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Equations for Schubert varieties Previous Next Contents

3.iv. Equations for Schubert varieties

Let X be a p by (m+p)-matrix, which we think of as coordinates for p-planes and F.  a complete flag. Then the condition that X lie in YF., (the intersection of X with Fm+p+1-ai has dimension at least i), is
           (m+p+1-i-ai)-minors of      for each    i=1,2,...,p.
Observe that in general, this gives many more equations than the codimension |a| of the Schubert variety Ya F..    Using these equations and a choice of local coordinates, we may convert the geometric statement of Conjecture 2 into a statement about certain polynomial systems, which will in general be overdetermined.