On a group ring identity related to the Alon-Jaeger-Tarsi conjecture
In this note we formulate a conjecture about two group ring identities and prove that it would imply the Alon-Jaeger-Tarsi conjecture.
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arXiv subjects
János Nagy
Publications and source records attributed to János Nagy.
Additive Combinatorics Using Equivariant Cohomology
We introduce a geometric method to study additive combinatorial problems. Using equivariant cohomology we reprove the Dias da Silva-Hamidoune theorem. We improve a result of Sun on the linear extension of the Erdős-Heilbronn conjecture. We generalize a theorem of G. Kós (the Grashopper problem) which in some sense is a simultaneous generalization of the Erdős-Heilbronn conjecture. We also prove a signed version of the Erdős-Heilbronn conjecture and the Grashopper problem. Most identities used are based on calculating the projective degree of an algebraic variety in two different ways.
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A sumset version of a conjecture of Pilz
Pilz's conjecture states that for any finite set $A=\{a_1,a_2,\dots,a_k\}$ of positive integers and positive integer $n$ in the union of the sets $\{a_1,2a_1,\dots,na_1\},\dots, \{a_k,2a_k,\dots,na_k\}$ (considered as a multiset) at least $n$ values appear an odd number of times. In this short note we consider a variant of this problem. Namely, we show that in the sumset $\{a_1,2a_1,\dots,na_1\}+\dots+\{a_k,2a_k,\dots,na_k\}$ (considered as a multiset) at least $n$ values appear an odd number of times.
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Irredundant hyperplane covers
We prove that if $G$ is an abelian group and $H_1x_1,\dots,H_{k}x_k$ is an irredundant (minimal) cover of $G$ with cosets, then $$|G:\bigcap_{i=1}^{k}H_{i}|=2^{O(k)}.$$ This bound is the best possible up to the constant hidden in the $O(\cdot)$ notation, and it resolves conjectures of Pyber (1996) and Szegedy (2007). We further show that if $G$ is an elementary $p$-group for some large prime $p$, and $H_1,\dots,H_k$ is a sequence of hyperplanes with many repetitions, then the bound above can be improved. As a consequence, we establish a substantial strengthening of the recently solved Alon-Jaeger-Tarsi conjecture: there exists $α>0$ such that for every invertible matrix $M\in\mathbb{F}_p^{n\times n}$ and any set of at most $p^α$ forbidden coordinates, one can find a vector $x\in\mathbb{F}_p^{n}$ such that neither $x$ nor $Mx$ have a forbidden coordinate.
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Invariants of relatively generic structures on normal surface singularities
In the present article we work out a relative setup of generic structures on surface singularities. We fix an analytic type on a subgraph of a rational homology sphere resolution graph $\mathcal{T}$ and we choose a relatively generic normal surface singularity $\tX$ with resolution graph $\mathcal{T}$. We provide formulae for the geometric genus and the analytical Poincaré series of $\tX$. We determine the base point structure of natural line bundles on $\tX$ and give a lower bound on the multiplicity of $\tX$ which is expected to be sharp. We prove similar results about cohomology numbers of relatively generic line bundles on every singularity with rational homology sphere link.
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Base points of natural line bundles on relatively generic surface singularities
In \cite{NNM} the author with A. Némethi computed the multiplicity of generic surface singularities, the formula is purely topological computable from the resolution graph of the surface singularity. In the present paper we extend the results partly to the relative case, when there is a pair of resolution graphs $\mathcal{T}_1 \subset \mathcal{T}$, a fixed singularity $\tX_1$ with resolution graph $\mathcal{T}_1$, a relatively generic singularity $\tX$ corresponding to the subsingularity $\tX_1$ with resolution graph $\mathcal{T}$. We determine the base points of the natural line bundles (under some mild conditions) on $\tX$ from the resolution graph $\mathcal{T}$ and the analytic invariants of the subsingularity $\tX_1$. For each base point $p$ we determine a lower bound for the number $t_p$ such that $p$ is $t_p$-simple and we compute from it a lower bound of the multiplicity of $\tX$, which is sharp in all known cases.
