Title: Nonsingular cubic hypersurfaces in P^9 are strongly rationally simply
connected.

Abstract:

Roughly speaking a smooth projective variety is called rationally simply
connected if it is rationally connected and there exists a countable sequence of
irreducible components of moduli spaces of rational curves which are themselves
birationally rationally connected. It turns out that this notion is not quite
strong enough to capture the essence of what should be a ``rationally simply
connected variety''. A more promising notion is that of a strongly rationally
simply connected variety where we ask, for every m \geq 0, that for
some irreducible
component M of M_{0, m}(X) the general fibre of the evaluation morphism

       ev_m : M_{0, m}(X) ----> X \times ... \times X

restricted to M is rationally connected. Work of Starr/Harris/de Jong shows
that this is related to the existence of very twisting scrolls in X.
In the talk we will discuss a method for verifying the existence of
very twisting surfaces on a variety, and apply it to the case of
smooth cubic hypersurfaces in P^9 - the "boundary" case. This allows
us to conclude that such hypersurfaces are indeed strongly rationally
simply connected.