Subsection 4.3.1
i1 : G24 = flagBundle({3,2}) o1 = G24 o1 : a flag bundle with subquotient ranks {3, 2} |
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 , H ] 1,1 1,2 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 |
Subsection 4.3.2
Exercise 4.34
How many lines meet 6 general 2-planes in ℙ4?
i3 : G14 = flagBundle({2,3}) o3 = G14 o3 : a flag bundle with subquotient ranks {2..3} |
i4 : sigma_1 = placeholderSchubertCycle({1,0},G14) o4 = H 2,1 QQ[][H , H , H , H , H ] 1,1 1,2 2,1 2,2 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 |
i5 : integral (sigma_1)^6 o5 = 5 |
Note that this is the degree of G(1,4) in the Plucker embedding, since σ1 is the hyperplane class.
Exercise 4.36 (a)
How many lines meet four general k-planes in ℙ2k+1?
i6 : numOfLines = k -> ( G := flagBundle({2,2*k}); sigma := placeholderSchubertCycle({k,0}, G); integral sigma^4) o6 = numOfLines o6 : FunctionClosure |
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 |
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
i8 : G37 = flagBundle({3,4}) o8 = G37 o8 : a flag bundle with subquotient ranks {3..4} |
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 |
i11 : integral sigma^2 o11 = 64 |
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 |
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 |
Exercise 4.44:
Compute σ2,12 in the Chow ring of G(3,6).
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 , H , H , H , H ] 1,1 1,2 1,3 2,1 2,2 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} |