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SpecialFanoFourfolds :: parameterCount

parameterCount -- count of parameters

Synopsis

Description

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 $\mathcal{C}_{20}$

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  - x x )
             4    3 5   2 4    1 5   2 3    1 4   2    0 5   1 2    0 4   1    0 3

o2 : Ideal of P5
i3 : X = specialCubicFourfold V
-- computing number of nodes using a probabilistic method... 
-- got 0 nodes

o3 = X

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 0.661521 seconds

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

o4 : Sequence
i5 : time discriminant X
     -- used 0.105782 seconds

o5 = 20

See also

Ways to use parameterCount :

For the programmer

The object parameterCount is a method function with options.