Example from Schubert: Conics on a quintic threefold

# Conics on a quintic threefold. This is the top Chern class of the
# quotient of the 5th symmetric power of the universal quotient on the
# Grassmannian of 2 planes in P^5 by the subbundle of quintic containing the
# tautological conic over the moduli space of conics.
>
> grass(3,5,c):         # 2-planes in P^4.

i1 : Gc = flagBundle({2,3}, VariableNames => {,c})

o1 = Gc

o1 : a flag bundle with subquotient ranks {2..3}

i2 : (Sc,Qc) = bundles Gc

o2 = (Sc, Qc)

o2 : Sequence

> B:=Symm(2,Qc):        # The bundle of conics in the 2-plane.

i3 : B = symmetricPower(2,Qc)

o3 = B

o3 : an abstract sheaf of rank 6 on Gc

> Proj(X,dual(B),z):    # X is the projective bundle of all conics.

i4 : X = projectiveBundle'(dual B, VariableNames => {,{z}})

o4 = X

o4 : a flag bundle with subquotient ranks {5, 1}

> A:=Symm(5,Qc)-Symm(3,Qc)&*o(-z):  # The rank 11 bundle of quintics
>                                   # restricted to the universal conic.

i5 : A = symmetricPower_5 Qc - symmetricPower_3 Qc ** OO(-z)

o5 = A

o5 : an abstract sheaf of rank 11 on X

> c11:=chern(rank(A),A):# its top Chern class.

i6 : c11 = chern(rank A, A)

            2 5
o6 = 609250c z
            3

                                                         QQ[][H   ..H   , c ..c ]
                                                               1,1   1,2   1   3
                         ---------------------------------------------------------------------------------------[H   ..H   , z]
                         (- H    - c , - H    - H   c  - c , - H   c  - H   c  - c , - H   c  - H   c , -H   c )  1,1   1,5
                             1,1    1     1,2    1,1 1    2     1,2 1    1,1 2    3     1,2 2    1,1 3    1,2 3
o6 : ----------------------------------------------------------------------------------------------------------------------------------------------
                                           2                                                           2
     (- H    - z - 4c , - H    - H   z + 5c  + 5c , - H    - H   z - 15c c  - 5c , - H    - H   z + 10c  + 20c c , - H    - H   z - 20c c , -H   z)
         1,1         1     1,2    1,1      1     2     1,3    1,2       1 2     3     1,4    1,3       2      1 3     1,5    1,4       2 3    1,5

> lowerstar(X,c11):     # push down to G(3,5).

i7 : X.StructureMap_* c11

            2
o7 = 609250c
            3

                                     QQ[][H   ..H   , c ..c ]
                                           1,1   1,2   1   3
o7 : ---------------------------------------------------------------------------------------
     (- H    - c , - H    - H   c  - c , - H   c  - H   c  - c , - H   c  - H   c , -H   c )
         1,1    1     1,2    1,1 1    2     1,2 1    1,1 2    3     1,2 2    1,1 3    1,2 3

> integral(Gc,");       # and integrate there.
                                     609250

i8 : integral oo

o8 = 609250