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SpecialFanoFourfolds :: specialCubicFourfold

specialCubicFourfold -- make a special cubic fourfold

Synopsis

Description

In the example below, we construct a cubic fourfold containg a rational scroll of degree 7 with 3 nodes.

i1 : K = ZZ/33331; ringP5 = K[x_0..x_5];
i3 : idS = ideal(x_0*x_2*x_3-2*x_1*x_2*x_3-x_1*x_3^2-x_2*x_3^2-x_0*x_1*x_4+2*x_1^2*x_4-x_1*x_2*x_4+x_2^2*x_4+2*x_0*x_3*x_4-x_1*x_3*x_4-x_1*x_4^2+x_1*x_3*x_5,
                 x_1^2*x_3-4*x_1*x_2*x_3-x_0*x_3^2-3*x_1*x_3^2-2*x_2*x_3^2+2*x_0^2*x_4-9*x_0*x_1*x_4+11*x_1^2*x_4-x_0*x_2*x_4-2*x_1*x_2*x_4+2*x_2^2*x_4+12*x_0*x_3*x_4-7*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2-6*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3-x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+2*x_1*x_3*x_5-x_3^2*x_5-x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2,
                 x_0*x_1*x_3-7*x_1*x_2*x_3-3*x_0*x_3^2-4*x_1*x_3^2-3*x_2*x_3^2+x_3^3+3*x_0^2*x_4-14*x_0*x_1*x_4+17*x_1^2*x_4-x_0*x_2*x_4-3*x_1*x_2*x_4+3*x_2^2*x_4+19*x_0*x_3*x_4-9*x_1*x_3*x_4-x_2*x_3*x_4-6*x_3^2*x_4+x_0*x_4^2-9*x_1*x_4^2+6*x_2*x_4^2-3*x_3*x_4^2-3*x_4^3-2*x_0*x_1*x_5+2*x_1^2*x_5+4*x_1*x_2*x_5+5*x_0*x_3*x_5+4*x_1*x_3*x_5-2*x_3^2*x_5-2*x_0*x_4*x_5-7*x_1*x_4*x_5+5*x_2*x_4*x_5+3*x_3*x_4*x_5-2*x_1*x_5^2,
                 x_0^2*x_3-12*x_1*x_2*x_3-6*x_0*x_3^2-6*x_1*x_3^2-5*x_2*x_3^2+2*x_3^3+5*x_0^2*x_4-24*x_0*x_1*x_4+29*x_1^2*x_4-x_0*x_2*x_4-5*x_1*x_2*x_4+5*x_2^2*x_4+32*x_0*x_3*x_4-14*x_1*x_3*x_4-2*x_2*x_3*x_4-10*x_3^2*x_4+x_0*x_4^2-15*x_1*x_4^2+10*x_2*x_4^2-5*x_3*x_4^2-5*x_4^3-3*x_0*x_1*x_5+3*x_1^2*x_5+6*x_1*x_2*x_5+8*x_0*x_3*x_5+7*x_1*x_3*x_5-3*x_3^2*x_5-3*x_0*x_4*x_5-11*x_1*x_4*x_5+8*x_2*x_4*x_5+5*x_3*x_4*x_5-3*x_1*x_5^2,
                 x_1*x_2^2+6*x_1*x_2*x_3+2*x_0*x_3^2+3*x_1*x_3^2+2*x_2*x_3^2-x_3^3-3*x_0^2*x_4+12*x_0*x_1*x_4-14*x_1^2*x_4-2*x_2^2*x_4-15*x_0*x_3*x_4+6*x_1*x_3*x_4+x_2*x_3*x_4+5*x_3^2*x_4+x_0*x_4^2+8*x_1*x_4^2-5*x_2*x_4^2+2*x_3*x_4^2+2*x_4^3+x_0*x_1*x_5-2*x_1^2*x_5-4*x_1*x_2*x_5-4*x_0*x_3*x_5-3*x_1*x_3*x_5+2*x_3^2*x_5+2*x_0*x_4*x_5+7*x_1*x_4*x_5-4*x_2*x_4*x_5-2*x_3*x_4*x_5+2*x_1*x_5^2,
                 x_0*x_2^2+10*x_1*x_2*x_3+3*x_0*x_3^2+5*x_1*x_3^2+4*x_2*x_3^2-x_3^3-5*x_0^2*x_4+19*x_0*x_1*x_4-22*x_1^2*x_4-x_0*x_2*x_4+3*x_1*x_2*x_4-4*x_2^2*x_4-24*x_0*x_3*x_4+9*x_1*x_3*x_4+x_2*x_3*x_4+8*x_3^2*x_4+2*x_0*x_4^2+11*x_1*x_4^2-7*x_2*x_4^2+4*x_3*x_4^2+3*x_4^3+2*x_0*x_1*x_5-4*x_1^2*x_5-7*x_1*x_2*x_5-7*x_0*x_3*x_5-5*x_1*x_3*x_5-x_2*x_3*x_5+3*x_3^2*x_5+4*x_0*x_4*x_5+12*x_1*x_4*x_5-7*x_2*x_4*x_5-3*x_3*x_4*x_5+4*x_1*x_5^2,
                 x_1^2*x_2+17*x_1*x_2*x_3+6*x_0*x_3^2+9*x_1*x_3^2+7*x_2*x_3^2-2*x_3^3-9*x_0^2*x_4+36*x_0*x_1*x_4-44*x_1^2*x_4+3*x_0*x_2*x_4+5*x_1*x_2*x_4-7*x_2^2*x_4-47*x_0*x_3*x_4+21*x_1*x_3*x_4+2*x_2*x_3*x_4+16*x_3^2*x_4+24*x_1*x_4^2-16*x_2*x_4^2+7*x_3*x_4^2+7*x_4^3+3*x_0*x_1*x_5-6*x_1^2*x_5-9*x_1*x_2*x_5-12*x_0*x_3*x_5-8*x_1*x_3*x_5+5*x_3^2*x_5+5*x_0*x_4*x_5+19*x_1*x_4*x_5-12*x_2*x_4*x_5-7*x_3*x_4*x_5+5*x_1*x_5^2,
                 x_0*x_1*x_2+29*x_1*x_2*x_3+11*x_0*x_3^2+15*x_1*x_3^2+12*x_2*x_3^2-4*x_3^3-16*x_0^2*x_4+62*x_0*x_1*x_4-74*x_1^2*x_4+5*x_0*x_2*x_4+9*x_1*x_2*x_4-12*x_2^2*x_4-80*x_0*x_3*x_4+35*x_1*x_3*x_4+4*x_2*x_3*x_4+27*x_3^2*x_4+40*x_1*x_4^2-27*x_2*x_4^2+12*x_3*x_4^2+12*x_4^3+5*x_0*x_1*x_5-10*x_1^2*x_5-16*x_1*x_2*x_5-21*x_0*x_3*x_5-14*x_1*x_3*x_5+9*x_3^2*x_5+9*x_0*x_4*x_5+33*x_1*x_4*x_5-21*x_2*x_4*x_5-12*x_3*x_4*x_5+9*x_1*x_5^2,
                 x_0^2*x_2+49*x_1*x_2*x_3+19*x_0*x_3^2+25*x_1*x_3^2+20*x_2*x_3^2-7*x_3^3-28*x_0^2*x_4+106*x_0*x_1*x_4-124*x_1^2*x_4+8*x_0*x_2*x_4+16*x_1*x_2*x_4-20*x_2^2*x_4-134*x_0*x_3*x_4+58*x_1*x_3*x_4+7*x_2*x_3*x_4+45*x_3^2*x_4+66*x_1*x_4^2-45*x_2*x_4^2+20*x_3*x_4^2+20*x_4^3+9*x_0*x_1*x_5-18*x_1^2*x_5-28*x_1*x_2*x_5-37*x_0*x_3*x_5-23*x_1*x_3*x_5+16*x_3^2*x_5+16*x_0*x_4*x_5+57*x_1*x_4*x_5-36*x_2*x_4*x_5-20*x_3*x_4*x_5+16*x_1*x_5^2,
