Linear series on metrized complexes of algebraic curves

A metrized complex of algebraic curves over an algebraically closed field κ is, roughly speaking, a finite metric graph Γ together with a collection of marked complete nonsingular algebraic curves C_v over κ, one for each vertex v of Γ; the marked points on C_v are in bijection with the edges of Γ incident to v. 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 κ 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 X defined over a non-Archimedean field K, together with a strongly semistable model X for X over the valuation ring R of K, we define a corresponding metrized complex CX of curves over the residue field κ of K and a canonical specialization map τ∗^CX from divisors on X to divisors on CX 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 CX. 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 g^d _r in a regular family of semistable curves is a limit g^d _r on the special fiber.