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

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