Real Rational Curves in Grassmannians

Abstract

Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some general fixed figures. For the problem of plane conics tangent to five general conics, the (surprising) answer is that all 3264 may be real. Similarly, given any problem of enumerating p-planes incident on some general fixed subspaces, there are real fixed subspaces such that each of the (finitely many) incident p-planes are real. We show that the problem of enumerating parameterized rational curves in a Grassmannian satisfying simple (codimension 1) conditions may have all of its solutions be real.