Recent work by many people has established a strong link between the properties of division algebras over a function field and the geometry of certain associated moduli spaces over the base field. I will explain this link and give one simple application: an effective form of Merkurjev's theorem for function fields of surfaces over finite fields.
Given a smooth proper family of varieties X-->B over an algebraically closed field k of characteristic zero, we will show that there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is a countable field such as the field of algebraic numbers. In fact, we will explain two approaches to this sort of statement - the first is a local p-adic approach, while the second argument is via Hodge theory. As time permits, we will discuss the analogous statement for higher codimension cycles. This is joint work with Bjorn Poonen and Claire Voisin.
I will discuss rational simple connectedness, an algebraic analogue of simple connectedness replacing continuous maps from the unit interval by regular morphisms from the projective line. Then I will discuss how to use this notion to complete the proof of Serre's "Conjecture II" for function fields of surfaces: every algebraic principal bundle for a simply connected, semisimple algebraic group scheme over and algebraic surface has a rational section. Due to tremendouos work of Merkurjev, Suslin, Bayer-Fluckiger, Parimala, Chernousov and Gille, the conjecture reduces to the case when the algebraic group scheme is constant (the "split" case), which is what we prove. This is joint work with de Jong and He.
A Mustafin variety is a degeneration of projective space induced by a point configuration in a Bruhat-Tits building. The special fiber is reduced and Cohen-Macaulay, and its irreducible components form interesting combinatorial patterns. For configurations that lie in one apartment, these patterns are regular mixed subdivisions of scaled simplices, and the Mustafin variety is a twisted Veronese variety built from such a subdivision. This connects our subject to tropical and toric geometry. For general configurations, the irreducible components of the special fiber are rational varieties, and any blow-up of projective space along a linear subspace arrangement can arise. We present a detailed study of Mustafin varieties for configurations in the Bruhat-Tits tree of PGL(2) and in the two-dimensional building of PGL(3). The latter yields the classification of Mustafin triangles into 38 combinatorial types. This is joint work with Dustin Cartwright, Mathias Haebich and Annette Werner. The audience is encouraged to look at the pictures in arXiv:1002.1418.
A major open problem initiated by the pioneering work of Harris and Mumford on Kodaira dimension of the moduli space of stable curves, is to refine our knowledge of its birational geometry by understanding geometry of its birational models. At the moment there are no reasonable conjectures describing the cone of effective divisors and its decomposition into Mori chambers encoding ample divisors on various birational models. In a joint work with Ana-Maria Castravet, we study the genus zero case. We introduce new combinatorial structures called hypertrees and show that they give exceptional divisors on the moduli space of stable rational curves with many remarkable properties. We conjecture that these divisors (along with the boundary divisors) generate the effective cone of the moduli space.
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