Example Scheme_SecantVariety (H122E56)

> P<a,b,c,d,e> := ProjectiveSpace(RationalField(),4);
> X := Scheme(P,[a*d + c*e, a*c + d*e,
>  a^2 - e^2, c^2*e - d^2*e,
>  b^2 + c*d + e^2]); // union of 3 irreducible curves
> Dimension(X);
1
> time SecantVariety(X : PatchIndex := 2); 
Scheme over Rational Field defined by
a^4*b^2 + a^4*e^2 - a^3*c^2*e - a^3*d^2*e + a^2*b^4 + 2*a^2*b^2*c*d + 
    1/4*a^2*c^4 + 1/2*a^2*c^2*d^2 + 1/4*a^2*d^4 - a^2*e^4 + a*c^2*e^3 + 
    a*d^2*e^3 - b^4*e^2 - 2*b^2*c*d*e^2 - b^2*e^4 - 1/4*c^4*e^2 - 
    1/2*c^2*d^2*e^2 - 1/4*d^4*e^2
Time: 26.890
> Dimension($1);
3
> time IsInSecantVariety(X,P![0,1,0,-3,0]);
true
Time: 0.000

Isomorphic Projection to Subspaces

The aim of the functions here is to try to find embeddings of projective schemes which lie in high-dimensional ambient spaces into lower dimensional spaces via projection.

Let X be a scheme in ordinary projective space that is assumed to be reduced but not necessarily irreducible or non-singular, d the dimension of X and n the dimension of its ambient space P. The conditions imply that the tangent variety and secant variety of X are finite unions of subschemes of dimension at most 2d + 1 (cf [Har77] IV.3), so while n > 2d + 1, we can find points in P lying in neither of these, unless possibly the base field is a finite field. For such a point, projection down to a hyperplane gives an isomorphism of X to its image (cf. op. cit.) and we can continue projecting down one dimension at a time until n = 2d + 1.

In fact, the tangent and secant varieties of the image of X will be their images under the projection also, which induces a finite map on them (it's quasi-finite since the inverse image of a point in the image hyperplane is a line through the projecting point pt, which any (closed) subscheme not containing pt must intersect finitely). Hence their images have the same dimensions. If the maximum of these dimensions is tsd ≤2d + 1 we can actually project down to a subspace of dimension tsd. Further, the fact that the tangent and secant images correspond under projection shows that if we find a linear change of variables for the coordinates of P, (x₁, ..., x_n) -> (y₁, ..., y_n), such that y₁, ..., y_tsd are a set of Noether normalising variables for the defining ideal of the union of the tangent and secant varieties in the coordinate ring of P, then we can project X isomorphically down to the tsd dimensional linear subspace y_{tsd + 1}=0, ..., y_n=0 directly.

This would be the most elegant solution. However, in practice it involves computing the full tangent and secant varieties which can be extremely time-consuming once we are in dimensions above around 3 or 4. By contrast, checking if a random point of P lies in the secant or tangent variety is reasonably fast. Therefore, we choose to follow the method of picking random points to project down one dimension at a time and finish when we reach dimension 2d + 1 (when the secant variety generally fills the ambient space anyway).

IsomorphicProjectionToSubspace(X) : Sch -> