The special Schubert calculus is real

Frank Sottile MSRI & University of Wisconsin-Madison Algebraic geometry Seminar University of Utah 9 December 1998

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 case known, all solutions may be real. The purpose of this talk is to present two new families of enumerative problems with this property. The first are those arising from special Schubert conditions on p-planes in C^m+p, and the second concerns parameterized rational curves in a Grassmannian satisfying certain simple (codimension 1) conditions.