surface(MultiprojectiveVariety,MultiprojectiveVariety) -- make a Hodge-special surface

i1 : K = ZZ/65521;

i2 : C = random PP_K^(1,3); -- random twisted cubic in P^3

o2 : ProjectiveVariety, curve in PP^3

i3 : j = parametrize PP_K(1,1,1,4); 

o3 : WeightedRationalMap (birational map from PP^3 to PP(1,1,1,4))

i4 : C = (rationalMap(ambient C,source j) * j) C;

o4 : ProjectiveVariety, curve in PP(1,1,1,4)

i5 : describe C

o5 = ambient:.............. PP(1,1,1,4)
     dim:.................. 1
     codim:................ 2
     degree:............... 12
     generators:........... 3^1 5^2 8^1 
     purity:............... true
     dim sing. l.:......... -1

i6 : S = random(8,C);

o6 : ProjectiveVariety, surface in PP(1,1,1,4)

i7 : describe S

o7 = ambient:.............. PP(1,1,1,4)
     dim:.................. 2
     codim:................ 1
     degree:............... 8
     generators:........... 8^1 
     purity:............... true
     dim sing. l.:......... -1

i8 : S = surface(C,S);

o8 : ProjectiveVariety, octic surface in PP(1,1,1,4) with rank 2 lattice
     defined by the intersection matrix | 2 3  | (det: -19)
                                        | 3 -5 |

i9 : discriminant S

o9 = -19

i10 : parameterCount(S,Verbose=>true)
C: curve in PP(1,1,1,4) cut out by 4 hypersurfaces of degrees 3^1 5^2 8^1 
S: surface in PP(1,1,1,4) defined by a form of degree 8
ambient: P = PP(1,1,1,4)
h^1(N_{C,P}) = 1
--warning: condition h^1(N_{C,P}) == 0 not satisfied
h^0(N_{C,P}) = 21
h^0(I_{C,P}(8)) = 36
h^0(N_{C,P}) + 35 = 56
h^0(N_{C,S}) = 0
dim{[S] : C ⊂ S ⊂ P} >= 56
dim P(H^0(O_P(8))) = 60
codim{[S] : C ⊂ S ⊂ P} <= 4

o10 = (4, (36, 21, 0))

o10 : Sequence

i11 : f := map(S,1,0)

o11 = multi-rational map consisting of one single rational map
      source variety: surface in PP(1,1,1,4) defined by a form of degree 8
      target variety: PP^2

o11 : WeightedRationalMap (rational map from S to PP^2)

i12 : f = quadricFibration f

o12 = multi-rational map consisting of one single rational map
      source variety: surface in PP(1,1,1,4) defined by a form of degree 8
      target variety: PP^2

o12 : QuadricFibration (rational map from S to PP^2)

i13 : discriminant f
-- starting computation of the generic fiber...
-- computation of the generic fiber successfully completed.
-- verifying the computation of the discriminant locus

o13 = curve in PP^2 defined by a form of degree 8

o13 : ProjectiveVariety, curve in PP^2