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 given general figures [5]. For the problem of plane conics tangent to five general (real) conics, the surprising answer is that all 3264 may be real [13]. Similarly, given any problem of enumerating p-planes incident on some given general subspaces, there are real subspaces such that each of the (finitely many) incident p-planes is real [17]. We show that the problem of enumerating parameterized rational curves in a Grassmannian satisfying simple (codimension 1) conditions may have all of its solutions real. This problem of enumerating rational curves in a Grassmannian arose in at least two distinct areas of mathematics. The number of such curves was predicted by the formula of Vafa and Intriligator [20, 8] from mathematical physics. It is also the number of complex dynamic compensators which stabilize a given linear system, and the enumeration was solved in this context [12, 11]. The question of real solutions also arose in systems theory [3]. This application will be discussed in Section 4.