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.