Vol. 12, No. 1, 2018

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Local positivity of linear series on surfaces

Alex Küronya and Victor Lozovanu

Vol. 12 (2018), No. 1, 1–34

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.

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Newton–Okounkov bodies, linear series on surfaces, local positivity
Mathematical Subject Classification 2010
Primary: 14C20
Secondary: 14J99, 32Q15, 52B99
Received: 30 May 2016
Revised: 7 April 2017
Accepted: 13 May 2017
Published: 13 March 2018
Alex Küronya
Institut für Mathematik
Johann-Wolfgang-Goethe Universität Frankfurt
D-60325 Frankfurt am Main
Budapest University of Technology and Economics
Department of Algebra
H-1111 Budapest
Victor Lozovanu
Institut für Algebraische Geometrie
Leibniz Universität Hannover
D-30167 Hannover