Cubic hypersurfaces are rationally simply connected -- Matt DeLand, February 6, 2009

Rational simple connectedness is an analogue of simple connectedness for complex varieties having important applications: every 2-parameter family of rationally simply connected varieties has a rational section, and a 1-parameter family has so many rational sections that they approximate every power series section to arbitrary order. Unfortunately the condition is quite difficult to verify and is known to hold only for homogeneous spaces and also for some projective hypersurfaces satisfying a list of hypotheses. My new approach works by studying a canonically defined foliation on the moduli space of rational curves on the variety. By proving integrability of this foliation, I prove that all smooth cubic hypersurfaces in CP^n are rationally simply connected for n at least 9 (it does not hold for n < 9).