Relatively very free curves and rational simple connectedness

Abstract

Rational connectedness is an algebro-geometric analogue of path connectedness depending crucially on the existence of special rational curves called very free curves. For a family of varieties parametrized by a base space, the relevant curves are relatively very free curves. Rational simple connectedness is an algebro-geometric analogue of simple connectedness that depends analogously on the existence of special families of rational curves called very twisting families. Until now, this has been a mysterious notion, frustrating progress. We clarify this by showing that relatively very free curves give very twisting families. We also introduce a new construction of relatively very free curves by studying a natural integrable foliation on the total space of the family. As a consequence, we prove that a general complete intersection in ℙⁿ is rationally simply connected if and only if it is 2-Fano, i.e., if the first and second graded pieces of the Chern character are positive. This is equivalent to a numerical condition on the type (d₁,...,d_c) of the complete intersection: that d₁² + ⋯ + d_c² ≤ n. This subsumes all previous results on rational simple connectedness of complete intersections. As a corollary, we prove the most general result concerning the Hassett–Tschinkel weak approximation conjecture for 2-Fano complete intersections. Further, combined with a result of de Jong, He and Starr, we prove that every smooth, 2-Fano complete intersection defined over the function field of a surface has a rational point.