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

Synopsis

Function: toGrass
Usage:

toGrass X
Inputs:
- X, an embedded projective variety, an ordinary Gushel-Mukai fourfold, or a del Pezzo variety of dimension at least 4 (e.g., a sixfold projectively equivalent to $\mathbb{G}(1,4)\subset\mathbb{P}^9$)
Outputs:
- a multi-rational map, an embedding of $X$ into the Grassmannian $\mathbb{G}(1,4)\subset\mathbb{P}^9$, Plücker embedded

Description

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
     -- used 11.5312 seconds

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(-----[x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 ]) defined by
                                33331   0    1    2    3    4    5    6    7    8
             {
              x0 x0  - x0 x0  + x0 x0 ,
                4  6     3  7     1  8
              
              x0 x0  - x0 x0  + x0 x0 ,
                4  5     2  7     0  8
              
              x0 x0  - x0 x0  + x0 x0  + x0 x0  - x0 x0 ,
                3  5     2  6     0  8     1  8     5  8
              
              x0 x0  - x0 x0  + x0 x0  + x0 x0  - x0 x0 ,
                1  5     0  6     0  7     1  7     5  7
              
              x0 x0  - x0 x0  + x0 x0  + x0 x0  - x0 x0  + x0 x0
                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:
     {
      5418x0  - 821x0  + 5588x0  - 3585x0  - 1758x0  - 15576x0  + 9147x0  - 14993x0  - 4736x0 ,
            0        1         2         3         4          5         6          7         8
      
      11632x0  - 4732x0  - 10523x0  - 11526x0  - 1991x0  - 1831x0  - 9701x0  + 12320x0  - 2015x0 ,
             0         1          2          3         4         5         6          7         8
      
      16371x0  - 7244x0  + 4935x0  + 15111x0  + 3749x0  - 12977x0  + 15511x0  + 7287x0  + 6751x0 ,
             0         1         2          3         4          5          6         7         8
      
      - 13960x0  - 3219x0  + 8239x0  - 10597x0  + 7747x0  + 273x0  - 6285x0  + 2934x0  - 4471x0 ,
               0         1         2          3         4        5         6         7         8
      
      - 12638x0  - 12017x0  - 2651x0  + 7012x0  - 9505x0  + 3559x0  - 2170x0  - 59x0  - 265x0 ,
               0          1         2         3         4         5         6       7        8
      
      - 4096x0  + 10456x0  - 2284x0  + 11208x0  + 5756x0  - 6263x0  + 599x0  + 7817x0  - 6486x0 ,
              0          1         2          3         4         5        6         7         8
      
      5896x0  + 11711x0  - 9239x0  + 9726x0  + 9682x0  + 2295x0  - 6875x0  - 16024x0  - 7246x0 ,
            0          1         2         3         4         5         6          7         8
      
      - 8687x0  + 14564x0  + 3651x0  - 6141x0  - 7924x0  + 3227x0  - 5479x0  + 13427x0  + 11982x0 ,
              0          1         2         3         4         5         6          7          8
      
      89x0  + 11710x0  + 1284x0  - 12079x0  + 11673x0  - 2256x0  + 12732x0  - 7459x0  - 5231x0 ,
          0          1         2          3          4         5          6         7         8
      
      - 25x0  + 12923x0  + 1000x0  + 871x0  + 15902x0  - 3782x0  - 7479x0  - 5250x0  + 11717x0
            0          1         2        3          4         5         6         7          8
     }

Ways to use this method: