Weak approximation and R-equivalence over function fields of curves
-- Jason Starr, September 11, 2009

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Hassett and Tschinkel conjecture that a one-parameter family of
projective schemes whose general member is rationally connected always
satisfies "weak approximation", i.e., every power series solution in the
parameter is approximated to arbitrary order by polynomial solutions in the
parameter.  I will explain joint work with Mike Roth proving this conjecture
when the fibers of the family over Laurent series fields are "R-connected",
the analogue of rational connectedness over non-algebraically closed fields.
This implies all the known cases of the Hassett-Tschinkel conjecture, and we
also give some new ones.  The proof uses "pseudo ideal sheaves", which
generalize ideal sheaves in the same way that Fulton's effective pseudo
divisors generalize effective divisors.