The Wronski map and Grassmanians of real codimension 2 subspaces

For an integer m ≥ 2, let GR = GR(m,m + 2) ⊂ RP , N = m(m + 3)/2, be the Plücker embedding of the Grassmanian of msubspaces in Rm+2 . We consider a central projection of GR into a projective space RP of the same dimension as GR . The topological degree of this projection can be properly defined, although GR may be non-orientable. We find that this degree is 0 for even m , and ±u((m+ 1)/2) for odd m , where u(d) is the d-th Catalan number. This implies that for odd m , the cardinality k of intersection of GR with a generic subspace L ⊂ RP of codimension 2m satisfies k ≥ u((m + 1)/2). It follows that such intersection is non-empty for every subspace of codimension 2m . The estimate is best possible. The result has applications to real enumerative geometry and to the problem of pole assignment by static output feedback in control theory. It implies that, for a generic linear system with an odd number m of inputs, 2 outputs, and state of dimension 2m , one can assign arbitrary symmetric set of 2m poles with a real gain matrix.