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Additive bases, coset covers, and non-vanishing linear maps
Recently, the first two authors proved the Alon-Jaeger-Tarsi conjecture on non-vanishing linear maps, for large primes. We extend their ideas to address several other related conjectures. We prove the weak Additive Basis conjecture proposed by Szegedy, making a significant step towards the Additive Basis conjecture of Jaeger, Linial, Payan, and Tarsi. In fact, we prove it in a strong form: there exists a set $A\subset\mathbb{F}_p^*$ of size $O(\log p)$ such that if $B\subset\mathbb{F}_p^{n}$ is the union of $p$ linear bases, then $A\cdot B=\{a\cdot v:a\in A, v\in B\}$ is an additive basis. An old result of Tomkinson states that if $G$ is a group, and $\{H_{i}x_{i}:i\in [k]\}$ is an irredundant coset cover of $G$, then $|G:\bigcap_{i\in [k]} H_{i}|\leq k!,$ and this bound is the best possible. It is a longstanding open problem whether the upper bound can be improved to $e^{O(k)}$ in case we restrict cosets to subgroups. Pyber proposed to study this question for abelian groups. We show that somewhat surprisingly, if $G$ is abelian, the upper bound can be improved to $e^{O(k\log \log k)}$ already in the case of general coset covers, making the first substantial improvement over the $k!$ bound. Finally, we prove a natural generalization of the Alon-Jaeger-Tarsi conjecture for multiple matrices.
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Normal reduction number of normal surface singularities
Let $(X,o)$ be a complex analytic normal surface singularity and let ${\mathcal O}_{X,o}$ be its local ring. We investigate the normal reduction number of ${\mathcal O}_{X,o}$ and related numerical analytical invariants via resolutions $\widetilde{X}\to X$ of $(X,o)$ and cohomology groups of different line bundles ${\mathcal L}\in {\rm Pic}(\widetilde{X})$. The normal reduction number is the universal optimal bound from which powers of certain ideals have stabilization properties. Here we combine this with stability properties of the iterated Abel maps. Some of the main results provide topological upper bounds for both stabilization properties.
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Hyperelliptic involutions on generic normal surface singularities
In the classical case of irreducible smooth algebraic curves every genus $2$ curve is hyperelliptic, or in other words there is a complete linear series $g_2^1$ on them. On the other hand if $g > 2$, then a generic smooth curve of genus $2$ is nonhyperelliptic. In this article we investigate the situation of normal surface singularities, so we fix a resolution graph $\mathcal{T}$ and a generic singularity with resolution $\tX$ corresponding to it in the sense of \cite{NNII}. We consider an integer effective cycle $Z$ on the resolution $\tX$ and investigate the existence of a complete linear series $g_2^1$ on it. The article has the main motivation that we will use heavily the results in it to compute the class of the image varieties of Abel maps in a following manuscript.
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The possible values of geometric genera of normal surface singularities
In this article we prove that the possible geometric genuses $p_g(\tX)$ corresponding to normal surface singularities $\tX$ with fixed negative definite resolution graph $\mathcal{T}$ form an interval of integers. Similarly let us have a resolution graph $\mathcal{T}$ and a fixed normal surface singularity $(X, 0)$ with resolution $\tX$ and resolution graph $\mathcal{T}$, furthermore consider a Chern class $l' \in L'$. We prove that the possible values of $h^1(\tX, \calL)$, where $\calL \in \pic^{l'}(\tX)$ form an interval of integers.
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The Alon-Jaeger-Tarsi conjecture via group ring identities
In this paper we resolve the Alon-Jaeger-Tarsi conjecture for sufficiently large primes. Namely, we show that for any finite field $\mathbb{F}$ of size $61<|\mathbb F|\ne 79$ and any nonsingular matrix $M$ over $\mathbb{F}$ there exists a vector $x$ such that neither $x$ nor $Ax$ has a 0 component.
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Brill-Noether problem on splice quotient singularities and duality of topological Poincaré series
In this manuscript we investigate the analouge of the Brill-Noether problem for smooth curves in the case of normal surface singularities. We determine the maximal possible value of $h^1$ of line bundles without fixed components in the Picard group $\pic^{l'}(\tX)$ in the following cases: for some special Chern classes $l'$ if $\tX$ is a resolution of a splice quotient singularity $(X, 0)$ and for arbitrary Chern classes in the case of weighted homogenous singularities. Motivated by this problem, we define the \emph{virtual cohomology numbers} $h^1_{virt}(l')$ for all Chern classes $l'$ such that $h^1_{virt}(0)$ is the canonical normalized Seiberg-Witten invariant and we generalize the duality formulae of Seiberg-Witten invariants obtained by the authors and A. Némethi in \cite{LNNdual}, for the virtual cohomology numbers.
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The multiplicity of generic normal surface singularities
We provide combinatorial/topological formula for the multiplicity of a complex analytic normal surface singularity whenever the analytic structure on the fixed topological type is generic.
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Class of images of Abel maps on normal surface singularities
In this paper we investigate Abel maps on normal surface singularities described in \cite{NNI}. We investigate the affine version of the class of the images of Abel maps on normal surface singularities. More precisely we consider the projective clousure of the image of an Abel map, its dual projective variety and we substract from its degree the multiplicity of the infinite hyperplane on the dual variety. In the case of generic singularities we prove explicit combinatorial formulas of this invariant, in the general case we prove an upper bound.
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Cohomology of natural line bundles on generic normal surface singularities
Let $\mathcal{T}$ be an arbitrary resolution graph and $(X, 0)$ a generic complex analytic normal surface singularity, and $\tilde{X}$ a generic resolution corresponding to it. Fix an effective integer cycle $Z$ supported on the exceptional curve and also an arbitrary Chern class $Z' \in L'$. In this article we aim to compute the cohomology numbers $h^1(\mathcal{O}_{Z}(Z'))$. Notice, that the case $Z'_v < 0, v \in |Z|$ was discussed in \cite{NNA2}, where the main theorem was, that in this special case these cohomology numbers equal to the cohomology numbers of the generic line bundle in $\text{Pic}^{Z'}(Z)$ However the condition $Z'_v < 0, v \in |Z|$ was crucial in the proof and without this assumption the statement is far from being true. In this article using the tecniques of relatively generic line bundles and relatively generic analytic structures from \cite{R} we give combinatorial algorithms to compute the cohomology numbers of natural line bundles $h^1(\calO_{Z}(Z'))$ for generic singularities in all cases.
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Invariants of relatively generic surface singularities II. Images of Abel maps
In \cite{R} the author investigated invariants of relatively generic structures on surface singularities generalising results of \cite{NNA1} and \cite{NNA2} about generic analytic structures and generic line bundles to the case of the relative setup, where we fix a given analytic type or line bundle on a smaller subgraph or more generally on a smaller cycle and we choose a relatively generic line bundle or analytic type on the large cycle and managed to compute some of it's invariants, like geometric genus or $h^1$ of natural line bundles. In \cite{NNAD} the authors investigated the images of Abel maps $c^{l'}(Z) : \eca^{l'}(Z) \to \pic^{l'}(Z)$, where $l' \in - S'(|Z|)$, especially the dimensions of the images of these maps and gave two algorithms to compute these invariants from cohomology numbers of cycles and from periodic constants of singularities we get from $\tX$ by blowing it up at generic points sequentially. Furthemore in \cite{NNAD} the authors gave explicit combinatorial formulas in the case of generic singularities. In this paper we want to generalise the theorems from \cite{NNAD} to the relatively generic case. In this case we fix a subsingularity $\tX_1$ for a subgraph $\mathcal{T}_1 \subset \mathcal{T}$ and a relatively generic singularity $\tX$ corresponding to $\tX_1$. Furthermore we fix a line bundle $\calL$ on $\tX_1$ and a Chern class $l' \in - S'$, such that $c^1(\calL) = R(l')$. Our main goal in the article is to compute $\dim(c^{l'}(\eca^{ l', \calL}(Z)))$ from invariants of the subsingularity $\tX_1$ and to conclude a few corollaries.
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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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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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