toGrass -- Gushel morphism from a GM fourfold to GG(1,4)

Usage:

toGrass X
Inputs:
- X, a Gushel-Mukai fourfold
Outputs:
- a multi-rational map, a linear morphism from $X$ into the Grassmannian $\mathbb{G}(1,4)\subset\mathbb{P}^9$, Plücker embedded, which is an embedding when $X$ is of ordinary type

Description

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

i2 : X = gushelMukaiFourfold(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));

o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0

i3 : time toGrass X
 -- used 3.9186s (cpu); 2.666s (thread); 0s (gc)

o3 = multi-rational map consisting of one single rational map
     source variety: 4-dimensional subvariety of PP^8 cut out by 6 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
              
               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(-----[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:
     {
      5601x  + 884x  + 6926x  + 5393x  - 10165x  - 11843x  + 12117x  + 1300x  - 13972x ,
           0       1        2        3         4         5         6        7         8
      
      4392x  + 8551x  - 13317x  - 11710x  - 16494x  + 6088x  + 12312x  + 2372x  + 11891x ,
           0        1         2         3         4        5         6        7         8
      
      15028x  - 7707x  + 6391x  + 6317x  - 6672x  - 8736x  + 3147x  - 12114x  + 2081x ,
            0        1        2        3        4        5        6         7        8
      
      - 9308x  + 16430x  + 917x  + 7026x  - 8520x  + 5348x  + 13827x  - 7807x  + 14387x ,
             0         1       2        3        4        5         6        7         8
      
      - 9504x  - 7275x  + 6009x  - 1633x  - 1645x  - 16019x  - 7002x  - 12528x  + 4972x ,
             0        1        2        3        4         5        6         7        8
      
      - 10876x  + 9715x  - 12400x  - 4684x  + 8317x  + 6409x  - 6918x  + 9359x  - 7053x ,
              0        1         2        3        4        5        6        7        8
      
      8706x  + 2673x  - 11266x  + 4856x  - 5395x  + 4047x  - 4444x  + 2552x  - 4720x ,
           0        1         2        3        4        5        6        7        8
      
      - 3448x  - 13149x  + 4340x  - 10249x  + 15560x  - 13187x  - 9194x  - 6006x  - 14639x ,
             0         1        2         3         4         5        6        7         8
      
      - 8335x  - 16563x  + 2051x  + 16566x  + 11099x  + 9338x  + 5412x  + 7284x  - 16611x ,
             0         1        2         3         4        5        6        7         8
      
      7744x  + 6441x  + 10349x  - 11882x  + 13915x  - 14776x  + 14074x  + 15588x  - 9667x
           0        1         2         3         4         5         6         7        8
     }

Ways to use toGrass:

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

For the programmer

The object toGrass is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SpecialFanoFourfolds/docs.m2:93:0.

toGrass -- Gushel morphism from a GM fourfold to GG(1,4)

Description

See also

Ways to use toGrass:

For the programmer