gushelMukaiFourfold(EmbeddedProjectiveVariety) -- random special Gushel-Mukai fourfold

• Function: gushelMukaiFourfold

• Usage:

  gushelMukaiFourfold S

  gushelMukaiFourfold (S%Y)

• Inputs:
  • S, an embedded projective variety, a smooth irreducible surface $S$ which is a subvariety of a del Pezzo fivefold/sixfold $Y$; alternatively, you can pass the ideal of $S$ in $Y$ (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 false, request verbose feedback
• Outputs:
  • a Gushel-Mukai fourfold, a random special Gushel-Mukai fourfold containing the given surface

i1 : Y = GG(ZZ/33331,1,4);

o1 : ProjectiveVariety, GG(1,4)

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

o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in Y)

i3 : X = gushelMukaiFourfold S;

o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0

i4 : discriminant X

o4 = 12