Pl\"ucker varieties and higher secants of Sato's Grassmannian

Abstract

Every Grassmannian, in its Pl\"ucker embedding, is defined by quadratic polynomials. We prove a vast, qualitative, generalisation of this fact to what we call Pl\"ucker varieties. A Pl\"ucker variety is in fact a family of varieties in exterior powers of vector spaces that, like the Grassmannian, is functorial in the vector space and behaves well under duals. A special case of our result says that for each fixed natural number k, the k-th secant variety of any Pl\"ucker-embedded Grassmannian is defined in bounded degree independent of the Grassmannian. Our approach is to take the limit of a Pl\"ucker variety in the dual of a highly symmetric space known as the infinite wedge, and to prove that up to symmetry the limit is defined by finitely many polynomial equations. For this we prove the auxilliary result that for every natural number p the space of p-tuples of infinite-by-infinite matrices is Noetherian modulo row and column operations. Our results have algorithmic counterparts: every bounded Pl\"ucker variety has a polynomial-time membership test, and the same holds for Zariski-closed, basis-independent properties of p-tuples of matrices.