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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 |

Ametrized complex of algebraic curvesis, 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.