A smooth cubic fourfold Y containing a degree-5 del Pezzo surface X is known to be rational, see for example S. L. Tregub's "Three constructions of rationality of a four-dimensional cubic", 1984. If H is the hyperplane class on Y, then 2H - X is a linear series which gives a birational map from Y to \mathbb{P}^4. We will reproduce the numerical calculations which suggest (but do not prove) this fact. We start by building part of the Chow ring of Y:
i1 : p = base(r,s) o1 = p o1 : an abstract variety of dimension 0 |
i2 : P5 = projectiveBundle(5,p) o2 = P5 o2 : a flag bundle with subquotient ranks {1, 5} |
i3 : Y = sectionZeroLocus OO_P5(3) -- cubic fourfold o3 = Y o3 : an abstract variety of dimension 4 |
i4 : x = chern(1,OO_Y(1)) -- hyperplane class o4 = H 2,1 QQ[r..s][H , H ..H ] 1,1 2,1 2,5 ------------------------------------------------------------------------------------------------------ (- H - H , - H H - H , - H 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 2,4 1,1 2,4 2,5 1,1 2,5 o4 : -------------------------------------------------------------------------------------------------------- H 2,5 |
We then build the Chow ring of the degree-5 del Pezzo:
i5 : S = intersectionRing p -- important that we use the same base ring o5 = S o5 : PolynomialRing |
i6 : B1 = S[e_1..e_4,h, Join => false] o6 = B1 o6 : PolynomialRing |
i7 : I1 = (ideal vars B1)^3 -- relations imposed by dimension 3 2 2 2 2 2 2 2 2 3 2 2 2 o7 = ideal (e , e e , e e , e e , e h, e e , e e e , e e e , e e h, e e , e e e , e e h, e e , e e h, e h , e , e e , e e , e h, 1 1 2 1 3 1 4 1 1 2 1 2 3 1 2 4 1 2 1 3 1 3 4 1 3 1 4 1 4 1 2 2 3 2 4 2 ---------------------------------------------------------------------------------------------------------------------------- 2 2 2 3 2 2 2 2 3 2 2 3 e e , e e e , e e h, e e , e e h, e h , e , e e , e h, e e , e e h, e h , e , e h, e h , h ) 2 3 2 3 4 2 3 2 4 2 4 2 3 3 4 3 3 4 3 4 3 4 4 4 o7 : Ideal of B1 |
i8 : I2 = ideal flatten (for i from 0 to 4 list (for j from i+1 to 4 list (B1_i * B1_j))) -- relations imposed by non-intersection o8 = ideal (e e , e e , e e , e h, e e , e e , e h, e e , e h, e h) 1 2 1 3 1 4 1 2 3 2 4 2 3 4 3 4 o8 : Ideal of B1 |
i9 : I3 = ideal for i from 1 to 4 list (e_i^2 + h^2) -- relations that make each exceptional divisor a (-1)-curve 2 2 2 2 2 2 2 2 o9 = ideal (e + h , e + h , e + h , e + h ) 1 2 3 4 o9 : Ideal of B1 |
i10 : I = trim (I1 + I2 + I3) 2 2 2 2 2 2 2 2 o10 = ideal (e h, e h, e h, e h, e + h , e e , e e , e e , e + h , e e , e e , e + h , e e , e + h ) 4 3 2 1 4 3 4 2 4 1 4 3 2 3 1 3 2 1 2 1 o10 : Ideal of B1 |
i11 : B = B1/I o11 = B o11 : QuotientRing |
i12 : integral B := b -> coefficient((B_4)^2, b) o12 = -*Function[stdio:12:17-12:41]*- o12 : FunctionClosure |
We build the canonical class and tangent class of X:
i13 : K = -(3*h - e_1 - e_2 - e_3 - e_4) o13 = e + e + e + e - 3h 1 2 3 4 o13 : B |
i14 : tX = 1 - K + 7*h^2 2 o14 = 7h - e - e - e - e + 3h + 1 1 2 3 4 o14 : B |
The pullback map from Y to X takes the hyperplane class to the anticanonical class on X. Because a projectiveBundle has extra generators, we end up also having to say where powers of the hyperplane class map to:
i15 : A = intersectionRing Y o15 = A o15 : QuotientRing |
i16 : f = map(B, A, {K, -K, K^2, -K^3, K^4, -K^5}) 2 o16 = map(B,A,{e + e + e + e - 3h, - e - e - e - e + 3h, 5h , 0, 0, 0, r, s}) 1 2 3 4 1 2 3 4 o16 : RingMap B <--- A |
Now we build the inclusion of X in Y, which assembles the above information into a variety:
i17 : i = inclusion(f, SubTangent => tX, SubDimension => 2, Base => p) o17 = i o17 : a map to a variety from a variety |
i18 : X = source i o18 = X o18 : an abstract variety of dimension 2 |
i19 : Z = target i o19 = Z o19 : an abstract variety of dimension 4 |
We blow up this inclusion so that we can work with the linear series 2H - X as a divisor.
i20 : (Ztilde, PN, PNmap, Zmap) = blowup(i) o20 = (Ztilde, PN, PNmap, Zmap) o20 : Sequence |
And now we calculate the Euler characteristic and degree of the line bundle 2H - E on Ztilde.
i21 : AZtilde = intersectionRing Ztilde o21 = AZtilde o21 : QuotientRing |
i22 : exc = chern(1,exceptionalDivisor Ztilde) -- exceptional class o22 = E 0 o22 : AZtilde |
i23 : EBA = intersectionRing Z o23 = EBA o23 : QuotientRing |
i24 : hyp = Zmap^* promote(x, EBA) -- hyperplane class of Ztilde o24 = H 2,1 o24 : AZtilde |
i25 : L = OO_Ztilde(2*hyp - exc) o25 = L o25 : an abstract sheaf of rank 1 on Ztilde |
i26 : chi L o26 = 5 o26 : S |
i27 : integral ((chern(1,L))^4) o27 = 1 o27 : S |
More generally, we can compute the Euler characteristic and degree of all line bundles of the form rH + sE on Ztilde:
i28 : (r', s') = ((r_A, s_A) / (elt -> promote(elt, EBA))) / Zmap^* o28 = (r, s) o28 : Sequence |
i29 : L = OO_Ztilde(r' * hyp + s' * exc) o29 = L o29 : an abstract sheaf of rank 1 on Ztilde |
i30 : chi L 1 4 5 2 2 5 3 7 4 3 3 5 2 5 2 23 3 15 2 35 5 2 9 29 o30 = -r - -r s - -r*s - --s + -r + -r s - -r*s - --s + --r + --r*s - --s + -r + --s + 1 8 4 3 24 4 4 4 12 8 12 24 4 12 o30 : S |
i31 : integral ((chern(1,L))^4) 4 2 2 3 4 o31 = 3r - 30r s - 40r*s - 7s o31 : S |