Singularity Theory and Algebraic Geometry
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Uniform Lech's inequality
Ma, L.; Smirnov, I.
(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, ... -
COHOMOLOGY OF CONTACT LOCI
Budur, N.; Fernández de Bobadilla, J.; Le, Q.; Nguyen, D. (2022-01-01)
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 ... -
Lower bounds on Hilbert-Kunz multiplicities and maximal F-signature
Jeffries, J.; Nakajima, Y.; Smirnov, I.; 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. (2022)
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) ... -
Moderately Discontinuous Homology
Fernández de Bobadilla, J.; Heinze, S.; Sampaio, J.E. (2021-01-01)
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 ... -
Linearization of holomorphic families of algebraic automor- phisms of the affine plane
Kuroda, S.; Kutzschebauch, F.; Pelka, T.R. (2022-01-03)
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 ... -
The Abel map for surface singularities III: Elliptic germs
Nagy, J.; Némethi, A. (2021-01-01)
The present note is part of a series of articles targeting the theory of Abel maps associated with complex normal surface singularities with rational homology sphere links (Nagy and Némethi in Math Annal 375(3):1427–1487, ... -
Delta invariant of curves on rational surfaces I. An analytic approach
Cogolludo-Agustín, J.I.; László, T.; Martín-Morales, J.; Némethi, A. (2021-01-01)
We prove that if (C, 0) is a reduced curve germ on a rational surface singularity (X, 0) then its delta invariant can be recovered by a concrete expression associated with the embedded topological type of the pair C X. ... -
Parametrization simple irreducible plane curve singularities in arbitrary characteristic
(2020-01-01)We study the classification of plane curve singularities in arbitrary characteristic. We first give a bound for the determinacy of a plane curve singularity with respect to pararametrization equivalence in terms of its ... -
Classification of Lipschitz simple function germs
(2020-07-01)It was shown by Henry and Parusiński in 2003 that the bi-Lipschitz right equivalence of function germs admits moduli. In this article, we introduce the notion of Lipschitz simple function germ and present the complete ... -
Image Milnor Number Formulas for Weighted-Homogeneous Map-Germs
Pallarés Torres, I.; Peñafort Sanchis, Guillermo (2021-07-05)
We give formulas for the image Milnor number of a weighted-homogeneous map-germ $(\mathbb{C}^n,0)\to(\mathbb{C}^{n+1},0)$, for $n=4$ and $5$, in terms of weights and degrees. Our expressions are obtained by a purely ... -
Classification of smooth factorial affine surfaces of Kodaira dimension zero with trivial units
(2021-07-31)We give a corrected statement of (Gurjar and Miyanishi 1988, Theorem 2), which classifies smooth affine surfaces of Kodaira dimension zero, whose coordinate ring is factorial and has trivial units. Denote the class of such ... -
Some contributions to the theory of singularities and their characteristic classes
Pallarés Torres, I. (2021-06-02)
In this Ph.D. thesis, we give some contributions to the theory of singularities, as well as to the theory of characteristic classes of singular spaces. The first part of this thesis is devoted to the theory of singularities ... -
Decomposition theorem and torus actions of complexity one
(2020)We algorithmically compute the intersection cohomology Betti numbers of any complete normal algebraic variety with a torus action of complexity one. -
The abel map for surface singularities II. Generic analytic structure
Nagy, J.; Némethi, A. (2019)
We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure ... -
Reflection maps
(2020)Given a reflection group G acting on a complex vector space V , a reflection map is the composition of an embedding X → V with the quotient map V → Cp of G. We show how these maps, which can highly singular, may be studied ... -
Local Topological Obstruction For Divisors
(2020)Given a smooth, projective variety $X$ and an effective divisor $D\,\subseteq\, X$, it is well-known that the (topological) obstruction to the deformation of the fundamental class of $D$ as a Hodge class, lies in ... -
Kato-matsumoto-type results for disentanglements
(2020)We consider the possible disentanglements of holomorphic map germs f : (Cn, 0) → (CN , 0), 0 < n < N, with nonisolated locus of instability Inst(f). The aim is to achieve lower bounds for their (homological) connectiv- ity ... -
On a conjecture of harris
(2019)For d ≥ 4, the Noether-Lefschetz locus NLd parametrizes smooth, degree d sur- faces in P3 with Picard number at least 2. A conjecture of Harris states that there are only finitely many irreducible components of the ... -
On the length of perverse sheaves on hyperplane arrangements
Budur, N.; Liu, Y. (2019)
Abstract. In this article we address the length of perverse sheaves arising as direct images of rank one local systems on complements of hyperplane arrangements. In the case of a cone over an essential line arrangement ...
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