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Seminar

Graduate student seminar: "Degenerations of projective space induced by an affine building" September 14, 2009
Parent Program: --
Location: MSRI: Baker Board Room
Speaker(s) Mathias Häbich
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Abstract/Media

Let K be a discretely valued field with ring of integers R. To every set
of vertices in a certain simplicial complex called the affine building
of SL_d(K), one can assign a scheme over R. Its special fiber is a
degeneration of projective (d-1)-space over K. This scheme has been
studied by Mumford[1] in the case d = 2 in his paper on uniformization
of p-adic curves, and for d > 2 special cases have bee studied by
Mustafin[2].

We want to understand the general case, combinatorically as well as
algebraically. Tropical Geometry seems to provide the right tools. The
combinatorial connection between the building and Tropical Geometry has
been explored by Joswig, Sturmfels and Yu[3].

[1] D. Mumford: The irreducibility of the space of curves of given
genus. Publ. Math. IHES 1969
[2] G. Mustafin: Nonarchimedian uniformization. Math. USSR Sbornik 1978
[3] Michael Joswig, Bernd Sturmfels, Josephine Yu: Affine buildings and
tropical convexity. Albanian J. of Math. 2007
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