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arXiv · 1609.06036

Newton-Okounkov Bodies over Discrete Valuation Rings and Linear Systems on Graphs

Eric Katz·Stefano Urbinati
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Abstract

The theory of Newton-Okounkov bodies attaches a convex body to a line bundle on a variety equipped with flag of subvarieties. This convex body encodes the asymptotic properties of sections of powers of the line bundle. In this paper, we study Newton-Okounkov bodies for schemes defined over discrete valuation rings. We give the basic properties and then focus on the case of toric schemes and semistable curves. We provide a description of the Newton-Okounkov bodies for semistable curves in terms of the Baker--Norine theory of linear systems on graphs, finding a connection with tropical geometry. We do this by introducing an intermediate object, the Newton-Okounkov linear system of a divisor on a curve. We prove that it is equal to the set of effective elements of the real Baker-Norine linear system of the specialization of that divisor on the dual graph of the curve. As a bonus, we obtain an asymptotic algebraic geometric description of the Baker-Norine linear system.

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Eric Katz, Stefano Urbinati. 2016-09-20. Newton-Okounkov Bodies over Discrete Valuation Rings and Linear Systems on Graphs. https://arxiv.org/abs/1609.06036

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