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SpecialFanoFourfolds :: specialGushelMukaiFourfold(Ideal)

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

Synopsis

Description

i1 : use Grass(1,4,ZZ/33331);
i2 : -- cubic scroll in G(1,4)
     S = ideal(2*p_(1,3)-p_(2,3)+p_(0,4)-2*p_(1,4)-p_(3,4),p_(0,3)-p_(2,3)-p_(1,4)-p_(3,4),2*p_(1,2)+p_(2,3)+p_(0,4)-2*p_(2,4)-p_(3,4),p_(0,2)+p_(0,4)-p_(1,4)-p_(2,4)-p_(3,4),2*p_(0,1)-p_(2,3)+p_(0,4)-2*p_(1,4)-p_(3,4),2*p_(2,3)*p_(1,4)-p_(2,3)*p_(2,4)+p_(0,4)*p_(2,4)-2*p_(1,4)*p_(2,4)-p_(2,3)*p_(3,4)-p_(0,4)*p_(3,4)+p_(2,4)*p_(3,4)+p_(3,4)^2,p_(0,4)^2-2*p_(0,4)*p_(1,4)+2*p_(1,4)^2-p_(0,4)*p_(2,4)+p_(1,4)*p_(2,4)+p_(1,4)*p_(3,4)-p_(2,4)*p_(3,4)-p_(3,4)^2,p_(2,3)*p_(0,4)-p_(2,3)*p_(2,4)-p_(1,4)*p_(2,4)-p_(0,4)*p_(3,4)+p_(1,4)*p_(3,4)+p_(3,4)^2);

                                                                ZZ
                                                              -----[p   , p   , p   , p   , p   , p   , p   , p   , p   , p   ]
                                                              33331  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
o2 : Ideal of ----------------------------------------------------------------------------------------------------------------------------------------------------------------
              (p   p    - p   p    + p   p   , p   p    - p   p    + p   p   , p   p    - p   p    + p   p   , p   p    - p   p    + p   p   , p   p    - p   p    + p   p   )
                2,3 1,4    1,3 2,4    1,2 3,4   2,3 0,4    0,3 2,4    0,2 3,4   1,3 0,4    0,3 1,4    0,1 3,4   1,2 0,4    0,2 1,4    0,1 2,4   1,2 0,3    0,2 1,3    0,1 2,3
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 to the method. 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)