arXiv subjects
Publications and source records attributed to Alex Küronya.
We study K-stability of smooth Fano threefolds of Picard rank $2$ and degree $22$ which can be obtained by blowing up a smooth complete intersection of two quadrics in $\mathbb{P}^5$ along a conic. We also describe the automorphism groups of these threefolds.
We show that the asymptotic behavior of the two main competing notions of rank of a linear series on a tropical curve is governed by asymptotic invariants, closely paralleling the theory of volumes in algebraic geometry. We introduce and study tropical notions of volume associated to both divisors and tropical modules. We prove optimal asymptotic results for each case. In addition, we show that the tropical volume is compatible with the tropicalization of curves.
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.
We study effective global generation properties of projectivizations of curve semistable vector bundles over curves and abelian varieties.
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 geometric heart of our approach is the spectral data morphism for Dolbeault moduli spaces on abelian varieties that naturally factors the Hitchin morphism and whose target is not an affine space of pluricanonical sections, but a suitable symmetric product.
We contemplate the range of convex Fujita numbers for minimal smooth projective surfaces according to their position in the Kodaira-Enriques classification.
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-parameter subgroups and centered at a non-toric point are finitely generated.
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 hyperkähler manifolds and work of Mukai, Pareschi and Yoshioka to obtain effective global generation statements for certain moduli spaces of sheaves on abelian surfaces. Among these is the statment that if $\cL$ is an ample line bundle on the Hilbert square $S^{[2]}$ of an abelian surface $S,$ then $\cL^{\otimes m}$ is globally generated for $m \geq 3.$
The homogeneous spectrum of a multigraded finitely generated algebra (in the sense of Brenner-Schröer) 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 for toric prevarieties and study its basic properties. With these tools at hand, we prove a generalization of Payne's and Foster--Gross--Payne's tropical limit theorem for divisorial schemes.
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 rationality of many Seshadri constants that have not been settled before.
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 Néron-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 possibly irrational number $\sqrt d$ as $m$ grows.
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.
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 denominators is equivalent to \emph{primitive bounded negativity}, which in turn is implied by bounded negativity. We connect this result to the study of semi-effective orders of divisors: Our second result shows that negative classes exist that become effective only after taking an arbitrarily large multiple.
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 give a rationality criterion for certain Seshadri constants. Along the way we introduce Newton--Okounkov bodies of abstract graded semigroups and determine conditions for their slices to be Newton--Okounkov bodies of subsemigroups.
We show that if $X$ is an abelian variety of dimension $g \geq 1$ and ${\mathcal E}$ is an M-regular coherent sheaf on $X$, the Castelnuovo-Mumford regularity of ${\mathcal E}$ with respect to an ample and globally generated line bundle ${\mathcal O}(1)$ on $X$ is at most $g$, and that equality is obtained when ${\mathcal E}^{\vee}(1)$ is continuously globally generated. As an application, we give a numerical characterization of ample semihomogeneous vector bundles for which this bound is attained.
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. As an application of our methods we confirm a conjecture of Gross and Popescu on abelian surfaces with a very ample primitive polarization of type $(1,d)$, whenever $d\geq 23$.
This is a survey article on Newton-Okounkov bodies in projective geometry focusing on the relationship between positivity of divisors and Newton-Okounkov bodies.
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 $X$.