                 x_1^3+47*x_1*x_2*x_3+18*x_0*x_3^2+23*x_1*x_3^2+19*x_2*x_3^2-7*x_3^3-24*x_0^2*x_4+97*x_0*x_1*x_4-117*x_1^2*x_4+8*x_0*x_2*x_4+16*x_1*x_2*x_4-19*x_2^2*x_4-127*x_0*x_3*x_4+54*x_1*x_3*x_4+7*x_2*x_3*x_4+42*x_3^2*x_4-x_0*x_4^2+62*x_1*x_4^2-42*x_2*x_4^2+19*x_3*x_4^2+19*x_4^3+9*x_0*x_1*x_5-16*x_1^2*x_5-25*x_1*x_2*x_5-33*x_0*x_3*x_5-23*x_1*x_3*x_5+14*x_3^2*x_5+14*x_0*x_4*x_5+51*x_1*x_4*x_5-33*x_2*x_4*x_5-19*x_3*x_4*x_5+14*x_1*x_5^2,
                 x_0*x_1^2+79*x_1*x_2*x_3+29*x_0*x_3^2+40*x_1*x_3^2+32*x_2*x_3^2-11*x_3^3-41*x_0^2*x_4+164*x_0*x_1*x_4-196*x_1^2*x_4+14*x_0*x_2*x_4+26*x_1*x_2*x_4-32*x_2^2*x_4-214*x_0*x_3*x_4+92*x_1*x_3*x_4+11*x_2*x_3*x_4+71*x_3^2*x_4-2*x_0*x_4^2+105*x_1*x_4^2-71*x_2*x_4^2+32*x_3*x_4^2+32*x_4^3+14*x_0*x_1*x_5-26*x_1^2*x_5-41*x_1*x_2*x_5-55*x_0*x_3*x_5-38*x_1*x_3*x_5+23*x_3^2*x_5+23*x_0*x_4*x_5+85*x_1*x_4*x_5-55*x_2*x_4*x_5-32*x_3*x_4*x_5+23*x_1*x_5^2,
                 x_0^2*x_1+133*x_1*x_2*x_3+48*x_0*x_3^2+68*x_1*x_3^2+54*x_2*x_3^2-18*x_3^3-70*x_0^2*x_4+278*x_0*x_1*x_4-330*x_1^2*x_4+24*x_0*x_2*x_4+44*x_1*x_2*x_4-54*x_2^2*x_4-361*x_0*x_3*x_4+156*x_1*x_3*x_4+18*x_2*x_3*x_4+120*x_3^2*x_4-4*x_0*x_4^2+177*x_1*x_4^2-120*x_2*x_4^2+54*x_3*x_4^2+54*x_4^3+23*x_0*x_1*x_5-44*x_1^2*x_5-69*x_1*x_2*x_5-93*x_0*x_3*x_5-63*x_1*x_3*x_5+39*x_3^2*x_5+39*x_0*x_4*x_5+144*x_1*x_4*x_5-93*x_2*x_4*x_5-54*x_3*x_4*x_5+39*x_1*x_5^2,
                 x_0^3+224*x_1*x_2*x_3+80*x_0*x_3^2+115*x_1*x_3^2+91*x_2*x_3^2-30*x_3^3-119*x_0^2*x_4+470*x_0*x_1*x_4-555*x_1^2*x_4+41*x_0*x_2*x_4+75*x_1*x_2*x_4-91*x_2^2*x_4-608*x_0*x_3*x_4+263*x_1*x_3*x_4+30*x_2*x_3*x_4+202*x_3^2*x_4-8*x_0*x_4^2+297*x_1*x_4^2-202*x_2*x_4^2+91*x_3*x_4^2+91*x_4^3+39*x_0*x_1*x_5-76*x_1^2*x_5-118*x_1*x_2*x_5-158*x_0*x_3*x_5-105*x_1*x_3*x_5+67*x_3^2*x_5+68*x_0*x_4*x_5+245*x_1*x_4*x_5-158*x_2*x_4*x_5-91*x_3*x_4*x_5+67*x_1*x_5^2);

