|Date: Tuesday, February 10, 2009
Title: Rational simple connectedness and Serre's "Conjecture II", Keeler Lectures
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.
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.
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