This method 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 = Grass(1,4,ZZ/33331); |
i2 : S = schubertCycle({2,0},G) + ideal(random(1,G), random(1,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 , - 2322p + 8971p - 15493p - 4330p -
3,4 0,1 0,3 1,3 0,4 1,4 3,4 0,1 0,2 1,2 0,3
----------------------------------------------------------------------------------------------------------------------------
1233p - 7781p + 8886p - 8356p - 1343p + 6377p , - 2520p - 16661p - 8928p - 7368p - 4492p -
1,3 2,3 0,4 1,4 2,4 3,4 0,1 0,2 1,2 0,3 1,3
----------------------------------------------------------------------------------------------------------------------------
4888p - 7955p + 4700p + 5969p - 6811p )
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 : time parameterCount X
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\subset X \subset Y} >= 38
dim P(H^0(O_Y(2))) = 39
codim{[X] : S\subset X \subset Y} <= 1
-- used 9.31324 seconds
o4 = (1, (28, 11, 0))
o4 : Sequence
|
i5 : time discriminant X
-- used 2.68774 seconds
o5 = 12
|