Big Divisor Research Articles

Compact moduli of K3 surfaces with a nonsymplectic automorphism

We construct a modular compactification via stable slc pairs for the moduli spaces of K3 surfaces with a nonsymplectic group of automorphisms under the assumption that some combination of the fixed loci of automorphisms defines an effective big divisor, and prove that it is semitoroidal.

Seshadri-type constants and Newton-Okounkov bodies for non-positive at infinity valuations of Hirzebruch surfaces

We consider flags E • = {X ⊃ E ⊃ {q}}, where E is an exceptional divisor defining a non-positive at infinity divisorial valuation νE of a Hirzebruch surface , q a point in E and X the surface given by νE , and determine an analogue of the Seshadri constant for pairs (νE , D), D being a big divisor on . The main result is an explicit computation of the vertices of the Newton-Okounkov bodies of pairs (E •, D) as above, showing that they are quadrilaterals or triangles and distinguishing one case from another.

On the number of vertices of Newton–Okounkov polygons

The Newton–Okounkov body of a big divisor D on a smooth surface is a numerical invariant in the form of a convex polygon. We study the geometric significance of the shape of Newton–Okounkov polygons of ample divisors, showing that they share several important properties of Newton polygons on toric surfaces. In concrete terms, sides of the polygon are associated to some particular irreducible curves, and their lengths are determined by the intersection numbers of these curves with D . As a consequence of our description, we determine the numbers k such that D admits some k -gon as a Newton–Okounkov body, elucidating the relationship of these numbers with the Picard number of the surface, which was first hinted at by work of Küronya, Lozovanu and Maclean.

Iitaka fibrations for dlt pairs polarized by a nef and log big divisor

AbstractWe study lc pairs polarized by a nef and log big divisor. After proving the minimal model theory for projective lc pairs polarized by a nef and log big divisor, we prove the effectivity of the Iitaka fibrations and some boundedness results for dlt pairs polarized by a nef and log big divisor.

Parallel Strategy to Factorize Fermat Numbers with Implementation in Maple Software

In accordance with the traits of parallel computing, the paper proposes a parallel algorithm to factorize the Fermat numbers through parallelization of a sequential algorithm. The kernel work to parallelize a sequential algorithm is presented by subdividing the computing interval into subintervals that are assigned to the parallel processes to perform the parallel computing. Maple experiments show that the parallelization increases the computational efficiency of factoring the Fermat numbers, especially to the Fermat number with big divisors.

Zariski’s conjecture and Euler–Chow series

We study the relations between the finite generation of Cox ring, the rationality of Euler–Chow series and Poincaré series and Zariski’s conjecture on dimensions of linear systems. We prove that if the Cox ring of a smooth projective variety is finitely generated, then all Poincaré series of the variety are rational. We also prove that the multi-variable Poincaré series associated to big divisors on a smooth projective surface are rational, assuming the rationality of multi-variable Poincaré series on curves.

A generalization of Δ-genus for big divisors on projective varieties

Let X be a complex projective variety, and let D be a big Cartier divisor on X. For this data, we define an invariant that refines the Δ-genus introduced by Fujita for nef and big divisors. We relate to the Okounkov body of D, and give some examples. In particular, we show that can be arbitrarily negative already on surfaces.

Finite Generation of the Algebra of Type A Conformal Blocks via Birational Geometry

Abstract We study the birational geometry of the moduli space of parabolic bundles over a projective line, in the framework of Mori’s program. We show that the moduli space is a Mori dream space. As a consequence, we obtain the finite generation of the algebra of type A conformal blocks. Furthermore, we compute the H-representation of the effective cone that was previously obtained by Belkale. For each big divisor, the associated birational model is described in terms of moduli space of parabolic bundles.

Okounkov Bodies Associated to Pseudoeffective Divisors II

We first prove some basic properties of Okounkov bodies and give a characterization of Nakayama and positive volume subvarieties of a pseudoeffective divisor in terms of Okounkov bodies. Next, we show that each valuative and limiting Okounkov bodies of a pseudoeffective divisor which admits the birational good Zariski decomposition is a rational polytope with respect to some admissible flag. This is an extension of the result of Anderson-Kuronya-Lozovanu about the rational polyhedrality of Okounkov bodies of big divisors with finitely generated section rings.

Newton–Okounkov bodies sprouting on the valuative tree

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 mapping the smooth, irreducible rational curve E to O. The main objective of this paper is to start a systematic study of the variation of these infinitesimal Newton-Okounkov bodies as (E, p) varies, focusing on the case X = P2.

On the ampleness and bigness of non-integral divisors

Given a Weil non-integral divisor D, it is natural to associate it the line bundle of its integral part $$\mathcal {O}_X([D])$$ . In this work we study which of the classical characterizations of ample and big divisors can be extended to non-integral divisors via the corresponding line bundles.

