Weak Approximation and R-equivalence over Function Fields of Curves

For a rationally connected fibration over a complex curve, and for a closed point of the curve, we prove that every power series section of the fibration near the point is approximated to arbitrary order by polynomial sections provided the “Laurent fiber”, i.e., the deleted power series neighborhood of the fiber, as a variety over the Laurent series field is R-connected – the analogue of rational connectedness when working over a non-algebraically closed field such as Laurent series. In other words, we prove the Hassett-Tschinkel conjecture when the Laurent fiber is R-connected. For varieties over the fraction field of a complete DVR, we introduce a “continuous variant” of R-connectedness called pseudo R-connectedness and we prove pseudo R-connectedness of the Laurent fiber also implies the Hassett-Tschinkel conjecture. Our theorem implies all of the known cases of the Hassett-Tschinkel conjecture, and we also prove some new cases. The key is a new object, a “pseudo ideal sheaf”, which generalizes Fulton’s notion of effective pseudo divisor.