Every separably rationally connected variety over the function field of a curve has a rational
point. Conjecturally, under some additional hypotheses, a rationally 1-connected variety over the function
field of a char 0 surface has a rational point. I will discuss this conjecture, a theorem, and a corollary: a
new proof of de Jong’s period-index theorem unconditional on the characteristic. This is joint work with de
Jong. Time permitting, I will also discuss connections to weak approximation over function fields of curves.

Lecture #12269

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