Newton-Okounkov Bodies over Discrete Valuation Rings

Eric Katz, Stefano Urbinati

Published 2016-09-20Version 1

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. 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 families of curves. We describe the Newton-Okounkov for semistable families of curves in terms of the Baker-Norine theory of linear systems on graphs, making a connection between asymptotics of linear systems and tropical geometry.