specialGushelMukaiFourfold(Ideal) -- random special Gushel-Mukai fourfold

Synopsis

Function: specialGushelMukaiFourfold
Usage:

specialGushelMukaiFourfold S
Inputs:
- S, an ideal, the ideal of a smooth irreducible surface in the coordinate ring of a del Pezzo fivefold or del Pezzo sixfold (e.g., an ideal in the ring Grass$(1,4)$)
Optional inputs:
- InputCheck => ..., default value 1, make a special Gushel-Mukai fourfold
- Verbose => ..., default value true, make a special Gushel-Mukai fourfold
Outputs:
- a special Gushel-Mukai fourfold, a random special Gushel-Mukai fourfold containing the given surface

Description

i1 : G = Grass(1,4,ZZ/33331);

i2 : -- cubic scroll in G(1,4)
     S = schubertCycle({2,0},G) + schubertCycle({1,0},G) + schubertCycle({1,0},G)

o2 = ideal (p    + 8480p    + 6727p    - 11656p    - 14853p    - 13522p   , p    + 8480p    - 15777p    - 11656p    + 664p    -
             1,2        1,3        2,3         1,4         2,4         3,4   0,2        0,3         2,3         0,4       2,4  
     ----------------------------------------------------------------------------------------------------------------------------
     11804p   , p    - 6727p    - 15777p    + 14853p    + 664p    - 14854p   , p    - 13957p    + 11800p    + 15640p    -
           3,4   0,1        0,3         1,3         0,4       1,4         3,4   0,1         0,2         1,2         0,3  
     ----------------------------------------------------------------------------------------------------------------------------
     14837p    - 3747p    - 9672p    - 15041p    + 12855p    - 4551p   , p    + 8778p    + 5117p    + 6948p    - 6159p    +
           1,3        2,3        0,4         1,4         2,4        3,4   0,1        0,2        1,2        0,3        1,3  
     ----------------------------------------------------------------------------------------------------------------------------
     10441p    - 175p    - 3457p    + 14533p    + 1182p   )
           2,3       0,4        1,4         2,4        3,4

o2 : Ideal of G

i3 : X = specialGushelMukaiFourfold S;

o3 : SpecialGushelMukaiFourfold (Gushel-Mukai fourfold containing a surface of degree 3 and sectional genus 0)

i4 : discriminant X

o4 = 12

Some random Gushel-Mukai fourfolds can also be obtained by passing strings. For instance, an object as above is also given as follows.

i5 : specialGushelMukaiFourfold("cubic scroll");

o5 : SpecialGushelMukaiFourfold (Gushel-Mukai fourfold containing a surface of degree 3 and sectional genus 0)