This method 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 C20
i1 : P5 = ZZ/33331[x_0..x_5]; |
i2 : V = trim minors(2,genericSymmetricMatrix(P5,3)) 2 2 2 o2 = ideal (x - x x , x x - x x , x x - x x , x - x x , x x - x x , x - 4 3 5 2 4 1 5 2 3 1 4 2 0 5 1 2 0 4 1 ------------------------------------------------------------------------ x x ) 0 3 o2 : Ideal of P5 |
i3 : X = specialCubicFourfold V o3 = -- special cubic fourfold -- ZZ ambient projective space: Proj(-----[x , x , x , x , x , x ]) 33331 0 1 2 3 4 5 surface: { 2 x - x x , 4 3 5 x x - x x , 2 4 1 5 x x - x x , 2 3 1 4 2 x - x x , 2 0 5 x x - x x , 1 2 0 4 2 x - x x 1 0 3 } fourfold: { 2 3 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 10866x x + 5398x + 5107x x x - 13804x x - 10808x x + 15262x x - 670x - 10866x x - 5398x x x + 15317x x + 8480x x x - 12849x x x + 13470x x - 15317x x - 9579x x - 5107x x + 5324x x x + 1031x x - 9398x x x - 11081x x x - 8829x x + 11818x x x + 9579x x x - 1564x x x + 2637x x - 5071x x + 8570x x - 456x x - 6898x + 10808x x - 5864x x x - 7653x x + 670x x x - 3107x x x + 9534x x + 2627x x x + 6635x x x - 7531x x x + 456x x + 11936x x x - 1039x x x + 16659x x x + 6898x x x + 3783x x - 9534x x - 16659x x - 3783x x 0 1 1 0 1 2 1 2 0 2 1 2 2 0 3 0 1 3 1 3 0 2 3 1 2 3 2 3 0 3 2 3 0 4 0 1 4 1 4 0 2 4 1 2 4 2 4 0 3 4 1 3 4 2 3 4 0 4 1 4 2 4 3 4 4 0 5 0 1 5 1 5 0 2 5 1 2 5 2 5 0 3 5 1 3 5 2 3 5 3 5 0 4 5 1 4 5 2 4 5 3 4 5 4 5 0 5 1 5 3 5 } o3 : SpecialCubicFourfold (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 0.355923 seconds o4 = 1 |
i5 : time discriminant X -- used 0.0658618 seconds o5 = 20 |