i1 : P = abstractProjectiveSpace' 3
o1 = P
o1 : a flag bundle with subquotient ranks {3, 1}
|
i2 : tangentBundle P o2 = a sheaf o2 : an abstract sheaf of rank 3 on P |
i3 : chern tangentBundle P
2 3
o3 = 1 + 4h + 6h + 4h
QQ[][H ..H , h]
1,1 1,3
o3 : ----------------------------------------------------
(- H - h, - H - H h, - H - H h, -H h)
1,1 1,2 1,1 1,3 1,2 1,3
|
i4 : todd P
11 2 3
o4 = 1 + 2h + --h + h
6
QQ[][H ..H , h]
1,1 1,3
o4 : ----------------------------------------------------
(- H - h, - H - H h, - H - H h, -H h)
1,1 1,2 1,1 1,3 1,2 1,3
|
i5 : chi OO_P(3) o5 = 20 |
To compute the Hilbert polynomial of a sheaf on projective space, we work over a base variety of dimension zero whose intersection ring contains a free variable $n$, instead of working over point:
i6 : pt = base n o6 = pt o6 : an abstract variety of dimension 0 |
i7 : Q = abstractProjectiveSpace'_4 pt
o7 = Q
o7 : a flag bundle with subquotient ranks {4, 1}
|
i8 : chi OO_Q(n)
1 4 5 3 35 2 25
o8 = --n + --n + --n + --n + 1
24 12 24 12
o8 : QQ[n]
|
The base variety may itself be a projective space:
i9 : S = abstractProjectiveSpace'(4, VariableName => h)
o9 = S
o9 : a flag bundle with subquotient ranks {4, 1}
|
i10 : P = abstractProjectiveSpace'(3, S, VariableName => H)
warning: clearing value of symbol H to allow access to subscripted variables based on it
: debug with expression debug 204 or with command line option --debug 204
o10 = P
o10 : a flag bundle with subquotient ranks {3, 1}
|
i11 : dim P o11 = 7 |
i12 : todd P
5 11 2 35 2 3 55 2 35 2 25 3 5 3 385 2 2 25 3 4 35 2 3 275 3 2
o12 = 1 + (2H + -h) + (--H + 5h*H + --h ) + (H + --h*H + --h H + --h ) + (-h*H + ---h H + --h H + h ) + (--h H + ---h H +
2 6 12 12 6 12 2 72 6 12 72
---------------------------------------------------------------------------------------------------------------------------
4 25 3 3 11 4 2 4 3
2h H) + (--h H + --h H ) + h H
12 6
QQ[][H ..H , h]
1,1 1,4
--------------------------------------------------------------------[H ..H , H]
(- H - h, - H - H h, - H - H h, - H - H h, -H h) 1,1 1,3
1,1 1,2 1,1 1,3 1,2 1,4 1,3 1,4
o12 : -----------------------------------------------------------------------------------
(- H - H, - H - H H, - H - H H, -H H)
1,1 1,2 1,1 1,3 1,2 1,3
|
The object abstractProjectiveSpace' is a method function with options.