Algebraic Geometry Program Seminar

Pluecker Varieties and Higher Secants of Sato's Grassmannian

Every Grassmannian, in its Pluecker embedding, is defined by quadratic polynomials. I will sketch a qualitative generalisation of this fact to "Pluecker varieties", which are families of varieties in exterior powers of vector spaces that, like Grassmannians, are functorial in the vector space and behave well under duals. A special case of our results is that for each fixed natural number k, the k-th secant variety of any Pluecker-embedded Grassmannian is defined in bounded degree independent of the Grassmannian. Our approach is to take the limit of a Pluecker 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. (joint work with Rob Eggermont)