Alex Küronya

Asymptotic cohomological functions on projective varieties

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 give a notion invariant under the numerical equivalence of divisors, and extend uniquely to continuous functions on the real Néron–Severi space. To illustrate the theory, we work out these invariants for abelian varieties, smooth surfaces, and certain homogeneous spaces.

Local positivity of linear series on surfaces

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 the main results is a description of moving Seshadri constants via infinitesimal Newton-Okounkov polygons. As an illustration of our ideas we reprove results of Ein-Lazarsfeld on Seshadri constants on surfaces.

Negative curves on algebraic surfaces

We study curves of negative self-intersection on algebraic surfaces. Our main result shows there exist smooth complex projective surfaces X, related to Hilbert modular surfaces, such that X contains reduced, irreducible curves C of arbitrarily negative self-intersection C 2 . Previously the only known examples of surfaces for which C 2 was not bounded below were in positive characteristic, and the general expectation was that no examples could arise over the complex numbers. Indeed, we show that the idea underlying the examples in positive characteristic cannot produce examples over the complex number field, and thus our complex examples require a different approach.

Vanishing sequences and Okounkov bodies

We define and study the vanishing sequence along a real valuation of sections of a line bundle on a normal 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 in terms of restricted volumes for divisorial valuations. We also show on an example that the associated concave function on the Okounkov body can be discontinuous at boundary points.

Infinitesimal Newton–Okounkov bodies and jet separation

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 simplices contained in Newton-Okoukov bodies and jet separation; our results also lead to a convex geometric description of moving Seshadri constants.

Recent developments and open problems in linear series

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 the workshop. A lot of arguments presented here are scattered in the literature or constitute folklore. It was one of our aims to have a usable and easily accessible collection of examples and results.

Computing Levi Decompositions in Lie algebras

Abstract. 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 experiences are discussed at the end of the paper.

A divisorial valuation with irrational volume

of an m-primary ideal a⊆ R. In fact, if qm = a for a fixed ideal a then it is evident that e(a)= vol(v). The volume of a valuation has implicitly been studied already in [4], but it was first explicitly defined in [5]. The terminology is intended to emphasize the relation with global invariants of linear series on projective varieties. A natural question is to what extent the properties of vol(v) mirror those of the Samuel multiplicity. Results of [5,9] assert that

An elementary approach to containment relations between symbolic and ordinary powers of certain monomial ideals

ABSTRACT We provide an elementary explanation of a surprising result of Ein–Lazarsfeld–Smith and Hochster–Huneke on the containment between symbolic and ordinary powers of ideals for a certain class of simple monomial ideals.

Mini-Workshop: Negative Curves on Algebraic Surfaces

Negative curves play a prominent role in the geometry of projective 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 the case of surfaces, classes of negative curves span extremal rays of the Mori cone. Any knowledge about them on a given surface reveals important information on linear series as well. Mathematics Subject Classification (2010): Primary 14C17; Secondary 14G35. Introduction by the Organisers The miniworkshop Negative curves on algebraic surfaces gathered together a variety of mathematicians interested in this subject with a wide spread of backgrounds and professional experience. Participants came from several European countries (France, Germany, Great Britain, Hungary, Norway, Poland, Sweden) and from the United States. Their expertise ranged from advanced graduate students through post-docs to established senior researchers. This variety of experience and background greatly contributed to generating stimulating discussions during the workshop and the working group sessions, leading to what we believe will be the basis for several research collaborations in the near future. 556 Oberwolfach Report 10/2014 The theme of the workshop The primary goal of our meeting was to understand curves on algebraic surfaces, a classical area of mathematics, giving rise to a vast array of research with connections to subjects ranging from differential geometry and number theory to seemingly unrelated fields such as ergodic theory. The workshop revolved around the following long standing conjecture which recently has attracted quite a lot of attention in the field of linear series as well (see e.g. [1], [3], [7], [8], [10], [12]). Conjecture 1 (Bounded Negativity Conjecture (BNC)). Let X be a smooth projective surface over the complex numbers. Then there exists a constant bX such that (C) ≥ −bX for all reduced effective curves C on X. The conjecture is known to hold in a number of cases (as proven in [1]), nevertheless, it is wide open in general. It is of great interest because negative curves appear naturally in different branches of algebraic geometry: a natural source of examples are irreducible components of exceptional divisors on resolutions of surface singularities. For a classical connection, elaborating on Nagata’s famous conjecture, Segre [15], Gimigliano [9], Harbourne [11] and Hirschowitz [13] came to a striking geometric conjecture (part of what is called the SHGH-conjecture in the literature) stating that the only negative curves on the blow up of projective plane P in s ≥ 10 points are the (−1) curves. Considerable recent work leading to exciting partial results (see [5], [6], [2], [14], [4]), has been devoted to proving the SHGH conjecture; the efforts include large amounts of computer experiments. In spite of these efforts, much of the area surrounding negative curves on surfaces remains to be explored. For instance, it is at present not even known if the selfintersection numbers of reduced curves on blow ups of P are bounded from below – as would be the case for blow ups of generic points in P, with bound −1, if the SHGH conjecture is true. The modus operandi of our workshop was to devote half of the available time to research in groups focusing on concrete subproblems. One of the working groups was in fact devoted to discussing exactly the above aspect of the conjecture. It turns out that one obtains interesting examples by looking at curves (on the blow ups of P) coming from arrangements of lines in the projective plane. It has been observed that there is an intriguing relation between highly negative curves on the one hand and counterexamples to a seemingly unrelated problem on containment relations between various symbolic and usual powers of ideals of planar points. A second working group took up the question of boundedness for Shimura curves on ball quotients, in an attempt to reproduce the finiteness results of [1] for Shimura curves with negative self-intersection in quaternionic Hilbert modular Mini-Workshop: Negative Curves on Algebraic Surfaces 557 surfaces. During the workshop we have learned that there is an unpublished result due to Margulis (communicated to us by Domingo Toledo and Misha Kapovich) that uses ergodic theoretic methods to show that there are only finitely many embedded totally geodesic subvarieties of at least half the dimension in certain locally symmetric varieties. This would cover at least the case of smooth Shimura curves on ball quotients. There seems to be a reasonable chance that these ideas can be extended to the case of arbitrary Shimura curves, with the help of Ratner type theorems, but the status of such a statement is unclear. A major difference to the vector bundle methods applied in [1] in the Hilbert modular case is that the ergodic theoretic way does not give rise to effective BNC bounds. The third working group considered the question of possible higher-dimensional generalizations of the Bounded Negativity Conjecture. Several results and examples were given; nevertheless, it appears that most of the natural generalizations are false. The structure of the workshop The aim of the workshop was twofold: to gather together experts working on the three different aspects mentioned above, and to stimulate collaboration by discussing open problems in the field. For this reason every day consisted of two main activities: research talks, two to three in the morning; and working group discussions, in the afternoon. A list of possible questions to work on during the workshop was distributed via email well ahead of the workshop. During the first day, final selections were made. Three main areas of interest emerged from the discussion: negative curves in arithmetic settings, bounded negativity for cycles on higher dimensional varieties, and local negativity and containment relations. Consequently three working groups were formed. The workshop was just the starting point to ignite collaborations on the chosen problems. The working groups continue their efforts. The outcome of these discussions will hopefully appear elsewhere. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”.