Alex Kuronya

Alex Kuronya
Goethe University Frankfurt · Institut für Mathematik

PhD

About

84
Publications
4,598
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1,022
Citations
Additional affiliations
September 2010 - September 2013
University of Freiburg
Position
  • Akademischer Mitarbeiter
September 2004 - August 2008
University of Duisburg-Essen
Position
  • Wissenschaftlicher Mitarbeiter
September 2004 - December 2014
Budapest University of Technology and Economics
Position
  • Professor (Associate)
Education
September 1999 - August 2004
University of Michigan
Field of study
  • Pure Mathematics

Publications

Publications (84)
Article
We study effective global generation of adjoint line bundles on smooth projective varieties. To measure the effectivity we introduce the concept of the convex Fujita number of a smooth projective variety and compute its value for a class of varieties with prescribed dimension d ≥ 2 and an arbitrary projective group as fundamental group.
Preprint
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We study effective global generation properties of projectivizations of curve semistable vector bundles over curves and abelian varieties.
Article
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We provide a combinatorial criterion for the finite generation of a valuation semigroup associated with an ample divisor on a smooth toric surface and a non-toric valuation of maximal rank. As an application, we construct a lattice polytope such that none of the valuation semigroups of the associated polarized toric variety coming from one-paramete...
Preprint
Let $X$ be a complex abelian variety. We prove an analogue of both the (cohomological) $P=W$ conjecture and the geometric $P=W$ conjecture connecting the finer topological structure of the Dolbeault moduli space of topologically trivial semistable Higgs bundles on $X$ and the Betti moduli space of characters of the fundamental group of $X$. The geo...
Preprint
Full-text available
We study effective global generation of adjoint line bundles on smooth projective varieties. To measure the effectivity we introduce the concept of the convex Fujita number of a smooth projective variety and compute its value for a class of varieties with prescribed dimension $d \geq 2$ and an arbitrary projective group as fundamental group.
Preprint
Full-text available
We provide a combinatorial criterion for the finite generation of a valuation semigroup associated with an ample divisor on a smooth toric surface and a non-toric valuation of maximal rank.
Preprint
The homogeneous spectrum of a multigraded finitely generated algebra (in the sense of Brenner-Schr\"oer) always admits an embedding into a toric variety that is not necessarily separated, a so-called toric prevariety. In order to have a convenient framework to study the tropicalization of homogeneous spectra we propose a tropicalization procedure f...
Article
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Let $$X \subset {\mathbb P}(1,1,1,m)$$ X ⊂ P ( 1 , 1 , 1 , m ) be a general hypersurface of degree md for some for $$d\ge 2$$ d ≥ 2 and $$m\ge 3$$ m ≥ 3 . We prove that the Seshadri constant $$\varepsilon ( {\mathcal O}_X(1), x)$$ ε ( O X ( 1 ) , x ) at a general point $$x\in X$$ x ∈ X lies in the interval $$\left[ \sqrt{d}- \frac{d}{m}, \sqrt{d}\r...
Article
We extend the theory of Newton–Okounkov bodies, originally developed by Boucksom–Chen, Kaveh–Khovanskii, and Lazarsfeld–Mustaţă for lattice semigroups, to the context of cancellative torsion–free semigroups.
Article
We show that the subgraph of the concave transform of a multiplicative filtration on a section ring is the Newton–Okounkov body of a certain semigroup, and if the filtration is induced by a divisorial valuation, then the associated graded algebra is the algebra of sections of a concrete line bundle in higher dimension. We use this description to gi...
Preprint
Let $X$ be a general hypersurface of degree $md$ in the weighted projective space with weights $1,1,1,m$ for some for $d\geq 2$ and $m\geq 3$. We prove that the Seshadri constant of the ample generator of the N\'eron-Severi space at a general point $x\in X$ lies in the interval $\left[\sqrt{d}- \frac d m, \sqrt{d}\right]$ and thus approaches the po...
Preprint
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We initiate a combinatorial study of Newton-Okounkov functions on toric varieties with an eye on the rationality of asymptotic invariants of line bundles. In the course of our efforts we identify a combinatorial condition which ensures a controlled behavior of the appropriate Newton-Okounkov function on a toric surface. Our approach yields the rati...
Article
In this paper, we study the question of whether on smooth projective surfaces the denominators in the volumes of big line bundles are bounded. In particular, we investigate how this condition is related to bounded negativity (i.e., the boundedness of self-intersections of irreducible curves). Our 1st result shows that boundedness of volume denomina...
Preprint
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We use homogeneous spectra of multigraded rings to construct toric embeddings of a large family of projective varieties which preserve some of the birational geometry of the underlying variety, generalizing the well-known construction associated to Mori Dream Spaces.
Article
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We characterize property $(N_p)$ on a polarized surface $(X,L)$ with trivial canonical bundle in terms of the (non)existence of certain forbidden subvarieties of $X$.
Preprint
Full-text available
In this paper we study the question of whether on smooth projective surfaces the denominators in the volumes of big line bundles are bounded. In particular we investigate how this condition is related to bounded negativity (i.e., the boundedness of self-intersections of irreducible curves). Our first result shows that boundedness of volume denomina...
Preprint
Full-text available
In this paper we study the question of whether on smooth projective surfaces the denominators in the volumes of big line bundles are bounded. In particular we investigate how this condition is related to bounded negativity (i.e., the boundedness of self-intersections of irreducible curves). Our first result shows that boundedness of volume denomina...
Preprint
Full-text available
We show that the subgraph of the concave transform of a multiplicative filtration on a section ring is the Newton--Okounkov body of a certain semigroup, and if the filtration is induced by a divisorial valuation, then the associated graded algebra is the algebra of sections of a concrete line bundle in higher dimension. We use this description to g...
Article
Full-text available
We study asymptotic invariants of linear series on surfaces with the help of Newton-Okounkov polygons. Our primary aim is to understand local positivity of line bundles in terms of convex geometry. We work out characterizations of ample and nef line bundles in terms of their Newton-Okounkov bodies, treating the infinitesimal case as well. One of th...
Chapter
The purpose of this short note is to draw more attention to a very general finite generation problem arising in valutation theory with exciting links to both algebra and geometry. In particular, we propose a few problems with the aim of connecting finite generation in local versus global settings.
Preprint
Full-text available
We prove a Fujita-type theorem for varieties with numerically trivial canonical bundle using properties of semihomogeneous bundles on abelian varieties. We combine our results with work of Riess on compact hyperk\"{a}hler manifolds and work of Mukai, Pareschi and Yoshioka to obtain effective global generation statements for certain moduli spaces of...
Preprint
Full-text available
We prove a Fujita-type theorem for varieties with numerically trivial canonical bundle. We deduce our result via a combination of algebraic and analytic methods, including the Kobayashi-Hitchin correspondence and positivity of direct image bundles. As an application, we combine our results with recent work of U. Riess on generalized Kummer varietie...
Article
There are several flavors of positivity in Algebraic Geometry. They range from conditions that determine vanishing of cohomology, to intersection theoretic properties, and to convex geometry. They offer excellent invariants that have been shown to govern the classification and the parameterization programs in Algebraic Geometry, and are finer than...
Article
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We ask when the CM (Castelnuovo–Mumford) regularity of a vector bundle on a projective variety X is numerical, and address the case when X is an abelian variety. We show that the continuous CM-regularity of a semihomogeneous vector bundle on an abelian variety X is a piecewise-constant function of Chern data, and we also use generic vanishing theor...
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The purpose of this paper is to charazterize asymptotic base loci of line bundles on projective varieties via Newton-Okounkov bodies.
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In this paper we explore the connection between asymptotic base loci and Newton-Okounkov bodies associated to infinitesimal flags. Analogously to the surface case, we obtain complete characterizations of augmented and restricted base loci. Interestingly enough, an integral part of the argument is a study of the relationship between certain simplice...
Article
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This is a survey article on Newton-Okounkov bodies in projective geometry focusing on the relationship between positivity of divisors and Newton-Okounkov bodies.
Preprint
This is a survey article on Newton-Okounkov bodies in projective geometry focusing on the relationship between positivity of divisors and Newton-Okounkov bodies.
Article
Given a smooth projective algebraic surface X, a point O in X and a big divisor D on X, we consider the set of all Newton-Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (E,p) which is infinitely near to O, in the sense that there is a sequence of blowups...
Article
In this paper we relate the SHGH Conjecture to the rationality of one-point Seshadri constants on blow ups of the projective plane.
Article