Local positivity in terms of Newton–Okounkov bodies

In recent years, the study of Newton–Okounkov bodies on normal varieties has become a central subject in the asymptotic theory of linear series, after its introduction by Lazarsfeld–Mustaţă and Kaveh–Khovanskii. One reason for this is that they encode all numerical equivalence information of divisor classes (by work of Jow). At the same time, they can be seen as local positivity invariants, and Küronya–Lozovanu have studied them in depth from this point of view.We determine what information is encoded by the set of all Newton–Okounkov bodies of a big divisor with respect to flags centered at a fixed point of a surface, by showing that it determines and is determined by the numerical equivalence class of the divisor up to negative components in the Zariski decomposition that do not go through the fixed point.

Restricted volumes of effective divisors

We study the restricted volume of effective divisors, its properties and the relationship with the related notion of reduced volume, defined via multiplier ideals, and with the asymptotic intersection number. We build upon the fundamental work of Lazarsfeld and Mustata relating the restricted volume of big divisors to the volume of the associated Okounkov body. We extend their constructions and results to the case of effective divisors, recovering some results of Kaveh and Khovanskii, proving a Fujita-type approximation in this larger setting and studying the restricted volume function. In order to relate the reduced volume and the asymptotic intersection number we investigate a boundedness property of asymptotic multiplier ideals and prove it holds, for instance, for finitely generated divisors. In this way we obtain also a complete picture for the canonical divisor of an arbitrary smooth projective variety and for nef divisors on varieties of dimension at most 3.

Welschinger invariants of real del Pezzo surfaces of degree ≥ 2

We compute the purely real Welschinger invariants, both original and modified, for all real del Pezzo surfaces of degree ≥ 2. We show that under some conditions, for such a surface X and a real nef and big divisor class D ∈ Pic (X), through any generic collection of - DKX - 1 real points lying on a connected component of the real part ℝX of X one can trace a real rational curve C ∈ |D|. This is derived from the positivity of appropriate Welschinger invariants. We furthermore show that these invariants are asymptotically equivalent, in the logarithmic scale, to the corresponding genus zero Gromov–Witten invariants. Our approach consists in a conversion of Shoval–Shustin recursive formulas counting complex curves on the plane blown up at seven points and of Vakil's extension of the Abramovich–Bertram formula for Gromov–Witten invariants into formulas computing real enumerative invariants.

Minkowski decomposition of Okounkov bodies on surfaces

We prove that the Okounkov body of a big divisor with respect to a general flag on a smooth projective surface whose pseudo-effective cone is rational polyhedral decomposes as the Minkowski sum of finitely many simplices and line segments arising as Okounkov bodies of nef divisors.

Swiss-Cheese Gravitino Dark Matter

We present a phenomenological model which we show can be obtained as a local realization of large volume D3/D7μ-Split SUSY on a nearly special Lagrangian three-cycle embedded in the big divisor of a Swiss-Cheese Calabi-Yau [Mansi Dhuria, Aalok Misra, arXiv:1207.2774 [hep-ph], Nucl. Phys. B867 (2013) 636–748]. After identification of the first generation of SM leptons and quarks with fermionic super-partners of four Wilson line moduli, we discuss the identification of gravitino as a potential dark matter candidate. We also show that it is possible to obtain a 125 GeV light Higgs in our setup.

On partially ample divisors

Partially ample divisors are defined by relaxing the different conditions that characterize the ample divisors. We prove that for nef and big divisors such notions coincide. We also prove that the partial ampleness of big divisors are preserved in the positive parts of the Fujita‐Zariski decomposition.

Nef and big divisors on toric weak Fano 3-folds

We show that a nef and big line bundle whose adjoint bundle has non-zero global sections on a nonsingular toric weak Fano 3-fold is normally generated. As a consequence, we see that any ample line bundle on a nonsingular toric waek Fano 3-fold is normally generated. As an application, we see that an ample line bundle on a Calabi-Yau hypersurface in a nonsingular toric Fano 4-fold is normally generated. Mathematics Subject Classification: Primary 14M25; Secondary 14M30, 14M45, 52B20

Nakamaye’s theorem on log canonical pairs

We generalize Nakamaye's description, via intersection theory, of the augmented base locus of a big and nef divisor on a normal pair with log-canonical singularities or, more generally, on a normal variety with non-lc locus of dimension at most 1. We also generalize Ein-Lazarsfeld-Mustata-Nakamaye-Popa's description, in terms of valuations, of the subvarieties of the restricted base locus of a big divisor on a normal pair with klt singularities.

Algebraic bounds on analytic multiplier ideals

Given a pseudo-effective divisor L we construct the diminished ideal of L, a continuous extension of the asymptotic multiplier ideal for big divisors to the pseudo-effective boundary. For most pseudo-effective divisors L the multiplier ideal of the metric of minimal singularities of L is contained in the diminished ideal. We also characterize abundant divisors using the diminished ideal, indicating that in this case the geometric and analytic information should coincide.