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