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## 4.ii.b The Degrees of Schubert Varieties

An intersection of two varieties X and Y is generically transverse if X and Y meet transversally along an open subset of every component of their intersection. When a and b are in Cn,k and satisfy a < b but there is no index c in Cn,k with a < c < b, then we say that b covers a. The following fact is elementary and due to Schubert.

Theorem 4.5   Let a in Cn,k and set Ha to be the hyperplane defined by pa=0. Then the intersection of Ha with Xa is generically transverse and equals the union of the Schubert varieties Xb for all b in Cn,k that are covered by a.

In fact this intersection is transverse along each Schubert cell Xbo, which is the difference of Xb and all of its Schubert subvarieties. We obtain the recursion for the degree d(a) of the Schubert variety Xa

 d(a)   = d(b) ,
(4.8)

the sum over all b in Cn,k that are covered by a. Since the minimal Schubert variety is a point (which has degree 1), this gives a conceptual formula for d(a). Let 0=(1,2,...,k) be the minimal element in the Bruhat order.

 d(a)   =   The number of saturated chains in the Bruhat order Cn,k from 0 to a . (4.9)

Figure 11 displays both the Bruhat order for k=3 and n=6 (on the left) and the degrees of the corresponding Schubert varieties (on the right).

 Figure 11: Bruhat order and degrees of Schubert varieties, k=3 and n=6

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