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András Némethi

Publications and source records attributed to András Némethi.

At least 37 records · Page 2Linked to original sources
arXiv2021

Local Newton nondegenerate Weil divisors in toric varieties

We introduce and develop the theory of Newton nondegenerate local Weil divisors $(X,0)$ in toric affine varieties. We characterize in terms of the toric combinatorics of the Newton diagram different properties of such singular germs: normality, Gorenstein property, or being an Cartier divisor in the ambient space. We discuss certain properties of their (canonical) resolution $\tilde{X}\to X $ and the corresponding canonical divisor. We provide combinatorial formulae for the delta--invariant $δ(X,0)$ and for the cohomology groups $H^i(\tilde X,\mathcal{O}_{\tilde X})$ for $i>0$. In the case $\dim(X,0)=2$, we provide the (canonical) resolution graph from the Newton diagram and we also prove that if such a Weil divisor is normal and Gorenstein, and the link is a rational homology sphere, then the geometric genus is given by the minimal path cohomology, a topological invariant of the link.

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arXiv2020

Local invariants of minimal generic curves on rational surfaces

Let (C,0) be a reduced curve germ in a normal surface singularity (X,0). The main goal is to recover the delta invariant of the abstract curve (C,0) from the topology of the embedding. We give explicit formulae whenever (C,0) is minimal generic and (X,0) is rational (as a continuation of previous works of the authors). Additionally we prove that if (X,0) is a quotient singularity, then the delta invariant of (C,0) only admits the values r-1 or r, where r is the number or irreducible components of (C,0). (r-1 realizes the extremal lower bound, valid only for `ordinary r-tuples'.)

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arXiv2020

The delta invariant of curves on rational surfaces II: Poincaré series and topological aspects

In this article we study abstract and embedded invariants of reduced curve germs via topological techniques. One of the most important numerical analytic invariants of an abstract curve is its delta invariant. Our primary goal is to develop delta invariant formulae for curves embedded in rational singularities in terms of embedded data. The topological machinery not only produces formulae, but it also creates deep connections with the theory of (analytical and topological) multivariable Poincaré series.

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arXiv2019

Delta invariant of curves on rational surfaces I. The analytic approach

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 (X,C). Furthermore, we also identify it with another (a priori) embedded analytic invariant, which is motivated by the theory of adjoint ideals. Finally, we connect our formulae with the local correction term at singular points of the global Riemann--Roch formula, valid for projective normal surfaces, introduced by Blache.

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arXiv2019

The Abel map for surface singularities II. Generic analytic structure

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 is generic (with respect to a fixed topological type), under the condition that the link is a rational homology sphere. The list of analytic invariants include: the geometric genus, the cohomology of certain natural line bundles, the cohomology of their restrictions on effective cycles (supported on the exceptional curve of a resolution), the cohomological cycle of natural line bundles, the multivariable Hilbert and Poincaré series associated with the divisorial filtration, the analytic semigroup, the maximal ideal cycle. The first part contains the definition of `generic structure' based on the work of Laufer. The second technical ingredient is the Abel map introduced by the authors. The results can be compared with certain parallel statements from the Brill-Noether theory and from the theory of Abel map associated with projective smooth curves, though the tools and machineries are very different.

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arXiv2019

The dimension of the image of the Abel map associated with normal surface singularities

