Real rational curves in Grassmannians

University of Toronto 19 January 2000 Frank Sottile University of Wisconsin-Madison

   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. Such questions arise also in applications; recently, Dietmaier has shown that all 40 positions of the Stewart platform in robotics may be real.

   The purpose of this talk is to describe a general method for constructing real solutions to some enumerative problems. This method is based upon homotopy continuation algorithms and is an analog of Viro's method, or toric deformations. We apply this method to show that a certain enumerative problem of counting rational curves on a Grassmannian coming from quantum cohomology can have all of its solutions be real.