Local positivity of linear series on surfaces

Küronya, Alex; Lozovanu, Victor

doi:10.2140/ant.2018.12.1

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Local positivity of linear series on surfaces

Alex Küronya and Victor Lozovanu

Vol. 12 (2018), No. 1, 1–34

DOI: 10.2140/ant.2018.12.1

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.

Keywords

Newton–Okounkov bodies, linear series on surfaces, local positivity

Mathematical Subject Classification 2010

Primary: 14C20

Secondary: 14J99, 32Q15, 52B99

Milestones

Received: 30 May 2016

Revised: 7 April 2017

Accepted: 13 May 2017

Published: 13 March 2018

Authors

Alex Küronya
Institut für Mathematik Johann-Wolfgang-Goethe Universität Frankfurt D-60325 Frankfurt am Main Germany	Budapest University of Technology and Economics Department of Algebra H-1111 Budapest Hungary

Victor Lozovanu
Institut für Algebraische Geometrie Leibniz Universität Hannover D-30167 Hannover Germany

Vol. 12, No. 1, 2018