o3 : Ideal of ringP5
i4 : idX = ideal(x_1^2*x_3+x_0*x_2*x_3-6*x_1*x_2*x_3-x_0*x_3^2-4*x_1*x_3^2-3*x_2*x_3^2+2*x_0^2*x_4-10*x_0*x_1*x_4+13*x_1^2*x_4-x_0*x_2*x_4-3*x_1*x_2*x_4+3*x_2^2*x_4+14*x_0*x_3*x_4-8*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2-7*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3-x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+3*x_1*x_3*x_5-x_3^2*x_5-x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2);

o4 : Ideal of ringP5
i5 : time X = specialCubicFourfold(idS,idX,NumNodes=>3)
     -- used 0.0313151 seconds

o5 = -- special cubic fourfold --
     ambient projective space: Proj(K[x , x , x , x , x , x ])
                                       0   1   2   3   4   5
     surface: {
                                     2      2              2               2                           2
               x x x  - 2x x x  - x x  - x x  - x x x  + 2x x  - x x x  + x x  + 2x x x  - x x x  - x x  + x x x ,
                0 2 3     1 2 3    1 3    2 3    0 1 4     1 4    1 2 4    2 4     0 3 4    1 3 4    1 4    1 3 5
               
                2                  2       2       2     2                  2                          2                            2        2       2       2       2     3             2                                    2                                               2
               x x  - 4x x x  - x x  - 3x x  - 2x x  + 2x x  - 9x x x  + 11x x  - x x x  - 2x x x  + 2x x  + 12x x x  - 7x x x  - 4x x  + x x  - 6x x  + 4x x  - 2x x  - 2x  - x x x  + x x  + 2x x x  + 3x x x  + 2x x x  - x x  - x x x  - 4x x x  + 3x x x  + 2x x x  - x x ,
                1 3     1 2 3    0 3     1 3     2 3     0 4     0 1 4      1 4    0 2 4     1 2 4     2 4      0 3 4     1 3 4     3 4    0 4     1 4     2 4     3 4     4    0 1 5    1 5     1 2 5     0 3 5     1 3 5    3 5    0 4 5     1 4 5     2 4 5     3 4 5    1 5
               
                                      2       2       2    3     2                   2                          2                                     2        2       2       2       2     3               2                                     2                                                 2
               x x x  - 7x x x  - 3x x  - 4x x  - 3x x  + x  + 3x x  - 14x x x  + 17x x  - x x x  - 3x x x  + 3x x  + 19x x x  - 9x x x  - x x x  - 6x x  + x x  - 9x x  + 6x x  - 3x x  - 3x  - 2x x x  + 2x x  + 4x x x  + 5x x x  + 4x x x  - 2x x  - 2x x x  - 7x x x  + 5x x x  + 3x x x  - 2x x ,
                0 1 3     1 2 3     0 3     1 3     2 3    3     0 4      0 1 4      1 4    0 2 4     1 2 4     2 4      0 3 4     1 3 4    2 3 4     3 4    0 4     1 4     2 4     3 4     4     0 1 5     1 5     1 2 5     0 3 5     1 3 5     3 5     0 4 5     1 4 5     2 4 5     3 4 5     1 5
               
                2                    2       2       2     3     2                   2                          2                                        2        2        2        2       2     3               2                                     2                                                  2
               x x  - 12x x x  - 6x x  - 6x x  - 5x x  + 2x  + 5x x  - 24x x x  + 29x x  - x x x  - 5x x x  + 5x x  + 32x x x  - 14x x x  - 2x x x  - 10x x  + x x  - 15x x  + 10x x  - 5x x  - 5x  - 3x x x  + 3x x  + 6x x x  + 8x x x  + 7x x x  - 3x x  - 3x x x  - 11x x x  + 8x x x  + 5x x x  - 3x x ,
                0 3      1 2 3     0 3     1 3     2 3     3     0 4      0 1 4      1 4    0 2 4     1 2 4     2 4      0 3 4      1 3 4     2 3 4      3 4    0 4      1 4      2 4     3 4     4     0 1 5     1 5     1 2 5     0 3 5     1 3 5     3 5     0 4 5      1 4 5     2 4 5     3 4 5     1 5
               
                  2                 2       2       2    3     2                   2       2                                     2        2       2       2       2     3              2                                     2                                                 2
               x x  + 6x x x  + 2x x  + 3x x  + 2x x  - x  - 3x x  + 12x x x  - 14x x  - 2x x  - 15x x x  + 6x x x  + x x x  + 5x x  + x x  + 8x x  - 5x x  + 2x x  + 2x  + x x x  - 2x x  - 4x x x  - 4x x x  - 3x x x  + 2x x  + 2x x x  + 7x x x  - 4x x x  - 2x x x  + 2x x ,
                1 2     1 2 3     0 3     1 3     2 3    3     0 4      0 1 4      1 4     2 4      0 3 4     1 3 4    2 3 4     3 4    0 4     1 4     2 4     3 4     4    0 1 5     1 5     1 2 5     0 3 5     1 3 5     3 5     0 4 5     1 4 5     2 4 5     3 4 5     1 5
               
