Every rationally connected variety over the function field of a curve has a rational point

By A. J. Jong, J. Starr. 2003

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Moduli of twisted sheaves (2004, methodology): “When X is a surface (over an algebraically closed field), a theorem of de Jong (which is re-proven in [46] using the theory developed here and the theorem of Graber-Harris-Starr-de Jong [15]) shows that per(X ) = ind(X ).”
Uniformization of $${\mathcal {G}}$$-bundles (2007, background): “The argument for the proof of the following proposition is a simple special case of the argument given by de Jong and Starr to produce sections of rationally connected fibrations ([9]):”
Varieties over a finite field with trivial Chow group of 0-cycles have a rational point (2002, background): “[6], where the finite field is replaced by k = F (curve) for F = C, resp.”
Weak approximation over function fields (2004, background): “We refer the reader to [ 11 ] for related results in positive characteristic (see also Remark 28).”