2012/04/16 by Omid Amini, Matthew Baker, Amini, Omid +1 · 1 citation
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1204.3508
openalex publication_date 2012/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.