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SpecialFanoFourfolds :: embed(Ideal)

embed(Ideal) -- embedding of a Gushel-Mukai fourfold/delPezzo fivefold/del Pezzo sixfold into Grass(1,4)

Synopsis

Description

i1 : P8 = ZZ/33331[x_0..x_8];
i2 : X = 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 : Ideal of P8
i3 : time embed X
     -- used 2.06716 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
             }
                                  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: {
                      - 4993x  - 12663x  - 3763x  - 3126x  + 14655x  - 238x  - 8671x  - 12430x  + 1526x ,
                             0         1        2        3         4       5        6         7        8
                      
                      5133x  - 11938x  - 15689x  - 4522x  - 13503x  + 1311x  + 15527x  + 1346x  + 12477x ,
                           0         1         2        3         4        5         6        7         8
                      
                      11488x  + 6557x  - 6832x  + 5721x  - 14068x  - 6739x  - 14843x  - 14656x  + 6516x ,
                            0        1        2        3         4        5         6         7        8
                      
                      - 9189x  - 5422x  + 16196x  - 13909x  + 6333x  - 2092x  + 7531x  - 10094x  - 244x ,
                             0        1         2         3        4        5        6         7       8
                      
                      - 5428x  - 9575x  - 1807x  - 8109x  + 11094x  + 10238x  - 6492x  + 16031x  - 5409x ,
                             0        1        2        3         4         5        6         7        8
                      
                      - 15275x  - 14785x  + 6472x  - 10604x  + 4939x  + 14324x  - 6771x  + 14676x  + 3755x ,
                              0         1        2         3        4         5        6         7        8
                      
                      1636x  + 14709x  + 8561x  + 1707x  - 9310x  + 2206x  + 14692x  + 10139x  - 15710x ,
                           0         1        2        3        4        5         6         7         8
                      
                      5271x  + 832x  + 2881x  + 6624x  - 9640x  + 10955x  + 7528x  + 12918x  - 6040x ,
                           0       1        2        3        4         5        6         7        8
                      
                      - 413x  - 15386x  + 11491x  + 8004x  + 15934x  - 14552x  - 5545x  - 3668x  + 9091x ,
                            0         1         2        3         4         5        6        7        8
                      
                      - 16457x  - 7752x  + 7998x  + 2366x  - 15097x  + 13610x  - 1851x  + 8333x  - 2342x
                              0        1        2        3         4         5        6        7        8
                     }

o3 : RationalMap (linear rational map from 5-dimensional subvariety of PP^8 to 6-dimensional subvariety of PP^9)

See also