# Conics on a quintic threefold. This is the top Chern class of the # quotient of the 5th symmetric power of the universal quotient on the # Grassmannian of 2 planes in P^5 by the subbundle of quintic containing the # tautological conic over the moduli space of conics. > > grass(3,5,c): # 2-planes in P^4.
i1 : Gc = flagBundle({2,3}, VariableNames => {,c}) o1 = Gc o1 : a flag bundle with subquotient ranks {2..3} |
i2 : (Sc,Qc) = bundles Gc o2 = (Sc, Qc) o2 : Sequence |
> B:=Symm(2,Qc): # The bundle of conics in the 2-plane.
i3 : B = symmetricPower(2,Qc) o3 = B o3 : an abstract sheaf of rank 6 on Gc |
> Proj(X,dual(B),z): # X is the projective bundle of all conics.
i4 : X = projectiveBundle'(dual B, VariableNames => {,{z}}) o4 = X o4 : a flag bundle with subquotient ranks {5, 1} |
> A:=Symm(5,Qc)-Symm(3,Qc)&*o(-z): # The rank 11 bundle of quintics > # restricted to the universal conic.
i5 : A = symmetricPower_5 Qc - symmetricPower_3 Qc ** OO(-z) o5 = A o5 : an abstract sheaf of rank 11 on X |
> c11:=chern(rank(A),A):# its top Chern class.
i6 : c11 = chern(rank A, A) 2 5 o6 = 609250c z 3 QQ[][H ..H , c ..c ] 1,1 1,2 1 3 ---------------------------------------------------------------------------------------[H ..H , z] (- H - c , - H - H c - c , - H c - H c - c , - H c - H c , -H c ) 1,1 1,5 1,1 1 1,2 1,1 1 2 1,2 1 1,1 2 3 1,2 2 1,1 3 1,2 3 o6 : ---------------------------------------------------------------------------------------------------------------------------------------------- 2 2 (- H - z - 4c , - H - H z + 5c + 5c , - H - H z - 15c c - 5c , - H - H z + 10c + 20c c , - H - H z - 20c c , -H z) 1,1 1 1,2 1,1 1 2 1,3 1,2 1 2 3 1,4 1,3 2 1 3 1,5 1,4 2 3 1,5 |
> lowerstar(X,c11): # push down to G(3,5).
i7 : X.StructureMap_* c11 2 o7 = 609250c 3 QQ[][H ..H , c ..c ] 1,1 1,2 1 3 o7 : --------------------------------------------------------------------------------------- (- H - c , - H - H c - c , - H c - H c - c , - H c - H c , -H c ) 1,1 1 1,2 1,1 1 2 1,2 1 1,1 2 3 1,2 2 1,1 3 1,2 3 |
> integral(Gc,"); # and integrate there. 609250
i8 : integral oo o8 = 609250 |