Subsection 4.3.1
We build arbitrary Schubert cycles using the command placeholderSchubertCycle. For example, on ${\mathbb G}(2,4)$, we can build the cycle $\sigma_{2,1,1}$ as follows:
i1 : G24 = flagBundle({3,2})
o1 = G24
o1 : a flag bundle with subquotient ranks {3, 2}
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i2 : sigma_(2,1,1) = placeholderSchubertCycle({2,1,1},G24)
2 2
o2 = H H - H
2,1 2,2 2,2
QQ[][H ..H , H ..H ]
1,1 1,3 2,1 2,2
o2 : ---------------------------------------------------------------------------------------------------------
(- H - H , - H - H H - H , - H - H H - H H , - H H - H H , -H H )
1,1 2,1 1,2 1,1 2,1 2,2 1,3 1,2 2,1 1,1 2,2 1,3 2,1 1,2 2,2 1,3 2,2
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Subsection 4.3.2
Exercise 4.34
How many lines meet 6 general 2-planes in ${\mathbb P}^4$?
The cycle of lines meeting a 2-plane in the Grassmannian ${\mathbb G}(1,4)$ is the Schubert cycle $\sigma_1$, so the number of lines meeting 6 general 2-planes is the degree of $(\sigma_1)^6$:
i3 : G14 = flagBundle({2,3})
o3 = G14
o3 : a flag bundle with subquotient ranks {2..3}
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i4 : sigma_1 = placeholderSchubertCycle({1,0},G14)
o4 = H
2,1
QQ[][H ..H , H ..H ]
1,1 1,2 2,1 2,3
o4 : ---------------------------------------------------------------------------------------------------------
(- H - H , - H - H H - H , - H H - H H - H , - H H - H H , -H H )
1,1 2,1 1,2 1,1 2,1 2,2 1,2 2,1 1,1 2,2 2,3 1,2 2,2 1,1 2,3 1,2 2,3
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i5 : integral (sigma_1)^6 o5 = 5 |
Note that this is the degree of ${\mathbb G}(1,4)$ in the Plucker embedding, since $\sigma_1$ is the hyperplane class.
Exercise 4.36 (a)
How many lines meet four general $k$-planes in ${\mathbb P}^{2k+1}$?
The cycles of lines meeting a $k$-plane in ${\mathbb G}(1,2k+1)$ is the Schubert cycle $\sigma_k$. We can build a function that calculates this value for any $k$, but we cannot use $k$ as a base parameter, since we need to build a different Grassmannian and Schubert cycle for each $k$.
i6 : numOfLines = k -> (
G := flagBundle({2,2*k});
sigma := placeholderSchubertCycle({k,0}, G);
integral sigma^4)
o6 = numOfLines
o6 : FunctionClosure
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Now we can calculate to our hearts' content:
i7 : for k from 1 to 5 do (
<< numOfLines(k) << " lines meet four " << k << "-planes in P" << 2*k+1 << "\n")
2 lines meet four 1-planes in P3
3 lines meet four 2-planes in P5
4 lines meet four 3-planes in P7
5 lines meet four 4-planes in P9
6 lines meet four 5-planes in P11
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Calculations slow down pretty quickly as $k$ gets large (the bottleneck is building the Chow ring), but we suspect the reader will have guessed the correct formula from the above data.
Linear Spaces on Quadrics
Exercise 4.43
A 2-plane in ${\mathbb P}^6$ is the same as a 3-plane in a 7-dimensional space. According to Proposition 4.42, the cycle of 3-planes contained in the zero-locus of a nondegenerate quadratic form on a 7-dimensional space is $2^3\sigma_{3,2,1}$ in $G(3,7)$. Hence we calculate:
i8 : G37 = flagBundle({3,4})
o8 = G37
o8 : a flag bundle with subquotient ranks {3..4}
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i9 : A37 = intersectionRing G37 o9 = A37 o9 : QuotientRing |
i10 : sigma = 8*placeholderSchubertCycle({3,2,1},G37)
2 2
o10 = 8H H H - 8H - 8H H
2,1 2,2 2,3 2,3 2,1 2,4
o10 : A37
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i11 : integral sigma^2 o11 = 64 |
More generally, we can ask: given 2 general quadrics in ${\mathbb P}^{2k+2}$, how many $k$-planes are contained in their intersection? We calculate:
i12 : numOfPlanes = k -> (
G:= flagBundle({k+1,k+2});
schubertlist := apply(k+1,i-> k+1-i); --the list {k+1,k,...,1}
sigma := (2^(k+1))*placeholderSchubertCycle(schubertlist, G);
integral sigma^2)
o12 = numOfPlanes
o12 : FunctionClosure
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i13 : numOfPlanes(2) --This was Exercise 4.43 o13 = 64 |
i14 : for k from 2 to 4 do (
<< numOfPlanes(k) << " " << k << "-planes in two quadrics in P" << 2*k+2 <<"\n")
64 2-planes in two quadrics in P6
256 3-planes in two quadrics in P8
1024 4-planes in two quadrics in P10
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Exercise 4.44:
Compute $\sigma_{2,1}^2$ in the Chow ring of $G(3,6)$.
This is easy with the function placeholderToSchubertBasis, which we already saw in Intersection Theory Section 4.2:
i15 : G36 = flagBundle({3,3})
o15 = G36
o15 : a flag bundle with subquotient ranks {2:3}
|
i16 : c = placeholderSchubertCycle({2,1,0},G36)
o16 = H H - H
2,1 2,2 2,3
QQ[][H ..H ]
1,1 2,3
o16 : --------------------------------------------------------------------------------------------------------------------------------------------------
(- H - H , - H - H H - H , - H - H H - H H - H , - H H - H H - H H , - H H - H H , -H H )
1,1 2,1 1,2 1,1 2,1 2,2 1,3 1,2 2,1 1,1 2,2 2,3 1,3 2,1 1,2 2,2 1,1 2,3 1,3 2,2 1,2 2,3 1,3 2,3
|
i17 : placeholderToSchubertBasis(c^2,G36)
o17 = s + 2s + s
{2, 2, 2} {3, 2, 1} {3, 3, 0}
o17 : QQ[][s , s , s , s , s , s , s , s , s , s , s , s , s , s , s , s , s , s , s , s ]
{0, 0, 0} {1, 0, 0} {1, 1, 0} {1, 1, 1} {2, 0, 0} {2, 1, 0} {2, 1, 1} {2, 2, 0} {2, 2, 1} {2, 2, 2} {3, 0, 0} {3, 1, 0} {3, 1, 1} {3, 2, 0} {3, 2, 1} {3, 2, 2} {3, 3, 0} {3, 3, 1} {3, 3, 2} {3, 3, 3}
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We see that $\sigma_{3,2,1}$ occurs with coefficient $2$ in $\sigma_{2,1}^2$.