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Linear series on metrized complexes of algebraic curves

A metrized complex of algebraic curves is a finite metric graph together with a collection of marked complete nonsingular algebraic curves, one for each vertex, the marked points being in bijection with incident edges. We establish a Riemann-Roch theorem for metrized complexes of curves which generalizes both the classical Riemann-Roch theorem and its graph-theoretic and tropical analogues due to Baker-Norine, Gathmann-Kerber, and Mikhalkin-Zharkov. We also establish generalizations of the second author's specialization lemma and its weighted graph analogue due to Caporaso and the first author, showing that the rank of a divisor cannot go down under specialization from curves to metrized complexes. As an application of these considerations, we formulate a generalization of the Eisenbud-Harris theory of limit linear series to semistable curves which are not necessarily of compact type.