UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

March 15, 2011

939 Evans Hall

3:45PM: Linear Systems on Curves and Graphs

Melody Chan

I will give an expository talk on the so-called Specialization Lemma, due to Matt Baker, which relates dimensions of linear systems on curves and linear systems on graphs. We will first discuss the intersection theory of divisors on regular fibered surfaces; this will be our main tool for moving between the algebro-geometric and combinatorial worlds. We will then review the theory of divisors on graphs and give an example of a plane quartic curve specializing to the complete graph on four vertices.

5:00PM: On the geometrization of the absolute Galois group

Mehrdad Shashahani

Let M be a compact orientable topological surface and Ma a Riemann surface whose underlying surface is M. Then as an algebraic curve Ma can be defined over a number field if and only if it admits of a non-constant meromorphic function f : Ma → PC1 with at most three critical values according to theorems of Belyi and Weil. Inspired by the work of Belyi, Grothendieck suggested that studying certain graphs (f−1([0,1])), on orientable topological surfaces provides a geometric and combinatorial framework for the action of the absolute Galois group G and may lead to a deeper understanding of G which is known essentially only through its action on algebraic numbers. Among significant developments in this direction are the works of Matzat and collaborators on the Inverse Galois Problem and the works of Drinfeld and Ihara which led to the definition of the Grothendieck- Teichmu ̈ller group and an injection from G into it. However since this maps fails to be surjective its usefulness in understanding the absolute Galois group is somewhat in doubt. This work attempts to look at this general problem in a more topological/geometric framework. There has been a dearth of examples on surfaces of positive genus where the complex structure defined by the dessin is explicitly exhibited by recovering the defining equation of the curve from the dessin. We describe a method by means of which a profusion of such examples can be constructed. Since the absolute Galois group G is an inverse system of finite groups and the goal is to translate the action of G into an action on dessins, it is reasonable to seek families of dessins and associated families of algebraic objects with the structure of inverse systems. In fact such infinite families are exhibited and the relationship of the corresponding algebraic curves and equivalence classes of representations of the cartographic groups are clarified. Finally the relationship between quadratic differentials, certain naturally defined measures on the topological surface and dessin theory is discussed. This presentation is based on the work of my student, Ali Kamalinejad.

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