AbstractSheaf RingElement -- twist by a divisor class

Synopsis

Operator: SPACE
Usage:

F(n)
Inputs:
- F, an abstract sheaf,
- n, a ring element, in (or promotable to) the intersection ring of the variety of $F$, homogeneous of degree 1 or 0
Outputs:
- an abstract sheaf, which depends on the degree of n. In the case where n has degree 0, the sheaf returned is the tensor product of F with the n-th (tensor) power of the tautological line bundle on the variety of F. In the case where n has degree 1, the sheaf returned is the tensor product of F with the line bundle whose first Chern class is n.

Description

i1 : X = abstractProjectiveSpace' 4

o1 = X

o1 : a flag bundle with subquotient ranks {4, 1}

i2 : OO_X(3)

o2 = a sheaf

o2 : an abstract sheaf of rank 1 on X

i3 : chi oo

o3 = 35

i4 : pt = base n

o4 = pt

o4 : an abstract variety of dimension 0

i5 : Y = abstractProjectiveSpace'(4,pt)

o5 = Y

o5 : a flag bundle with subquotient ranks {4, 1}

i6 : OO_Y(n)

o6 = a sheaf

o6 : an abstract sheaf of rank 1 on Y

i7 : chi oo

      1 4    5 3   35 2   25
o7 = --n  + --n  + --n  + --n + 1
     24     12     24     12

o7 : QQ[n]

The notation OO(n) is an abbreviation for OO_X(n), where X is the variety whose intersection ring n is in. By default, the first Chern class of the tautological line bundle on a projective space or projective bundle is called h, so we may use OO(h) as alternative notation for OO_Y(1):

i8 : A = intersectionRing Y

o8 = A

o8 : QuotientRing

i9 : chern OO_Y(1)

o9 = 1 + h

o9 : A

i10 : OO(h)

o10 = a sheaf

o10 : an abstract sheaf of rank 1 on Y

i11 : chern oo

o11 = 1 + h

o11 : A

Caveat

Beware the low parsing precedence of the adjacency operator SPACE.

Ways to use this method: