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Schubert2 > Examples from Schubert, translated > Example from Schubert: Conics on a quintic threefold

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 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 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 , c ]
                                              1,1   1,2   1   2   3
               ------------------------------------------------------------------------------[H   , H   , H   , H   , H   , z]
               (H    + c , H    + H   c  + c , H   c  + H   c  + c , H   c  + H   c , H   c )  1,1   1,2   1,3   1,4   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 , c ]
                                    1,1   1,2   1   2   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