A cubic fourfold containing a degree-5 del Pezzo surface -- an example

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 
o7 = ideal (e , e e , e e , e e , e h, e e , e e e , e e e , e e h, e e ,
             1   1 2   1 3   1 4   1    1 2   1 2 3   1 2 4   1 2    1 3 
     ------------------------------------------------------------------------
                       2            2   3   2     2     2      2         
     e e e , e e h, e e , e e h, e h , e , e e , e e , e h, e e , e e e ,
      1 3 4   1 3    1 4   1 4    1     2   2 3   2 4   2    2 3   2 3 4 
     ------------------------------------------------------------------------
               2            2   3   2     2      2            2   3   2  
     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,
      2 3    2 4   2 4    2     3   3 4   3    3 4   3 4    3     4   4  
     ------------------------------------------------------------------------
        2   3
     e h , h )
      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       
o10 = ideal (e h, e h, e h, e h, e  + h , e e , e e , e e , e  + h , e e ,
              4    3    2    1    4        3 4   2 4   1 4   3        2 3 
      -----------------------------------------------------------------------
             2    2         2    2
      e e , e  + h , e e , e  + h )
       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 = FunctionClosure[stdio:12:14-12:42]

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     
o30 = -r  - -r s  - -r*s  - --s  + -r  + -r s - -r*s  - --s  + --r  + --r*s -
      8     4       3       24     4     4      4       12      8     12     
      -----------------------------------------------------------------------
       5 2   9    29
      --s  + -r + --s + 1
      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