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

Synopsis

Function: parameterCount
Usage:

parameterCount X
Inputs:
- X, a special cubic fourfold, a special cubic fourfold containing a surface $S$
Optional inputs:
- Verbose => ..., default value false, request verbose feedback
Outputs:
- an integer, an upper bound for the codimension in the moduli space of cubic fourfolds of the locus of cubic fourfolds that contain a surface belonging to the same irreducible component of the Hilbert scheme containing $[S]$
- a sequence, the triple of integers: $(h^0(I_{S/P^5}(3)), h^0(N_{S/P^5}), h^0(N_{S/X}))$

Description

This function implements a parameter count explained in the paper Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces, by H. Nuer.

Below, we show that the closure of the locus of cubic fourfolds containing a Veronese surface has codimension at most one (hence exactly one) in the moduli space of cubic fourfolds. Then, by the computation of the discriminant, we deduce that the cubic fourfolds containing a Veronese surface describe the Hassett's divisor $\mathcal{C}_{20}$

i1 : K = ZZ/33331; V = PP_K^(2,2);

o2 : ProjectiveVariety, surface in PP^5

i3 : X = specialCubicFourfold V;

o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0

i4 : time parameterCount(X,Verbose=>true)
 -- used 0.822841s (cpu); 0.550266s (thread); 0s (gc)
S: Veronese surface in PP^5
X: smooth cubic hypersurface in PP^5
(assumption: h^1(N_{S,P^5}) = 0)
h^0(N_{S,P^5}) = 27
h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
in particular, h^0(I_{S,P^5}(3)) is minimal
h^0(N_{S,P^5}) + 27 = 54
h^0(N_{S,X}) = 0
dim{[X] : S ⊂ X} >= 54
dim P(H^0(O_(P^5)(3))) = 55
codim{[X] : S ⊂ X} <= 1

o4 = (1, (28, 27, 0))

o4 : Sequence

i5 : discriminant X

o5 = 20

Ways to use this method: