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The theory of Newton–Okounkov bodies attaches a convex body to a line bundle on a variety equipped with a flag of subvarieties. This convex body encodes the asymptotic properties of sections of powers of the line bundle. In this article, 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.

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