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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 C_{n,k}
and satisfy a < b
but there is no index c in C_{n,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
C_{n,k} and set
H_{a} to be the hyperplane defined
by
p_{a}=0.
Then the intersection of
H_{a} with
X_{a} is generically transverse and equals the union of
the Schubert varieties
X_{b} for all
b in
C_{n,k} that are covered by
a.
In fact this intersection is transverse along each Schubert cell
X_{b}^{o}, which is the difference of
X_{b} and all of its Schubert subvarieties.
We obtain the recursion for the degree d(a)
of the Schubert variety X_{a}
d(a) = 

d(b) , 

(4.8) 
the sum over all b in C_{n,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 C_{n,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 
Next: 4.ii.c. Proof of Theorem 4.4
Up: 4.ii The Special Schubert Calculus
Previous: 4.ii.a. Geometry of Grassmann Varieties