraic Geometry in Prime Characteristics, ICTP Trieste, May 2-5, 2023.
BCAM Severo Ochoa course "Introduction to multiplicity theory" Spring 2023.
RGAS School on Singularities, IMUS Sevilla, January 8-12, 2024.