parameterCount -- count of parameters

Synopsis

• Usage:

  parameterCount(S,X)

• Inputs:
  • S, an embedded projective variety
  • X, an embedded projective variety, such that $S\subseteq X$
• Optional inputs:
  • Verbose => ..., default value false
• Outputs:
  • a count of parameters to estimate the dimensions of the corresponding Hilbert schemes

Description

See parameterCount(SpecialCubicFourfold) and parameterCount(SpecialGushelMukaiFourfold) for more precise applications of this function.

The following calculation shows that the family of complete intersections of 3 quadrics in $\mathbb{P}^5$ containing a rational normal quintic curve has codimension 1 in the space of all such complete intersections.

i1 : K = ZZ/33331; S = PP_K^(1,5);

o2 : ProjectiveVariety, curve in PP^5

i3 : X = random({{2},{2},{2}},S);

o3 : ProjectiveVariety, surface in PP^5

i4 : time parameterCount(S,X,Verbose=>true)
S: rational normal curve of degree 5 in PP^5
X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3 hypersurfaces of degree 2
(assumption: h^1(N_{S,P^5}) = 0)
h^0(N_{S,P^5}) = 32
h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
in particular, h^0(I_{S,P^5}(2)) is minimal
dim GG(2,9) = 21
h^0(N_{S,P^5}) + dim GG(2,9) = 53
h^0(N_{S,X}) = 0
dim{[X] : S ⊂ X} >= 53
dim GG(2,P(H^0(O_(P^5)(2)))) = 54
codim{[X] : S ⊂ X} <= 1
     -- used 0.214627 seconds

o4 = (1, (10, 32, 0))

o4 : Sequence

See also

• parameterCount(SpecialCubicFourfold) -- count of parameters in the moduli space of GM fourfolds
• parameterCount(SpecialGushelMukaiFourfold) -- count of parameters in the moduli space of GM fourfolds
• normalSheaf -- normal sheaf