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Linear series on metrized complexes of algebraic curves

Algebra & Discrete Mathematics

Speaker: Matt Baker, UC Berkeley
Location: 2112 MSB
Start time: Tue, May 8 2012, 2:10PM

A metrized complex of algebraic curves is, roughly speaking, a finite metric graph together with a collection of marked complete nonsingular algebraic curves Cv, one for each vertex of the graph. The marked points on Cv 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.