Producing algebraic curves in projective families via Floer theory

We will discuss the existence of rational (multi)sections and unirulings for projective families f:X→CP1 with at most two singular fibres. In particular, we will discuss two ingredients that are used to construct the above algebraic curves. 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.