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 0.347314 seconds o4 = (1, (28, 27, 0)) o4 : Sequence |
i5 : time discriminant X -- used 0.000688759 seconds o5 = 20 |