> DIM:=4:
> A:=bundle(2,c): # a bundle with Chern classes c1,c2 and rank 2
> B:=bundle(3,d): # a bundle with Chern classes d1,d2,d3 and rank 3
i1 : base(4, Bundle => (A,2,c), Bundle => (B,3,d))
o1 = a variety
o1 : an abstract variety of dimension 4
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> chern(A);
2
1 + c1 t + c2 t
i2 : chern A
o2 = 1 + c + c
1 2
o2 : QQ[c ..c , d ..d ]
1 2 1 3
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> segre(B);
2 2 3 3
1 + d1 t + (d1 - d2) t + (d1 - 2 d1 d2 + d3) t
4 2 2 4
+ (d1 - 3 d2 d1 + 2 d1 d3 + d2 ) t
i3 : segre B
2 3 4 2 2
o3 = 1 + d + (d - d ) + (d - 2d d + d ) + (d - 3d d + d + 2d d )
1 1 2 1 1 2 3 1 1 2 2 1 3
o3 : QQ[c ..c , d ..d ]
1 2 1 3
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> chern(A&*B); # The Chern class of the tensor product
2 2 2
1 + (2 d1 + 3 c1) t + (d1 + 5 c1 d1 + 3 c1 + 2 d2 + 3 c2) t +
3 2 2 3
(6 c1 c2 + 2 d1 d2 + c1 + 2 d3 + 4 c1 d2 + 2 c1 d1 + 4 d1 c2 + 4 d1 c1 ) t
2 2 2
+ (3 c1 d1 d2 + 6 d1 c1 c2 + 3 c2 + 3 c2 c1 + 2 d1 d3 + d2 + 3 c1 d3
2 2 2 2 3 4
+ 2 c2 d1 + 3 c1 d2 + c1 d1 + d1 c1 ) t
i4 : chern(A**B)
2 2 3 2
o4 = 1 + (3c + 2d ) + (3c + 3c + 5c d + d + 2d ) + (c + 6c c + 4c d +
1 1 1 2 1 1 1 2 1 1 2 1 1
------------------------------------------------------------------------
2 2 2 3
4c d + 2c d + 4c d + 2d d + 2d ) + (3c c + 3c + c d + 6c c d +
2 1 1 1 1 2 1 2 3 1 2 2 1 1 1 2 1
------------------------------------------------------------------------
2 2 2 2 2
c d + 2c d + 3c d + 3c d d + d + 3c d + 2d d )
1 1 2 1 1 2 1 1 2 2 1 3 1 3
o4 : QQ[c ..c , d ..d ]
1 2 1 3
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> chern(3,symm(3,dual(A)));
3
- 6 c1 - 30 c1 c2
i5 : chern_3 symmetricPower_3 dual A
3
o5 = - 6c - 30c c
1 1 2
o5 : QQ[c ..c , d ..d ]
1 2 1 3
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> segre(2,Hom(wedge(2,A),wedge(2,B)));
2 2
3 d1 - 8 c1 d1 + 6 c1 - d2
i6 : segre_2 Hom(exteriorPower_2 A,exteriorPower_2 B)
2 2
o6 = 6c - 8c d + 3d - d
1 1 1 1 2
o6 : QQ[c ..c , d ..d ]
1 2 1 3
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