Singularity Theory and Algebraic Geometry
http://hdl.handle.net/20.500.11824/18
2022-05-19T05:41:02Z
2022-05-19T05:41:02Z
Lower bounds on Hilbert-Kunz multiplicities and maximal F-signature
Jeffries, J.
Nakajima, Y.
Smirnov, I.
Watanabe, K.
Yoshida, K.
http://hdl.handle.net/20.500.11824/1468
2022-04-13T22:19:55Z
2022-01-01T00:00:00Z
Lower 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.
2022-01-01T00:00:00Z
Uniform Lech's inequality
Ma, L.
Smirnov, I.
http://hdl.handle.net/20.500.11824/1467
2022-04-13T22:19:54Z
2022-01-01T00:00:00Z
Uniform 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
Moderately Discontinuous Homology
Fernández de Bobadilla, J.
Heinze, S.
Sampaio, J.E.
http://hdl.handle.net/20.500.11824/1443
2022-03-08T23:19:35Z
2021-01-01T00:00:00Z
Moderately Discontinuous Homology
Fernández de Bobadilla, J.; Heinze, S.; Sampaio, J.E.
We introduce a new metric homology theory, which we call Moderately Discontinuous Homology, designed to capture Lipschitz properties of metric singular subanalytic germs. The main novelty of our approach is to allow “moderately discontinuous” chains, which are specially advantageous for capturing the subtleties of the outer metric phenomena. Our invariant is a finitely generated graded abelian group (Formula presented.) for any (Formula presented.) and homomorphisms (Formula presented.) for any (Formula presented.). Here (Formula presented.) is a “discontinuity rate”. The homology groups of a subanalytic germ with the inner or outer metric are proved to be finitely generated and only finitely many homomorphisms (Formula presented.) are essential. For (Formula presented.) Moderately Discontinuous Homology recovers the homology of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for (Formula presented.) -Homology recovers the homology of the punctured germ. Hence, our invariant can be seen as an algebraic invariant interpolating from the germ to its tangent cone. Our homology theory is a bi-Lipschitz subanalytic invariant, is invariant by suitable metric homotopies, and satisfies versions of the relative and Mayer-Vietoris long exact sequences. Moreover, fixed a discontinuity rate b we show that it is functorial for a class of discontinuous Lipschitz maps, whose discontinuities are b-moderated; this makes the theory quite flexible. In the complex analytic setting we introduce an enhancement called Framed MD homology, which takes into account information from fundamental classes. As applications we prove that Moderately Discontinuous Homology characterizes smooth germs among all complex analytic germs, and recovers the number of irreducible components of complex analytic germs and the embedded topological type of plane branches. Framed MD homology recovers the topological type of any plane curve singularity and relative multiplicities of complex analytic germs. © 2020 Wiley Periodicals LLC.
2021-01-01T00:00:00Z
Linearization of holomorphic families of algebraic automor- phisms of the affine plane
Kuroda, S.
Kutzschebauch, F.
Pelka, T.R.
http://hdl.handle.net/20.500.11824/1419
2022-02-04T00:19:27Z
2022-01-03T00:00:00Z
Linearization of holomorphic families of algebraic automor- phisms of the affine plane
Kuroda, S.; Kutzschebauch, F.; Pelka, T.R.
Let $G$ be a reductive group. We prove that a family of polynomial actions of $G$ on $\mathbb{C}^2$, holomorphically parametrized by an open Riemann surface, is linearizable. As an application, we show that a particular class of reductive group actions on $\mathbb{C}^3$ is linearizable. The main step of our proof is to establish a certain restrictive Oka property for groups of equivariant algebraic automorphisms of $\mathbb{C}^2$.
2022-01-03T00:00:00Z