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Mathematics Colloquium


Date:  Tuesday, February 10, 2009

Title:  Rational simple connectedness and Serre's "Conjecture II", Keeler Lectures

Abstract:  LECTURE I: ational simple connectedness and Serre's "Conjecture II" Rational connectedness and rational simple connectedness are algebro-geometric analogues of path connectedness and simple connectedness. Just as a fibration with 2-dimensional base and simply connected fibers admits a continuous section, also an algebraic fibration over a surface with rationally simply connected fibers admits a rational section (assuming some extra hypotheses). Following a strategy of Ph. Gille, this implies a conjecture of Serre: an (algebraic) principal fiber bundle over a surface for a simply connected, semisimple group has a rational section. This is joint work with A. J. de Jong and Xuhua He. LECTURE 2. Abel maps for fibrations over a curve The theorem from Lecture 1 regarding fibrations over a surface follows from a result about fibrations over a curve. For a rationally simply connected fibration over a curve the Abel map from the parameter space of sections to the Jacobian of the curve has rationally connected fibers if the homology class of the fiber is sufficiently positive. LECTURE 3. Weak approximation and R-equivalence Hassett and Tschinkel asked: does a rationally connected fibration over a curve satisfy "weak approximation"? In other words, is every power series section approximated to arbitrary order by rational sections? Mike Roth and I prove that a rationally connected fibration over a curve satisfies weak approximation if the associated fibers over Laurent series fields are each "R-connected" -- an analogue of rationally connected for varieties over non-algebraically closed fields. This gives new proofs of known weak approximation results. It also suggests most RC fibrations do not satisfy weak approximation. The proof uses a new notion, pseudo ideal sheaves, which are is an analogue of Fulton's pseudo-divisors (the effective ones) for cycles of higher codimension.

Speaker:  Jason Starr
Institution:  SUNY Stony Brook University


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