# parameterCount -- count of parameters

## Synopsis

• Usage:
parameterCount(S,X)
• Inputs:
• X, , such that $S\subseteq X$
• Optional inputs:
• Verbose => ..., default value false, request verbose feedback
• 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 0.93851 seconds o4 = (1, (10, 32, 0)) o4 : Sequence