Let $(X,o)$ be a complex normal surface singularity with rational homology sphere link and let $\widetilde{X}$ be one of its good resolutions. Fix an effective cycle $Z$ supported on the exceptional curve and also a possible Chern class $l'\in H^2(\widetilde{X},\mathbb{Z})$. Define ${\rm Eca}^{l'}(Z)$ as the space of effective Cartier divisors on $Z$ and $c^{l'}(Z):{\rm Eca}^{l'}(Z)\to {\rm Pic}^{l'}(Z)$, the corresponding Abel map. In this note we provide two algorithms, which provide the dimension of the image of the Abel map. Usually, $\dim {\rm Pic}^{l'}(Z)=p_g$, $\dim\,{\rm Im} (c^{l'}(Z))$ and ${\rm codim}\,{\rm Im} (c^{l'}(Z))$ are not topological, they are in subtle relationship with cohomologies of certain line bundles. However, we provide combinatorial formulae for them whenever the analytic structure on $\widetilde{X}$ is generic. The ${\rm codim}\,{\rm Im} (c^{l'}(Z))$ is related with $\{h^1(\widetilde{X},\mathcal{L})\}_{\mathcal{L}\in {\rm Im} (c^{l'}(Z))}$; in order to treat the `twisted' family $\{h^1(\widetilde{X},\mathcal{L}_0\otimes \mathcal{L})\}_{\mathcal{L}\in {\rm Im} (c^{l'}(Z))}$ we need to elaborate a generalization of the Picard group and of the Abel map. The above algorithms are also generalized.

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arXiv2019

Motivic Poincaré series of cusp surface singularities

We target multivariable series associated with resolutions of complex analytic normal surface singularities. In general, the equivariant multivariable analytical and topological Poincaré series are well-defined and have good properties only if the link is a rational homology sphere. We wish to create a model when this assumption is not valid: we analyse the case of cusps. For such germs we define even the motivic versions of these two series, we prove that they are equal, and we provide explicit combinatorial expression for them. This is done via a motivic multivariable series associated with the space of effective Cartier divisors of the reduced exceptional curve.

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arXiv2019

The Abel map for surface singularities III. Elliptic germs

If $(\widetilde{X},E)\to (X,o)$ is the resolution of a complex normal surface singularity and $c_1:{\rm Pic}(\widetilde{X})\to H^2(\widetilde{X},{\mathbb Z})$ is the Chern class map, then ${\rm Pic}^{l'}(\widetilde{X}):= c_1^{-1}(l')$ has a (Brill--Noether type) stratification $W_{l', k}:= \{{\mathcal L}\in {\rm Pic}^{l'}(\widetilde{X})\,:\, h^1({\mathcal L})=k\}$. In this note we determine it for elliptic singularities together with the stratification according to the cycle of fixed components. For elliptic singularities we also characterize the End Curve Condition and Weak End Curve Condition in terms of the Abel map, we provide several characterization of them, and finally we show that they are equivalent.

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arXiv2019

The boundary of the Milnor fibre of certain non-isolated singularities

Let $ Φ: ({\mathbb C}^2, 0) \to ( {\mathbb C}^3, 0) $ be a finitely determined complex analytic germ and let $(\{f=0\},0)$ be the reduced equation of its image, a non-isolated hypersurface singularity. We provide the plumbing graph of the boundary of the Milnor fibre of $f$ from the double-point-geometry of $Φ$.

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arXiv2019

On the topology of elliptic singularities

For any elliptic normal surface singularity with rational homology sphere link we consider a new elliptic sequence, which differs from the one introduced by Laufer and S. S.-T. Yau. However, we show that their length coincide. Using the properties of both sequences we succeed to connect the common length with the geometric genus and also with several topological invariants, e.g. with the Seiberg--Witten invariant of the link.

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arXiv2018

On the geometry of strongly flat semigroups and their generalizations

Our goal is to convince the readers that the theory of complex normal surface singularities can be a powerful tool in the study of numerical semigroups, and, in the same time, a very rich source of interesting affine and numerical semigroups. More precisely, we prove that the strongly flat semigroups, which satisfy the maximality property with respect to the Diophantine Frobenius problem, are exactly the numerical semigroups associated with negative definite Seifert homology spheres via the possible 'weights' of the generic $S^1$-orbit. Furthermore, we consider their generalization to the Seifert rational homology sphere case and prove an explicit (up to a Laufer computation sequence) formula for their Frobenius number. The singularities behind are the weighted homogeneous ones, whose several topological and analytical properties are exploited.