The aim of this note is to establish a somewhat surprising connection between functions on Newton-Okounkov bodies and Seshadri constants of line bundles on algebraic surfaces.
Article
In this note we relate the SHGH Conjecture to the rationality of one-point Seshadri constants on blow ups of the projective plane, and explain how rationality of Seshadri constants can be tested with the help of functions on Newton--Okounkov bodies.
Preprint
Given a smooth projective algebraic surface X, a point O in X and a big divisor D on X, we consider the set of all Newton-Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (E,p) which is infinitely near to O, in the sense that there is a sequence of blowups...
Article
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The purpose of this note is to find an elemenary explanation of a surprising result of Ein--Lazarsfeld--Smith \cite{ELS} and Hochster--Huneke \cite{HH} on the containment between symbolic and ordinary powers of ideals in simple cases. This line of research has been very active ever since, see for instance \cites{BC,HaH,DST} and the references there...
Article
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Building on the theory of infinitesimal Newton-Okounkov bodies and previous work of Lazarsfeld-Pareschi-Popa, we present a Reider-type theorem for higher syzygies of ample line bundles on abelian surfaces.
Chapter
Contemporary research in algebraic geometry is the focus of this collection, which presents articles on modern aspects of the subject. The list of topics covered is a roll-call of some of the most important and active themes in this thriving area of mathematics: the reader will find articles on birational geometry, vanishing theorems, complex geome...
Article
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Negative curves play a prominent role in the geometry of pro-jective surfaces. They occur naturally as the irreducible components of exceptional loci of resolutions of surface singularities, at the same time, they are closely related to the geometry of the effective cone, and thus form an important building block of the Minimal Model Program. In th...
Article
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The aim of this note is to shed some light on the relationships among some notions of positivity for vector bundles that arose in recent decades. Our purpose is to study several of the positivity notions studied for vector bundles with some notions of asymptotic base loci that can be defined on the variety itself, rather than on the projectivizatio...
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We prove a sharp inequality relating the Castelnuovo--Mumford regularity of a coherent ideal sheaf to its log-canonical threshold.
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It is well known that multi-point Seshadri constants for a small number $s$ of points in the projective plane are submaximal. It is predicted by the Nagata conjecture that their values are maximal for $s\geq 9$ points. Tackling the problem in the language of valuations one can make sense of $s$ points for any positive real $s\geq 1$. We show somewh...
Article
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We study the relationship between positivity of restriction of line bundles to general complete intersections and vanishing of their higher cohomology. As a result, we extend classical vanishing theorems of Kawamata-Viehweg and Fujita to possibly non-nef divisors. Résumé. - Nous étudions la relation entre la positivité des restrictions aux intersec...
Article
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We study curves of negative self-intersection on algebraic surfaces. We obtain results for smooth complex projective surfaces X on the number of reduced, irreducible curves C of negative self-intersection C^2. The only known examples of surfaces for which C^2 is not bounded below are in positive characteristic, and the general expectation is that n...
Article
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The volume of a Cartier divisor is an asymptotic invariant, which measures the rate of growth of sections of powers of the divisor. It extends to a continuous, homogeneous, and log-concave function on the whole N\'eron--Severi space, thus giving rise to a basic invariant of the underlying projective variety. Analogously, one can also define the vol...
Article
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We give an overview of partial positivity conditions for line bundles, mostly from a cohomological point of view. Although the current work is to a large extent of expository nature, we present some minor improvements over the existing literature and a new result: a Kodaira-type vanishing theorem for effective q-ample Du Bois divisors and log canon...
Article
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We define and study the vanishing sequence along a real valuation of sections of a line bundle on a projective variety. Building on previous work of the first author with Huayi Chen, we prove an equidistribution result for vanishing sequences of large powers of a big line bundle, and study the limit measure. In particular, the latter is described i...