                  2                  2       2       2    3     2                   2                          2                                     2         2        2       2       2     3               2                                              2                                                  2
               x x  + 10x x x  + 3x x  + 5x x  + 4x x  - x  - 5x x  + 19x x x  - 22x x  - x x x  + 3x x x  - 4x x  - 24x x x  + 9x x x  + x x x  + 8x x  + 2x x  + 11x x  - 7x x  + 4x x  + 3x  + 2x x x  - 4x x  - 7x x x  - 7x x x  - 5x x x  - x x x  + 3x x  + 4x x x  + 12x x x  - 7x x x  - 3x x x  + 4x x ,
                0 2      1 2 3     0 3     1 3     2 3    3     0 4      0 1 4      1 4    0 2 4     1 2 4     2 4      0 3 4     1 3 4    2 3 4     3 4     0 4      1 4     2 4     3 4     4     0 1 5     1 5     1 2 5     0 3 5     1 3 5    2 3 5     3 5     0 4 5      1 4 5     2 4 5     3 4 5     1 5
               
                2                    2       2       2     3     2                   2                           2                                        2          2        2       2     3               2                                      2                                                   2
               x x  + 17x x x  + 6x x  + 9x x  + 7x x  - 2x  - 9x x  + 36x x x  - 44x x  + 3x x x  + 5x x x  - 7x x  - 47x x x  + 21x x x  + 2x x x  + 16x x  + 24x x  - 16x x  + 7x x  + 7x  + 3x x x  - 6x x  - 9x x x  - 12x x x  - 8x x x  + 5x x  + 5x x x  + 19x x x  - 12x x x  - 7x x x  + 5x x ,
                1 2      1 2 3     0 3     1 3     2 3     3     0 4      0 1 4      1 4     0 2 4     1 2 4     2 4      0 3 4      1 3 4     2 3 4      3 4      1 4      2 4     3 4     4     0 1 5     1 5     1 2 5      0 3 5     1 3 5     3 5     0 4 5      1 4 5      2 4 5     3 4 5     1 5
               
                                        2        2        2     3      2                   2                            2                                        2          2        2        2      3                2                                        2                                                    2
               x x x  + 29x x x  + 11x x  + 15x x  + 12x x  - 4x  - 16x x  + 62x x x  - 74x x  + 5x x x  + 9x x x  - 12x x  - 80x x x  + 35x x x  + 4x x x  + 27x x  + 40x x  - 27x x  + 12x x  + 12x  + 5x x x  - 10x x  - 16x x x  - 21x x x  - 14x x x  + 9x x  + 9x x x  + 33x x x  - 21x x x  - 12x x x  + 9x x ,
                0 1 2      1 2 3      0 3      1 3      2 3     3      0 4      0 1 4      1 4     0 2 4     1 2 4      2 4      0 3 4      1 3 4     2 3 4      3 4      1 4      2 4      3 4      4     0 1 5      1 5      1 2 5      0 3 5      1 3 5     3 5     0 4 5      1 4 5      2 4 5      3 4 5     1 5
               
                2                     2        2        2     3      2                     2                             2                                         2          2        2        2      3                2                                         2                                                      2
               x x  + 49x x x  + 19x x  + 25x x  + 20x x  - 7x  - 28x x  + 106x x x  - 124x x  + 8x x x  + 16x x x  - 20x x  - 134x x x  + 58x x x  + 7x x x  + 45x x  + 66x x  - 45x x  + 20x x  + 20x  + 9x x x  - 18x x  - 28x x x  - 37x x x  - 23x x x  + 16x x  + 16x x x  + 57x x x  - 36x x x  - 20x x x  + 16x x ,
                0 2      1 2 3      0 3      1 3      2 3     3      0 4       0 1 4       1 4     0 2 4      1 2 4      2 4       0 3 4      1 3 4     2 3 4      3 4      1 4      2 4      3 4      4     0 1 5      1 5      1 2 5      0 3 5      1 3 5      3 5      0 4 5      1 4 5      2 4 5      3 4 5      1 5
               
                3                   2        2        2     3      2                    2                             2                                         2        2        2        2        2      3                2                                         2                                                      2
               x  + 47x x x  + 18x x  + 23x x  + 19x x  - 7x  - 24x x  + 97x x x  - 117x x  + 8x x x  + 16x x x  - 19x x  - 127x x x  + 54x x x  + 7x x x  + 42x x  - x x  + 62x x  - 42x x  + 19x x  + 19x  + 9x x x  - 16x x  - 25x x x  - 33x x x  - 23x x x  + 14x x  + 14x x x  + 51x x x  - 33x x x  - 19x x x  + 14x x ,
                1      1 2 3      0 3      1 3      2 3     3      0 4      0 1 4       1 4     0 2 4      1 2 4      2 4       0 3 4      1 3 4     2 3 4      3 4    0 4      1 4      2 4      3 4      4     0 1 5      1 5      1 2 5      0 3 5      1 3 5      3 5      0 4 5      1 4 5      2 4 5      3 4 5      1 5
               
