Subsection 5.4.1 - Universal bundles on projective space
We have two different methods in Schubert2 for producing projective spaces. We have already seen one method: build $\mathbb{P}^n$ as a Grassmannian:
i1 : P3 = flagBundle({1,3})
o1 = P3
o1 : a flag bundle with subquotient ranks {1, 3}
|
i2 : (S,Q) = P3.Bundles o2 = (S, Q) o2 : Sequence |
In this setting, the the bundle $O(1)$ is the dual of the universal subbundle $S$.
i3 : O1 = dual(S) o3 = O1 o3 : an abstract sheaf of rank 1 on P3 |
i4 : chern O1
o4 = 1 + H
2,1
QQ[][H , H ..H ]
1,1 2,1 2,3
o4 : ----------------------------------------------------------------
(- H - H , - H H - H , - H H - H , -H H )
1,1 2,1 1,1 2,1 2,2 1,1 2,2 2,3 1,1 2,3
|
Now, Schubert2 also comes with a built-in function abstractProjectiveSpace for making projective spaces. Using {/tt abstractProjectiveSpace} to build $\mathbb{P}^n$ is nice, because the resulting Chow ring is presented as a truncated polynomial ring in one variable, rather than as a ring with $n+1$ generators. But, be careful: this built-in actually produces the projective space of 1-quotients. For example:
i5 : P3' = abstractProjectiveSpace 3
o5 = P3'
o5 : a flag bundle with subquotient ranks {1, 3}
|
i6 : (S',Q') = P3'.Bundles o6 = (S', Q') o6 : Sequence |
i7 : chern S'
o7 = 1 - H
2,1
QQ[][h, H ..H ]
2,1 2,3
o7 : -------------------------------------------------------
(- h - H , - h*H - H , - h*H - H , -h*H )
2,1 2,1 2,2 2,2 2,3 2,3
|
i8 : chern Q' -- Q' is O(1) on P3'
o8 = 1 + H + H + H
2,1 2,2 2,3
QQ[][h, H ..H ]
2,1 2,3
o8 : -------------------------------------------------------
(- h - H , - h*H - H , - h*H - H , -h*H )
2,1 2,1 2,2 2,2 2,3 2,3
|
For the rest of this section, we will use the flagBundle method to produce $\mathbb{P}^n$, in order to be consistent with the choices in the book.
Subsection 5.4.2
The tangent bundle to projective space comes built-in in Schubert2. It can be accessed via the tangentBundle method:
i9 : T = tangentBundle(P3) o9 = T o9 : an abstract sheaf of rank 3 on P3 |
i10 : chern T
o10 = 1 + 4H + 6H + 4H
2,1 2,2 2,3
QQ[][H , H ..H ]
1,1 2,1 2,3
o10 : ----------------------------------------------------------------
(- H - H , - H H - H , - H H - H , -H H )
1,1 2,1 1,1 2,1 2,2 1,1 2,2 2,3 1,1 2,3
|
We can also produce the tangent bundle to $\mathbb{P}^n$ ourselves by using the Euler exact sequence:
i11 : TP3 = (4 * O1) - 1 o11 = T o11 : an abstract sheaf of rank 3 on P3 |
i12 : chern T == chern TP3 o12 = true |
i13 : rank T == rank TP3 o13 = true |
Note how Schubert2 treats integers in a bundle computation as copies of a trivial bundle. See AbstractSheaf * AbstractSheaf (missing documentation) and AbstractSheaf - AbstractSheaf (missing documentation), for example, for more information.