Back

Ilya Smirnov

Ikerbasque Research Fellow

Singularity Theory and Algebraic Geometry

T +34 946 567 842
F +34 946 567 842
E ismirnov@bcamath.org

Information of interest

Other links

Ikerbasque profile

I 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. 

 

  • Colength, Multiplicity, and Ideal Closure Operations II 

    Ma, L.; Pham, H.Q.; Smirnov, I.Autoridad BCAM (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 ...

  • Hilbert--Kunz multiplicity of quadrics via Ehrhart theory 

    Pak, I.; Shapiro, B.A.; Smirnov, I.Autoridad BCAM; Yoshida, K. (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 ...

  • Lech-Mumford constant and stability of local rings 

    Smirnov, Ilya; Ma, L.; Smirnov, I.Autoridad BCAM (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 ...

  • Stability and deformation of F-singularities 

    De Stefani, A.; Smirnov, I.Autoridad BCAM (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 ...

  • Effective generic freeness and applications to local cohomology 

    Cid-Ruiz, Y.; Smirnov, I.Autoridad BCAM (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 ...

  • The theory of F-rational signature 

    Smirnov, I.Autoridad BCAM; Tucker, K. (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 ...

  • COLENGTH, MULTIPLICITY, AND IDEAL CLOSURE OPERATIONS II 

    Ma, L.; Pham, H.Q.; Smirnov, I.Autoridad BCAM (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 ...

  • Lower bounds on Hilbert-Kunz multiplicities and maximal F-signature 

    Jeffries, J.; Nakajima, Y.; Smirnov, I.Autoridad BCAM; Watanabe, K.; Yoshida, K. (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 ...

  • Uniform Lech's inequality 

    Ma, L.; Smirnov, I.Autoridad BCAM (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, ...

More information

 

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.