Twisting surfaces and what you can do with them


Jason Starr and Joe Harris introduced (very) twisting surfaces in a paper titled
``RATIONAL CURVES ON HYPERSURFACES OF LOW DEGREE II''.
It turns out that these are (so far) the best analogue of the notion of a
(very) free rational curve on a smooth projective variety. In the presence of
a very twisting surface (say ruled by lines, i.e., a twisting scroll) a
certain type of question about moduli spaces of higher degree rational curves
on the variety can be answered. We will introduce the notion of a very twisting
surface in its most elementary form and we will try to indicate how to start
using it to prove results of this kind.

We will also briefly discuss some of the initial results of the preprint
``Low degree complete intersections are rationally simply connected''
which are needed to set the stage.