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arXiv2018

The Abel map for surface singularities I. Generalities and examples

Let $(X,o)$ be a complex normal surface singularity. We fix one of its good resolutions $\widetilde{X}\to X$, an effective cycle $Z$ supported on the reduced exceptional curve, and any possible (first Chern) class $l'\in H^2(\widetilde{X},\mathbb{Z})$. With these data we define the variety ${\rm ECa}^{l'}(Z)$ of those effective Cartier divisors $D$ supported on $Z$ which determine a line bundles $\mathcal{O}_Z(D)$ with first Chern class $l'$. Furthermore, we consider the affine space ${\rm Pic}^{l'}(Z)\subset H^1(\mathcal{O}_Z^*)$ of isomorphism classes of holomorphic line bundles with Chern class $l'$ and the Abel map $c^{l'}(Z):{\rm ECa}^{l'}(Z)\to {\rm Pic}^{l'}(Z)$. The present manuscript develops the major properties of this map, and links them with the determination of the cohomology groups $H^1(Z,\mathcal{L})$, where we might vary the analytic structure $(X,o)$ (supported on a fixed topological type/resolution graph) and we also vary the possible line bundles ${\mathcal{L}}\in {\rm Pic}^{l'}(Z)$. The case of generic line bundles of ${\rm Pic}^{l'}(Z)$ and generic line bundles of the image of the Abel map will have priority roles. Rewriting the Abel map via Laufer duality based on integration of forms on divisors, we can make explicit the Abel map and its tangent map. The case of superisolated and weighted homogeneous singularities are exemplified with several details. The theory has similar goals (but rather different techniques) as the theory of Abel map or Brill--Noether theory of reduced smooth projective curves.

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arXiv2018

Combinatorial duality for Poincaré series, polytopes and invariants of plumbed 3-manifolds

Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its Seiberg-Witten invariant can be computed as the `periodic constant' of the topological multivariable Poincaré series (zeta function). This involves a complicated regularization procedure (via quasipolynomials measuring the asymptotic behaviour of the coefficients). We show that the (a Gorenstein type) symmetry of the zeta function combined with Ehrhart-Macdonald-Stanley reciprocity (of Ehrhart theory of polytopes) provide a simple expression for the periodic constant. Using these dualities we also find a multivariable polynomial generalization of the Seiberg-Witten invariant, and we compute it in terms of lattice points of certain polytopes. All these invariants are also determined via lattice point counting, in this way we establish a completely general topological analogue of formulae of Khovanskii and Morales valid for singularities with non-degenerate Newton principal part.

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arXiv2018

Codimension 2 embeddings, algebraic surgery and Seifert forms

We study the cobordism of manifolds with boundary, and its applications to codimension 2 embeddings $M^m\subset N^{m+2}$, using the method of the algebraic theory of surgery. The first main result is a splitting theorem for cobordisms of algebraic Poincaré pairs, which is then applied to describe the behaviour on the chain level of Seifert surfaces of embeddings $M^{2n-1} \subset S^{2n+1}$ under isotopy and cobordism. The second main result (update: which is false) is that the $S$-equivalence class of a Seifert form is an isotopy invariant of the embedding, generalizing the Murasugi--Levine result for knots and links. The third main result is a generalized Murasugi--Kawauchi inequality giving an upper bound on the difference of the Levine--Tristram signatures of cobordant embeddings.

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arXiv2018

Euler characteristics of Hilbert schemes of points on simple surface singularities

We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition of the (equivariant) Hilbert scheme of the orbifold into affine space strata indexed by a certain combinatorial set, the set of Young walls. The generating series of Euler characteristics of Hilbert schemes of points of the singular surface of type A or D is computed in terms of an explicit formula involving a specialized character of the basic representation of the corresponding affine Lie algebra; we conjecture that the same result holds also in type E. Our results are consistent with known results in type A, and are new for type D.

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arXiv2017

Analytic singularities supported by a specific integral homology sphere link

The main question we target is the following: If one fixes a topological type of a complex normal surface singularity then what are the possible analytic types supported by it, and/or, what are the possible values of the geometric genus? We answer the question for a specific (in some sense pathological) topological type, which supports rather different analytic structures. These structures are listed together with some of their key analytic invariants.

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