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

TL;DR

This paper extends the theory of Newton-Okounkov bodies to schemes over discrete valuation rings, linking algebraic geometry with graph theory and tropical geometry through semistable curves and linear systems.

Contribution

It introduces Newton-Okounkov bodies for schemes over discrete valuation rings and connects them with Baker--Norine linear systems on graphs, revealing new geometric insights.

Findings

Newton-Okounkov bodies are described for semistable curves.

A correspondence between Newton-Okounkov linear systems and Baker-Norine systems is established.

An asymptotic algebraic geometric description of Baker-Norine systems is provided.

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.