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Example from Schubert: The number of elliptic cubics on a sextic 4-fold

We thank Rahul Pandharipande for this classic Schubert example. See Enumerative Geometry of Calabi-Yau 4-Folds.

> grass(3,6,c):
i1 : Gc = flagBundle({3,3}, 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(3,Qc):
i3 : B = symmetricPower_3 Qc

o3 = B

o3 : an abstract sheaf of rank 10 on Gc
> Proj(X,dual(B),z):
i4 : X = projectiveBundle'(dual B, VariableNames => {,{z}})

o4 = X

o4 : a flag bundle with subquotient ranks {9, 1}
> A:=Symm(6,Qc)-Symm(3,Qc)&@o(-z):
i5 : A = symmetricPower_6 Qc - symmetricPower_3 Qc ** OO(-z)

o5 = A

o5 : an abstract sheaf of rank 18 on X
> c18:=chern(rank(A),A):
> lowerstar(X,c18):
> integral(Gc,%);
Ans =  2734099200
i6 : integral chern A

o6 = 2734099200