Real Rational Curves in Grassmannians

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. Introduction 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 [5]. For the problem of plane conics tangent to five general conics, the (surprising) answer is that all 3264 may be real [12]. 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 [16]. 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. This problem of enumerating rational curves on 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 [19, 7] from mathematical physics. It is also the number of complex dynamic compensators which stabilize a particular linear system, and the enumeration was solved in this context [11, 10]. The question of real solutions also arises in systems theory [3]. Our proof, while exploiting techniques from systems theory, has no direct implications for the problem of real dynamic output compensation. 1. Statement of results We work with complex algebraic varieties and ask when a priori complex solutions to an enumerative problem are real. Fix integers m, p > 1 and q ≥ 0. Set n := m+p. Date: 23 April 1999. 1991 Mathematics Subject Classification. 14P99, 14N10, 14M15, 14Q20, 93B55, 65H10.