toGrass(EmbeddedProjectiveVariety) -- embedding of an ordinary Gushel-Mukai fourfold or a del Pezzo variety into GG(1,4)

i1 : x = gens ring PP_(ZZ/33331)^8;

i2 : X = projectiveVariety ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8);

o2 : ProjectiveVariety, 5-dimensional subvariety of PP^8

i3 : time toGrass X
warning: clearing value of symbol x to allow access to subscripted variables based on it
       : debug with expression   debug 9868   or with command line option   --debug 9868
 -- used 9.999s (cpu); 5.33148s (thread); 0s (gc)

o3 = multi-rational map consisting of one single rational map
     source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
     target variety: GG(1,4) ⊂ PP^9

o3 : MultirationalMap (rational map from X to GG(1,4))

i4 : show oo

o4 = -- multi-rational map --
                                  ZZ
     source: subvariety of Proj(-----[x , x , x , x , x , x , x , x , x ]) defined by
                                33331  0   1   2   3   4   5   6   7   8
             {
              x x  - x x  + x x ,
               4 6    3 7    1 8
              
              x x  - x x  + x x ,
               4 5    2 7    0 8
              
              x x  - x x  + x x  + x x  - x x ,
               3 5    2 6    0 8    1 8    5 8
              
              x x  - x x  + x x  + x x  - x x ,
               1 5    0 6    0 7    1 7    5 7
              
              x x  - x x  + x x  + x x  - x x  + x x
               1 2    0 3    0 4    1 4    2 7    0 8
             }
                                  ZZ
     target: subvariety of Proj(-----[x   , x   , x   , x   , x   , x   , x   , x   , x   , x   ]) defined by
                                33331  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
             {
              x   x    - x   x    + x   x   ,
               2,3 1,4    1,3 2,4    1,2 3,4
              
              x   x    - x   x    + x   x   ,
               2,3 0,4    0,3 2,4    0,2 3,4
              
              x   x    - x   x    + x   x   ,
               1,3 0,4    0,3 1,4    0,1 3,4
              
              x   x    - x   x    + x   x   ,
               1,2 0,4    0,2 1,4    0,1 2,4
              
              x   x    - x   x    + x   x
               1,2 0,3    0,2 1,3    0,1 2,3
             }
     -- rational map 1/1 -- 
     map 1/1, one of its representatives:
     {
      5418x  - 821x  + 5588x  - 3585x  - 1758x  - 15576x  + 9147x  - 14993x  - 4736x ,
           0       1        2        3        4         5        6         7        8
      
      11632x  - 4732x  - 10523x  - 11526x  - 1991x  - 1831x  - 9701x  + 12320x  - 2015x ,
            0        1         2         3        4        5        6         7        8
      
      16371x  - 7244x  + 4935x  + 15111x  + 3749x  - 12977x  + 15511x  + 7287x  + 6751x ,
            0        1        2         3        4         5         6        7        8
      
      - 13960x  - 3219x  + 8239x  - 10597x  + 7747x  + 273x  - 6285x  + 2934x  - 4471x ,
              0        1        2         3        4       5        6        7        8
      
      - 12638x  - 12017x  - 2651x  + 7012x  - 9505x  + 3559x  - 2170x  - 59x  - 265x ,
              0         1        2        3        4        5        6      7       8
      
      - 4096x  + 10456x  - 2284x  + 11208x  + 5756x  - 6263x  + 599x  + 7817x  - 6486x ,
             0         1        2         3        4        5       6        7        8
      
      5896x  + 11711x  - 9239x  + 9726x  + 9682x  + 2295x  - 6875x  - 16024x  - 7246x ,
           0         1        2        3        4        5        6         7        8
      
      - 8687x  + 14564x  + 3651x  - 6141x  - 7924x  + 3227x  - 5479x  + 13427x  + 11982x ,
             0         1        2        3        4        5        6         7         8
      
      89x  + 11710x  + 1284x  - 12079x  + 11673x  - 2256x  + 12732x  - 7459x  - 5231x ,
         0         1        2         3         4        5         6        7        8
      
      - 25x  + 12923x  + 1000x  + 871x  + 15902x  - 3782x  - 7479x  - 5250x  + 11717x
           0         1        2       3         4        5        6        7         8
     }