parameterCount(SpecialGushelMukaiFourfold) -- count of parameters in the moduli space of GM fourfolds

Synopsis

• Function: parameterCount

• Usage:

  parameterCount X

• Inputs:
  • X, a special Gushel-Mukai fourfold, a special GM fourfold containing a surface $S$ and contained in a del Pezzo fivefold $Y$
• Optional inputs:
  • Verbose => ..., default value false, request verbose feedback
• Outputs:
  • an integer, an upper bound for the codimension in the moduli space of GM fourfolds of the locus of GM fourfolds that contain a surface belonging to the same irreducible component of the Hilbert scheme of $Y$ that contains $[S]$
  • a sequence, the triple of integers: $(h^0(I_{S/Y}(2)), h^0(N_{S/Y}), h^0(N_{S/X}))$

Description

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 6.13387 seconds

o4 = (1, (28, 11, 0))

o4 : Sequence

i5 : discriminant X

o5 = 12

See also

• parameterCount(SpecialCubicFourfold) -- count of parameters in the moduli space of GM fourfolds
• normalSheaf -- normal sheaf