Unirulings and sections of projective families over P1

Abstract:  We will discuss the following results: If f: X -> P1 is a morphism of smooth projective varieties over the complex numbers with at most a single singular fibre, then X is chain uniruled by sections of f.  The same conclusion holds, but with genus zero multisections instead of sections, if f has at most two singular fibres, and the first Chern class of X is supported in a single fibre of f.

In particular, we will discuss two ingredients that are used to prove the above results.  The first is local symplectic cohomology groups associated to compact subsets of convex symplectic domains.  The second is a degeneration to the normal cone argument that allows one to produce closed curves in X from open curves (which are produced using local symplectic cohomology) in the complement of X by a singular fibre.