Singularity Theory and Algebraic Geometry
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The theory of Frational signature
(2024) 
Moderately discontinuous homotopy
(202212) 
Monodromy conjecture for semiquasihomogeneous hypersurfaces
(2021)We give a proof the monodromy conjecture relating the poles of motivic zeta functions with roots of bfunctions for isolated quasihomogeneous hypersurfaces, and more generally for semiquasihomogeneous hypersurfaces. We ... 
Polar exploration of complex surface germs
(2021)We prove that the topological type of a normal surface singularity pX, 0q provides finite bounds for the multiplicity and polar multiplicity of pX, 0q, as well as for the combinatorics of the families of generic hyperplane ... 
The dimension of the image of the Abel map associated with normal surface singularities
(2019)Let (X, o) be a complex normal surface singularity with rational homology sphere link and let Xe be one of its good resolutions. Fix an effective cycle Z supported on the exceptional curve and also a possible Chern class ... 
Monodromy conjecture for log generic polynomials
(2020)A log generic hypersurface in P n with respect to a birational modification of P n is by definition the image of a generic element of a high power of an ample linear series on the modification. A log verygeneric ... 
Uniform Lech's inequality
(2022)Let (R,m) be a Noetherian local ring, and let M be a finitely generated Rmodule 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
(20220101)We construct a spectral sequence converging to the cohomology with compact support of the mth contact locus of a complex polynomial. The first page is explicitly described in terms of a log resolution and coincides with ... 
Lower bounds on HilbertKunz multiplicities and maximal Fsignature
(2022)ABSTRACT. Hilbert–Kunz multiplicity and Fsignature 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
(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 mprimary ideals, that is, there exists ε > 0 such that e(I) ... 
Moderately Discontinuous Homology
(20210101)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
(20220103)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
(20210101)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
(20210101)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
(20200101)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
(20200701)It was shown by Henry and Parusiński in 2003 that the biLipschitz 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 WeightedHomogeneous MapGerms
(20210705)We give formulas for the image Milnor number of a weightedhomogeneous mapgerm $(\mathbb{C}^n,0)\to(\mathbb{C}^{n+1},0)$, 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
(20210731)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
(20210602)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.