1,720,965 research outputs found
Regularity of smooth curves in biprojective spaces
No full textAbstractMaclagan and Smith [D. Maclagan, G.G. Smith, Multigraded Castelnuovo–Mumford regularity, J. Reine Angew. Math. 571 (2004) 179–212] developed a multigraded version of Castelnuovo–Mumford regularity. Based on their definition we will prove in this paper that for a smooth curve C⊆Pa×Pb (a,b⩾2) of bidegree (d1,d2) with nondegenerate birational projections the ideal sheaf IC|Pa×Pb is (d2−b+1,d1−a+1)-regular. We also give an example showing that in some cases this bound is the best possible
Positivity of Line Bundles and Newton-Okounkov Bodies
No full textThe purpose of this paper is to describe asymptotic base loci of line bundles on projective varieties in terms of Newton-Okounkov bodies. As a result, we obtain equivalent characterizations of ampleness and nefness via convex geometry
A Reider-type theorem for higher syzygies on abelian surfaces
No full textBuilding 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
Local positivity of linear series on surfaces
No full textWe 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
Convex bodies appearing as Okounkov bodies of divisors
No full textAbstractBased on the work of Okounkov (Okounkov, 1996 [15], 2003 [16]), Lazarsfeld and Mustaţă (2009) [13] and Kaveh and Khovanskii (preprint) [10] 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 be said about the set of convex bodies that appear as Okounkov bodies? We show first that the set of convex bodies appearing as Okounkov bodies of big line bundles on smooth projective varieties with respect to admissible flags is countable. We then give a complete characterisation of the set of convex bodies that arise as Okounkov bodies of R-divisors on smooth projective surfaces. Such Okounkov bodies are always polygons, satisfying certain combinatorial criteria. Finally, we construct two examples of non-polyhedral Okounkov bodies. In the first one, the variety we deal with is Fano and the line bundle is ample. In the second one, we find a Mori dream space variety such that under small perturbations of the flag the Okounkov body remains non-polyhedral
Okounkov bodies of finitely generated divisors
No full textWe 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
Volume functions of linear series
No full textThe 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 in- variant of the underlying projective variety. Analogously, one can also define the volume function of a possibly non-complete multigraded linear series. In this paper we will address the question of characterizing the class of functions arising on the one hand as volume functions of multigraded linear series and on the other hand as volume functions of projective varieties. In the multigraded setting, inspired by the work of Lazarsfeld and Mustat¸˘a [16] on Ok- ounkov bodies, we show that any continuous, homogeneous, and log-concave function ap- pears as the volume function of a multigraded linear series. By contrast we show that there exists countably many functions which arise as the volume functions of projective varieties. We end the paper with an example, where the volume function of a projective variety is given by a transcendental formula, emphasizing the complicated nature of the volume in the classical case
From Convex Geometry of Certain Valuations to Positivity Aspects in Algebraic Geometry
No full textA few years ago Okounkov associated a convex set (Newton–Okounkov body) to a divisor, encoding the asymptotic vanishing behaviour of all sections of all powers of the divisor along a fixed flag. This brought to light the following guiding principle “use convex geometry, through the theory of these bodies, to study the geometrical/algebraic/arithmetic properties of divisors on smooth projective varieties”. The main goal of this survey article is to explain some of the philosophical underpinnings of this principle with a view towards studying local positivity and syzygetic properties of algebraic varieties
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