Singularity Theory and Algebraic Geometryhttp://hdl.handle.net/20.500.11824/182023-04-22T20:17:52Z2023-04-22T20:17:52ZUniform Lech's inequalityMa, L.Smirnov, I.http://hdl.handle.net/20.500.11824/15332023-04-21T08:42:47Z2022-01-01T00:00:00ZUniform Lech's inequality
Ma, L.; Smirnov, I.
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, this set is bounded above by a finite constant depending only on M. The lower bound extends a classical inequality of Lech, and the upper bound answers a question of Stückrad–Vogel in the affirmative. As an application, we obtain results on uniform behavior of the lengths of Koszul homology module
2022-01-01T00:00:00ZCOHOMOLOGY OF CONTACT LOCIBudur, N.Fernández de Bobadilla, J.Le, Q.Nguyen, D.http://hdl.handle.net/20.500.11824/14942023-04-21T08:42:34Z2022-01-01T00:00:00ZCOHOMOLOGY OF CONTACT LOCI
Budur, N.; Fernández de Bobadilla, J.; Le, Q.; Nguyen, D.
We construct a spectral sequence converging to the cohomology with compact support of the m-th contact locus of a complex polynomial. The first page is explicitly described in terms of a log resolution and coincides with the first page of McLean's spectral sequence converging to the Floer cohomology of the m-th iterate of the monodromy, when the polynomial has an isolated singularity. Inspired by this connection, we conjecture that if two germs of holomorphic functions are embedded topologically equivalent, then the Milnor fibers of their tangent cones are homotopy equivalent.
2022-01-01T00:00:00ZLower bounds on Hilbert-Kunz multiplicities and maximal F-signatureJeffries, J.Nakajima, Y.Smirnov, I.Watanabe, K.Yoshida, K.http://hdl.handle.net/20.500.11824/14682023-04-21T08:42:48Z2022-01-01T00:00:00ZLower bounds on Hilbert-Kunz multiplicities and maximal F-signature
Jeffries, J.; Nakajima, Y.; Smirnov, I.; Watanabe, K.; Yoshida, K.
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 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.
2022-01-01T00:00:00ZUniform Lech's inequalityMa, L.Smirnov, I.http://hdl.handle.net/20.500.11824/14672023-04-21T08:42:48Z2022-01-01T00:00:00ZUniform Lech's inequality
Ma, L.; Smirnov, I.
Let (R,m) be a Noetherian local ring of dimension d ≥ 2. We prove that if e(Rred) > 1, then the classical Lech’s inequality can be improved uniformly for all m-primary ideals, that is, there exists ε > 0 such that e(I) ≤ d!(e(R) − ε)l(R/I) for all m-primary ideals I ⊆ R. This answers a question raised in [3]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of I.
2022-01-01T00:00:00Z