                  2                   2        2        2      3      2                     2                              2                                          2         2         2        2        2      3                 2                                         2                                                      2
               x x  + 79x x x  + 29x x  + 40x x  + 32x x  - 11x  - 41x x  + 164x x x  - 196x x  + 14x x x  + 26x x x  - 32x x  - 214x x x  + 92x x x  + 11x x x  + 71x x  - 2x x  + 105x x  - 71x x  + 32x x  + 32x  + 14x x x  - 26x x  - 41x x x  - 55x x x  - 38x x x  + 23x x  + 23x x x  + 85x x x  - 55x x x  - 32x x x  + 23x x ,
                0 1      1 2 3      0 3      1 3      2 3      3      0 4       0 1 4       1 4      0 2 4      1 2 4      2 4       0 3 4      1 3 4      2 3 4      3 4     0 4       1 4      2 4      3 4      4      0 1 5      1 5      1 2 5      0 3 5      1 3 5      3 5      0 4 5      1 4 5      2 4 5      3 4 5      1 5
               
                2                      2        2        2      3      2                     2                              2                                            2         2         2         2        2      3                 2                                         2                                                       2
               x x  + 133x x x  + 48x x  + 68x x  + 54x x  - 18x  - 70x x  + 278x x x  - 330x x  + 24x x x  + 44x x x  - 54x x  - 361x x x  + 156x x x  + 18x x x  + 120x x  - 4x x  + 177x x  - 120x x  + 54x x  + 54x  + 23x x x  - 44x x  - 69x x x  - 93x x x  - 63x x x  + 39x x  + 39x x x  + 144x x x  - 93x x x  - 54x x x  + 39x x ,
                0 1       1 2 3      0 3      1 3      2 3      3      0 4       0 1 4       1 4      0 2 4      1 2 4      2 4       0 3 4       1 3 4      2 3 4       3 4     0 4       1 4       2 4      3 4      4      0 1 5      1 5      1 2 5      0 3 5      1 3 5      3 5      0 4 5       1 4 5      2 4 5      3 4 5      1 5
               
                3                    2         2        2      3       2                     2                              2                                            2         2         2         2        2      3                 2                                            2                                                        2
               x  + 224x x x  + 80x x  + 115x x  + 91x x  - 30x  - 119x x  + 470x x x  - 555x x  + 41x x x  + 75x x x  - 91x x  - 608x x x  + 263x x x  + 30x x x  + 202x x  - 8x x  + 297x x  - 202x x  + 91x x  + 91x  + 39x x x  - 76x x  - 118x x x  - 158x x x  - 105x x x  + 67x x  + 68x x x  + 245x x x  - 158x x x  - 91x x x  + 67x x
                0       1 2 3      0 3       1 3      2 3      3       0 4       0 1 4       1 4      0 2 4      1 2 4      2 4       0 3 4       1 3 4      2 3 4       3 4     0 4       1 4       2 4      3 4      4      0 1 5      1 5       1 2 5       0 3 5       1 3 5      3 5      0 4 5       1 4 5       2 4 5      3 4 5      1 5
              }
     fourfold: {
                 2                           2       2       2     2                   2                          2                            2        2       2       2       2     3             2                                    2                                               2
                x x  + x x x  - 6x x x  - x x  - 4x x  - 3x x  + 2x x  - 10x x x  + 13x x  - x x x  - 3x x x  + 3x x  + 14x x x  - 8x x x  - 4x x  + x x  - 7x x  + 4x x  - 2x x  - 2x  - x x x  + x x  + 2x x x  + 3x x x  + 3x x x  - x x  - x x x  - 4x x x  + 3x x x  + 2x x x  - x x
                 1 3    0 2 3     1 2 3    0 3     1 3     2 3     0 4      0 1 4      1 4    0 2 4     1 2 4     2 4      0 3 4     1 3 4     3 4    0 4     1 4     2 4     3 4     4    0 1 5    1 5     1 2 5     0 3 5     1 3 5    3 5    0 4 5     1 4 5     2 4 5     3 4 5    1 5
               }

o5 : SpecialCubicFourfold (Cubic fourfold containing a surface of degree 7 and sectional genus 0)
i6 : time discriminant X
     -- used 0.368703 seconds

o6 = 26

Ways to use specialCubicFourfold :