Nesting maps of Grassmannians

Abstract

Let F be a field and i < j be integers between 1 and n. A map of Grassmannians f : Gr(i, F^n) --> Gr(j, F^n) is called nesting, if l is contained in f(l) for every l in Gr(i, F^n). We show that there are no continuous nesting maps over C and no algebraic nesting maps over any algebraically closed field F, except for a few obvious ones. The continuous case is due to Stong and Grover-Homer-Stong; the algebraic case in characteristic zero can also be deduced from their results. In this paper we give new proofs that work in arbitrary characteristic. As a corollary, we give a description of the algebraic subbundles of the tangent bundle to the projective space P^n over F. Another application can be found in a recent paper math.AC/0306126 of George Bergman.