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Polynomial formulation of hypersurface Schubert conditions Previous Next Contents

2.i. Polynomial formulation of hypersurface Schubert conditions

A hypersurface Schubert condition on  p-planes in Cm+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 Cm+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,...,Kmp are in general position, then there are

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