2016/09/20 by Eric Katz, Katz, Eric, Stefano Urbinati +1
Computer Science · Mathematics · #Polynomial and algebraic computation #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.1609.06036
The theory of Newton-Okounkov bodies attaches a convex body to a line bundle\non a variety equipped with flag of subvarieties. This convex body encodes the\nasymptotic properties of sections of powers of the line bundle. In this paper,\nwe study Newton-Okounkov bodies for schemes defined over discrete valuation\nrings. We give the basic properties and then focus on the case of toric schemes\nand semistable curves. We provide a description of the Newton-Okounkov bodies\nfor semistable curves in terms of the Baker--Norine theory of linear systems on\ngraphs, finding a connection with tropical geometry. We do this by introducing\nan intermediate object, the Newton-Okounkov linear system of a divisor on a\ncurve. We prove that it is equal to the set of effective elements of the real\nBaker-Norine linear system of the specialization of that divisor on the dual\ngraph of the curve. As a bonus, we obtain an asymptotic algebraic geometric\ndescription of the Baker-Norine linear system.\n