Specialization of linear systems from curves to graphs
Matthew H. Baker · Algebra & Number Theory · 2008
We investigate the interplay between linear systems on curves and graphs in the context of specialization of divisors on an arithmetic surface.We also provide some applications of our results to graph theory, arithmetic geometry, and tropical geometry.ޑ the set of "rational points" of (see Section 1D).R a complete discrete valuation ring with field of fractions K and algebraically closed residue field k.K a fixed algebraic closure of K .X a smooth, proper, geometrically connected curve over K .X a proper model for X over R. For simplicity, we assume unless otherwise stated that X is regular, that the irreducible components of X k are all smooth, and that all singularities of X k are ordinary double points.Unless otherwise specified, by a smooth curve we will always mean a smooth, proper, geometrically connected curve over a field, and by an arithmetic surface we will always mean a proper flat scheme X over a discrete valuation ring such that the generic fiber of X is a smooth curve.We will usually, but not always, be MSC2000: primary 14H25, 05C99; secondary 14H51, 14H55.