Subsection 5.4.3
To treat tangent bundles to hypersurfaces in Schubert2, we have to be a little more careful. If X is a hypersurface in ℙn, we cannot hope to construct the Chow ring to X. Even for the case of an elliptic curve E (a degree-3 hypersurface in ℙ2), the construction of A1(E) amounts to completely understanding the group law on E and all points of E (so in particular, this ring is never finitely generated over C), and the situation quickly gets worse for higher dimensions and degrees.
However, for classes on X which are obtained by restricting classes on ℙn to X, we can easily understand a great deal via the projection formula, which in this particular case tells us that if i:X →ℙn is the inclusion, then
i*(α|X) = α∩[X]
So, if for example we are interested in calculating the degree of α|X, we can equivalently calculate the degree of α∩[X]. In this way we “push the problem forward” to ℙn.
i1 : P3 = flagBundle({1,3}) o1 = P3 o1 : a flag bundle with subquotient ranks {1, 3} |
i2 : O1 = dual(P3.Bundles#0) o2 = O1 o2 : an abstract sheaf of rank 1 on P3 |
i3 : T = tangentBundle(P3) o3 = T o3 : an abstract sheaf of rank 3 on P3 |
i4 : NX = O1^**4 -- the fourth tensor power of O(1), i.e. O(4) o4 = NX o4 : an abstract sheaf of rank 1 on P3 |
i5 : X = chern(1,NX) -- the fundamental class [X] of X o5 = 4H 2,1 QQ[][H , H , H , H ] 1,1 2,1 2,2 2,3 o5 : ---------------------------------------------------------------- (- 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 |
i6 : TX = chern(T - NX) * X o6 = 4H + 24H 2,1 2,3 QQ[][H , H , H , H ] 1,1 2,1 2,2 2,3 o6 : ---------------------------------------------------------------- (- 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 |
i7 : integral TX -- The Euler characteristic of a quartic surface o7 = 24 |
This works because we have
c(TX) = (c(TP)|X)/(c(NX)) = (c(TP))/(c(OP(X)))|X.
i8 : eulerChar = n -> ( S := base d; Pn := flagBundle({1,n},S); TPn := tangentBundle(Pn); O1 := dual(Pn.Bundles#0); NX := O1^**d; TX := chern(TPn - NX)*chern(1,NX); integral TX) o8 = eulerChar o8 : FunctionClosure |
i9 : eulerChar(4) -- The Euler characteristic of a degree-d hypersurface in P4 4 3 2 o9 = - d + 5d - 10d + 10d o9 : QQ[d] |
i10 : sub(eulerChar(4),{d=>4/1}) -- The Euler characteristic of quartic threefold o10 = -56 o10 : QQ |
i11 : middleBetti = n -> ( euC := eulerChar(n); ((-1)^(n-1)) * (euC - 2*ceiling((n-1)/2))) o11 = middleBetti o11 : FunctionClosure |
i12 : middleBetti(4) -- The middle Betti number of a degree-d hypersurface in P4 4 3 2 o12 = d - 5d + 10d - 10d + 4 o12 : QQ[d] |
i13 : sub(middleBetti(4), {d => 5/1}) -- The middle Betti number of a quintic threefold o13 = 204 o13 : QQ |
i14 : for n from 3 to 5 do ( for e from 2 to 5 do ( euC := sub(eulerChar(n),{d=>e/1}); midB := sub(middleBetti(n),{d=>e/1}); << "n: " << n << " d: " << e << " chi: " << euC << " middle Betti: " << midB << endl)) n: 3 d: 2 chi: 4 middle Betti: 2 n: 3 d: 3 chi: 9 middle Betti: 7 n: 3 d: 4 chi: 24 middle Betti: 22 n: 3 d: 5 chi: 55 middle Betti: 53 n: 4 d: 2 chi: 4 middle Betti: 0 n: 4 d: 3 chi: -6 middle Betti: 10 n: 4 d: 4 chi: -56 middle Betti: 60 n: 4 d: 5 chi: -200 middle Betti: 204 n: 5 d: 2 chi: 6 middle Betti: 2 n: 5 d: 3 chi: 27 middle Betti: 23 n: 5 d: 4 chi: 188 middle Betti: 184 n: 5 d: 5 chi: 825 middle Betti: 821 |
Exercise 5.11: Betti numbers of smooth complete intersections
c(TX) = c(TP)/(∏i=1k c(OP(di)))|X
