Polynomial formulation of hypersurface Schubert conditions

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 2 Oct 2005. For the current production of this journal, please refer to http://www.tandfonline.com/loi/uexm20.

2.i. Polynomial formulation of hypersurface Schubert conditions

A hypersurface Schubert condition on p-planes in C^m+p is the condition that a p-plane H meet a fixed m-plane K non-trivially. We write this in local coordinates for the Grassmannian of p-planes in C^m+p. Local coordinates are furnished by a m by p-matrix F of indeterminates, where H is the row space of the p by (m+p)-matrix obtained by concatenating an identity matrix with F. If we consider K to be the row space of a m by (m+p)-matrix, then the condition that H meet K nontrivially is

According to the Theorem of Schubert [Sc], and Kleiman's Transversality Theorem [Kl], if K₁,...,K_mp are in general position, then there are

p-planes H which meet each K_i nontrivially. In particular, the resulting system of mp determinants in the mp indeterminates given by F has d_m,p solutions.