Jason Starr
SUNY Stony Brook

Title: Pseudo ideal sheaves and weak approximation over function fields of curves

Abstract: By Graber-Harris-Starr, a rationally connected fibration over a curve has a rational section.  Hassett and Tschinkel conjecture more, that rationally connected fibrations satisfy "weak approximation": every formal power series section is approximated to arbitrary order by regular sections. They prove weak approximation at places of good reduction.  Mike Roth and I formulate a sharper conjecture relating weak appoximation to R-equivalence of rational points over Laurent series fields.  Using a new notion, "pseudo ideal sheaves", we prove our conjecture in some cases, thus giving a new proof of the Hassett-Tschinkel conjecture at places of good reduction.  By this approach, Zhiyu Tian proved the sharper conjecture at places of potentially good reduction, also settling the Hassett-Tschinkel conjecture for Fano hypersurfaces at places of potentially good reduction.