next | previous | forward | backward | up | top | index | toc | Macaulay2 website
SpecialFanoFourfolds :: parameterCount(SpecialCubicFourfold)

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

Synopsis

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;
-- calculated number of nodes (got 0 nodes)

o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
i4 : time parameterCount X
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\subset X} >= 54
dim P(H^0(O_(P^5)(3))) = 55
codim{[X] : S\subset X} <= 1
     -- used 0.541961 seconds

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

o4 : Sequence
i5 : time discriminant X
     -- used 0.000776729 seconds

o5 = 20

See also