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SpecialFanoFourfolds :: toGrass(EmbeddedProjectiveVariety)

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

Synopsis

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 4.19382 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:
     {
      16439x0  - 15415x0  + 12561x0  + 4254x0  + 495x0  - 13354x0  - 825x0  + 15281x0  + 1997x0 ,
             0          1          2         3        4          5        6          7         8
      
      10439x0  + 9146x0  + 9651x0  + 12408x0  + 2676x0  - 4974x0  - 11705x0  + 7350x0  - 4938x0 ,
             0         1         2          3         4         5          6         7         8
      
      3657x0  - 818x0  + 11119x0  - 16662x0  - 3171x0  - 7420x0  + 12406x0  + 12027x0  + 2941x0 ,
            0        1          2          3         4         5          6          7         8
      
      13119x0  + 12780x0  - 10829x0  - 7832x0  + 2991x0  - 3720x0  + 352x0  + 2273x0  + 4176x0 ,
             0          1          2         3         4         5        6         7         8
      
      - 4440x0  + 2856x0  - 9941x0  + 12086x0  - 2496x0  + 8959x0  + 8783x0  - 2870x0  - 2179x0 ,
              0         1         2          3         4         5         6         7         8
      
      4335x0  + 16441x0  - 1178x0  + 4576x0  + 5667x0  - 8818x0  + 11045x0  + 6670x0  - 762x0 ,
            0          1         2         3         4         5          6         7        8
      
      - 11737x0  - 8776x0  + 979x0  - 1641x0  - 4056x0  - 7754x0  - 1583x0  + 13067x0  + 2051x0 ,
               0         1        2         3         4         5         6          7         8
      
      - 16516x0  - 13248x0  - 13540x0  - 2613x0  + 3561x0  - 7295x0  + 950x0  - 15222x0  - 4048x0 ,
               0          1          2         3         4         5        6          7         8
      
      1460x0  + 15243x0  - 8672x0  - 14049x0  - 6732x0  + 14081x0  + 16258x0  + 10586x0  + 6989x0 ,
            0          1         2          3         4          5          6          7         8
      
      16575x0  + 15122x0  + 9850x0  + 9473x0  + 1065x0  - 3036x0  - 2350x0  - 15831x0  - 6227x0
             0          1         2         3         4         5         6          7         8
     }

See also

Ways to use this method: