Real rational curves in Grassmannians

Frank Sottile University of Wisconsin-Madison Differential Geometry Seminar 22 March 1999

In the last century, Schubert defined enumerative geometry to be concerned with `counting geometric figures of some kind having specified position with respect to some general fixed figures'. More recently, Fulton asked how many solutions to such a problem can be real; in particular can the fixed figures be chosen so that all solutions are real.

For the problem of the 3264 conics tangent to 5 general plane conics, Ronga, Tognoli, and Vust found the (surprising) answer that all may be real. In fact, for every known example, all solutions may be real. The purpose of this talk is to present two new families of enumerative problems with this property. One arise from the classical Schubert calculus of enumerative geometry on the Grassmannian, and the second from the quantum Schubert calculus enumerating rational curves on the Grassmannian. The picture above shows the first non-trivial case of both.