Linear series on metrized complexes of algebraic curves

A metrized complex of algebraic curves over an algebraically closed field is, roughly speaking, a finite metric graph together with a collection of marked complete nonsingular algebraic curves over , one for each vertex of ; the marked points on are in bijection with the edges of incident to . We define linear equivalence of divisors and establish a Riemann-Roch theorem for metrized complexes of curves which combines the classical Riemann-Roch theorem over with its graph-theoretic and tropical analogues from Amini and Caporaso (Adv Math 240:1-23, 2013); Baker and Norine (Adv Math 215(2):766-788, 2007); Gathmann and Kerber (Math Z 259(1):217-230, 2008) and Mikhalkin and Zharkov (Tropical curves, their Jacobians and Theta functions. Contemporary Mathematics 203-231, 2007), providing a common generalization of all of these results. For a complete nonsingular curve defined over a non-Archimedean field , together with a strongly semistable model for over the valuation ring of , we define a corresponding metrized complex of curves over the residue field of and a canonical specialization map from divisors on to divisors on which preserves degrees and linear equivalence. We then establish generalizations of the specialization lemma from Baker (Algebra Number Theory 2(6):613-653, 2008) and its weighted graph analogue from Amini and Caporaso (Adv Math 240:1-23, 2013), showing that the rank of a divisor cannot go down under specialization from to . As an application, we establish a concrete link between specialization of divisors from curves to metrized complexes and the theory of limit linear series due to Eisenbud and Harris (Invent Math 85:337-371, 1986). Using this link, we formulate a generalization of the notion of limit linear series to curves which are not necessarily of compact type and prove, among other things, that any degeneration of a in a regular family of semistable curves is a limit on the special fiber.