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SpecialFanoFourfolds :: embed

embed -- embedding of a Gushel-Mukai fourfold into Grass(1,4)

Synopsis

Description

i1 : P8 = ZZ/33331[x_0..x_8];
i2 : time X = specialGushelMukaiFourfold(ideal(x_6-x_7,x_5,x_3-x_4,x_1,x_0-x_4,x_2*x_7-x_4*x_8),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,x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8));
     -- used 1.79983 seconds

o2 : SpecialGushelMukaiFourfold (Gushel-Mukai fourfold containing a surface of degree 2 and sectional genus 0)
i3 : time embed X
     -- used 4.155e-6 seconds

o3 = -- 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
              
               2           2                                 2                                  2                  2                                         2                    2
              x  + x x  + x  + x x  + 2x x  + x x  + x x  + x  - x x  - x x  - 2x x  - x x  - 2x  + x x  + x x  + x  + 2x x  + x x  + 2x x  + x x  + x x  + x  - 3x x  + 2x x  - x  + x x  + x x  - 3x x  + 2x x  + x x  - x x
               0    0 1    1    0 2     0 3    1 3    2 3    3    0 4    1 4     2 4    3 4     4    0 5    2 5    5     0 6    1 6     2 6    3 6    5 6    6     4 7     5 7    7    1 8    3 8     4 8     5 8    6 8    7 8
             }
                                  ZZ
     target: subvariety of Proj(-----[t   , t   , t   , t   , t   , t   , t   , t   , t   , t   ]) defined by
                                33331  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
             {
              t   t    - t   t    + t   t   ,
               2,3 1,4    1,3 2,4    1,2 3,4
              
              t   t    - t   t    + t   t   ,
               2,3 0,4    0,3 2,4    0,2 3,4
              
              t   t    - t   t    + t   t   ,
               1,3 0,4    0,3 1,4    0,1 3,4
              
              t   t    - t   t    + t   t   ,
               1,2 0,4    0,2 1,4    0,1 2,4
              
              t   t    - t   t    + t   t
               1,2 0,3    0,2 1,3    0,1 2,3
             }
     defining forms: {
                      - 9249x  + 3767x  + 164x  - 6955x  - 13415x  + 5189x  - 9034x  + 1398x  + 6379x ,
                             0        1       2        3         4        5        6        7        8
                      
                      - 9037x  + 15555x  + 13722x  + 7428x  - 1280x  - 12690x  + 101x  - 124x  - 1300x ,
                             0         1         2        3        4         5       6       7        8
                      
                      5752x  + 10216x  + 6915x  - 9854x  + 5406x  + 4875x  + 242x  + 5553x  - 3485x ,
                           0         1        2        3        4        5       6        7        8
                      
                      - 13975x  + 2023x  + 11207x  - 15085x  + 5060x  + 15069x  - 7126x  + 14018x  + 12225x ,
                              0        1         2         3        4         5        6         7         8
                      
                      - 730x  + 5597x  - 5903x  + 15684x  + 1908x  + 2809x  + 1043x  + 15326x  + 9438x ,
                            0        1        2         3        4        5        6         7        8
                      
                      - 6716x  + 7618x  - 2084x  + 1301x  - 1630x  - 7544x  - 5283x  - 5208x  - 1196x ,
                             0        1        2        3        4        5        6        7        8
                      
                      - 5177x  + 16076x  + 4914x  - 13188x  - 8068x  - 15760x  + 15536x  - 6172x  + 3400x ,
                             0         1        2         3        4         5         6        7        8
                      
                      492x  + 12248x  - 6423x  + 16363x  + 7703x  - 6141x  + 8379x  + 3553x  - 10525x ,
                          0         1        2         3        4        5        6        7         8
                      
                      4274x  + 554x  + 3301x  + 16195x  - 16199x  + 14479x  + 12627x  - 3599x  + 15273x ,
                           0       1        2         3         4         5         6        7         8
                      
                      - 10607x  + 1565x  - 8200x  + 5233x  + 1920x  - 10028x  + 9532x  - 9674x  - 11506x
                              0        1        2        3        4         5        6        7         8
                     }

o3 : RationalMap (linear rational map from 4-dimensional subvariety of PP^8 to 6-dimensional subvariety of PP^9)

See also

Ways to use embed :