Article
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We study topological properties of functions on Okounkov bodies as introduced by Boucksom-Chen and Witt-Nystr\"om. We note that they are continuous over the whole Okounkov body whenever the body is polyhedral, on the other hand, we exhibit an example that shows that continuity along the boundary does not hold in general.
Article
Based on the work of Okounkov (\cite{Ok96}, \cite{Ok03}), Lazarsfeld and Musta\c t\u a (\cite{LM08}) and Kaveh and Khovanskii (\cite{KK08}) have independently associated a convex body, called the Okounkov body, to a big divisor on a smooth projective variety with respect to a complete flag. In this paper we consider the following question: what can...
Article
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We show that the Okounkov body of a big divisor with finitely generated section ring is a rational simplex, for an appropriate choice of flag; furthermore, when the ambient variety is a surface, the same holds for every big divisor. Under somewhat more restrictive hypotheses, we also show that the corresponding semigroup is finitely generated.
Article
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There are two main examples where a version of the Minimal Model Program can, at least conjecturally, be performed successfully: the first is the classical MMP associated to the canonical divisor, and the other is Mori Dream Spaces. In this paper we formulate a framework which generalises both of these examples. Starting from divisorial rings which...
Chapter
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In the week 3--9, October 2010, the Mathematisches Forschungsinstitut at Oberwolfach hosted a mini workshop Linear Series on Algebraic Varieties. These notes contain a variety of interesting problems which motivated the participants prior to the event, and examples, results and further problems which grew out of discussions during and shortly after...
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Based on a recent work by Thomas Bauer reproving the existence of Zariski decompositions for surfaces, we construct a b-divisorial analogue of Zariski decomposition in all dimensions.
Article
We establish a generalization of the Briancon-Skoda theorem about integral closures of ideals for graded systems of ideals satisfying a certain geometric condition.
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We consider a Cartier divisor L on a d-dimensional complex projective variety X. It is well-known that the dimensions of the cohomomology groups H^i(X,O_X(mL)) grow at most like m^d, and it is natural to ask when one of these actually has maximal growth. For i = 0, this happens by definition exactly when L is big. Here we focus on the question of w...
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In this paper we define certain analogues of the volume of a divisor - called asymptotic cohomological functions - and investigate their behaviour on the Neron--Severi space. We establish that asymptotic cohomological functions are invariant with respect to the numerical equivalence of divisors, and that they give rise to continuous functions on th...
Article
We study functions on the class group of a toric variety measuring the rates of growth of the cohomology groups of multiples of divisors. We show that these functions are piecewise polynomial with respect to finite polyhedral chamber decompositions. As applications, we express the self-intersection number of a T-Cartier divisor as a linear combinat...
Article
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The purpose of this paper is to investigate the behaviour of certain asymptotic invariants of line bundles on projective surfaces. In particular, we describe the volume of line bundles and their destabilizing numbers.
Thesis
In the present thesis we consider the asymptotic behavior of the cohomology groups of divisors on projective varieties. Based on the concept of the volume of a divisor, we construct a sequence of asymptotic invariants, called asymptotic cohomological functions, which are defined on the Neron-Severi space. These invariants measure the asymptotic gro...
Article
In this paper we present a divisorial valuation with irrational volume using an algebro-geometric construction.
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 We consider the algorithmic problem of computing Levi decompositions in Lie algebras and Wedderburn–Malcev decompositions in associative algebras over the field of rational numbers. We propose deterministic polynomial time algorithms for both problems. The methods are based on the corresponding classical existence theorems. Computational experienc...
Article
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We consider certain cohomological invariants called asymptotic cohomological functions, which are associated to irreducible projective varieties. Asymptotic cohomological functions are generalizations of the concept of the volume of a line bundle—the asymptotic growth of the number of global sections—to higher cohomology. We establish that they giv...

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