Mathematics Colloquia and Seminars

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A metrized complex of algebraic curves is, roughly speaking, a finite metric graph together with a collection of marked complete nonsingular algebraic curves C_v, one for each vertex of the graph. The marked points on C_v correspond bijectively to the edges of the graph incident to v. We establish a Riemann-Roch theorem for metrized complexes of algebraic curves which generalizes both the classical Riemann-Roch theorem and its tropical and graph-theoretic analogues. We also show that the rank of a divisor cannot go down under specialization from curves to metrized complexes. As an application of these ideas, we formulate a partial generalization of the Eisenbud-Harris theory of limit linear series to semistable curves which are not necessarily of compact type. This is joint work with Omid Amini.