Local positivity of linear series on surfaces

Abstract

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.

29 pages

References in corpus (2)

• Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties
• Seshadri constants via Okounkov functions and the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture

Cited by in corpus (18)

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• Toric degenerations and symplectic geometry of smooth projective varieties
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• Okounkov bodies associated to abundant divisors and Iitaka fibrations
• Local positivity in terms of Newton-Okounkov bodies
• Okounkov bodies and Zariski decompositions on surfaces
• Asymptotic base loci via Okounkov bodies
• Higher syzygies on abelian surfaces
• Seshadri constants and Okounkov bodies revisited
• Seshadri-type constants and Newton-Okounkov bodies for non-positive at infinity valuations of Hirzebruch surfaces
• On the valuative Nagata conjecture
• Extended Okounkov bodies and multi-point Seshadri constants
• Newton-Okounkov bodies sprouting on the valuative tree
• Newton-Okoukov bodies and symplectic embeddings into non-toric rational surfaces
• On the finite generation of valuation semigroups on toric surfaces
• Numerical and enumerative results on Okounkov bodies
• Local numerical equivalences and Okounkov bodies in higher dimensions