This function implements a parameter count explained in the paper On some families of Gushel-Mukai fourfolds.
Below, we show that the closure of the locus of GM fourfolds containing a cubic scroll has codimension at most one (hence exactly one) in the moduli space of GM fourfolds.
i1 : G = GG(ZZ/33331,1,4); o1 : ProjectiveVariety, GG(1,4) |
i2 : S = (schubertCycle({2,0},G) * random({{1},{1}},0_G))%G o2 = S o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G) |
i3 : X = specialGushelMukaiFourfold S; o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0 |
i4 : time parameterCount(X,Verbose=>true) S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2) X: GM fourfold containing S Y: del Pezzo fivefold containing X h^1(N_{S,Y}) = 0 h^0(N_{S,Y}) = 11 h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2)); in particular, h^0(I_{S,Y}(2)) is minimal h^0(N_{S,Y}) + 27 = 38 h^0(N_{S,X}) = 0 dim{[X] : S ⊂ X ⊂ Y} >= 38 dim P(H^0(O_Y(2))) = 39 codim{[X] : S ⊂ X ⊂ Y} <= 1 -- used 2.60835 seconds o4 = (1, (28, 11, 0)) o4 : Sequence |
i5 : discriminant X o5 = 12 |