We can use then Schubert2 to produce a closed-form formula for the degree of the top Chern class of the tangent bundle to a complete intersection of k hypersurfaces in ℙn:i15 : eulerChar = (n,k) -> ( S := base(e_1 .. e_k); Pn := flagBundle({1,n},S); TPn := tangentBundle(Pn); O1 := dual(Pn.Bundles#0); N := sum(1..k, i-> O1^**(e_i)); --the denominator in the above formula X := product(1..k, i->chern(1,O1^**(e_i))); --fundamental class of X TX := chern(TPn - N) * X; integral TX) o15 = eulerChar o15 : FunctionClosure |
i16 : eulerChar(4,2) -- Euler char of a complete intersection surface in P4 3 2 2 3 2 2 o16 = e e + e e + e e - 5e e - 5e e + 10e e 1 2 1 2 1 2 1 2 1 2 1 2 o16 : QQ[e , e ] 1 2 |
i17 : middleBetti = (n,k) -> ( euC := eulerChar(n,k); ((-1)^(n-k)) * (euC - 2*ceiling((n-k)/2))) o17 = middleBetti o17 : FunctionClosure |
i18 : sub(middleBetti(4,2),{e_1=>2,e_2=>3/1}) -- complete intersection of a quadric and cubic in P4 o18 = 22 o18 : QQ |
i19 : sub(middleBetti(5,3),{e_1=>2,e_2=>2,e_3=>2/1}) -- three quadrics in P5 o19 = 22 o19 : QQ |
i20 : sub(eulerChar(4,2),{e_1=>2,e_2=>3/1}) -- complete intersection of a quadric and cubic in P4 o20 = 24 o20 : QQ |
i21 : sub(eulerChar(5,3),{e_1=>2,e_2=>2,e_3=>2/1}) -- three quadrics in P5 o21 = 24 o21 : QQ |
Exercise 5.12: Betti numbers of the quadric line complex
i22 : sub(middleBetti(5,2),{e_1=>2,e_2=>2/1}) o22 = 4 o22 : QQ |
Exercise 5.13: Calculate the Euler characteristic of a smooth hypersurface of bidegree (a,b) in ℙm ×ℙn
i23 : P2 = flagBundle({1,2}) o23 = P2 o23 : a flag bundle with subquotient ranks {1..2} |
i24 : P2xP3 = flagBundle({1,3},P2,VariableNames => K) o24 = P2xP3 o24 : a flag bundle with subquotient ranks {1, 3} |
i25 : intersectionRing(P2xP3) QQ[][H , H , H ] 1,1 2,1 2,2 ---------------------------------------------[K , K , K , K ] (- H - H , - H H - H , -H H ) 1,1 2,1 2,2 2,3 1,1 2,1 1,1 2,1 2,2 1,1 2,2 o25 = --------------------------------------------------------------------- (- K - K , - K K - K , - K K - K , -K K ) 1,1 2,1 1,1 2,1 2,2 1,1 2,2 2,3 1,1 2,3 o25 : QuotientRing |
Note that if we didn’t use the VariableNames options this ring would be horrible to look at, since classes pulled back from ℙ2 and ℙ3 would both be named H.
i26 : eulerChar = (n,m) -> ( S := base(a,b); Pn := flagBundle({1,n},S); PnxPm := flagBundle({1,m},Pn); T := tangentBundle(PnxPm); O1Pn := dual(Pn.Bundles#0); f := PnxPm / Pn; -- the first projection map from P2xP3 to P2 O10 := f^* O1Pn; -- we pull back O_P2(1) to get O(1,0) O01 := dual(PnxPm.Bundles#0); -- O(0,1) NX := (O10^**a)**(O01^**b); -- O(a,b) X := chern(1,NX); -- Chow class of divisor of type (a,b) TX := chern(T - NX) * X; -- pushed-forward total chern class of tangent bundle to X integral TX) -- chi of a smooth hypersurface of bidegree (a,b) in PnxPm o26 = eulerChar o26 : FunctionClosure |
i27 : eulerChar(4,4) -- chi of a smooth hypersurface of bidegree (a,b) in P4xP4 4 4 4 3 3 4 4 2 3 3 2 4 4 o27 = - 70a b + 175a b + 175a b - 150a b - 500a b - 150a b + 50a b + ----------------------------------------------------------------------- 3 2 2 3 4 4 3 2 2 3 4 500a b + 500a b + 50a*b - 5a - 200a b - 600a b - 200a*b - 5b + ----------------------------------------------------------------------- 3 2 2 3 2 2 25a + 300a b + 300a*b + 25b - 50a - 200a*b - 50b + 50a + 50b o27 : QQ[a, b] |
i28 : sub(eulerChar(2,3),{a=>1,b=>0/1}) -- is P1xP3, should be 8 by Kunneth o28 = 8 o28 : QQ |
i29 : sub(eulerChar(1,1),{a=>1,b=>1/1}) -- a conic in P2, should be 2 o29 = 2 o29 : QQ |
i30 : sub(eulerChar(1,1),{a=>2,b=>1/1}) -- a twisted cubic, should be 2 o30 = 2 o30 : QQ |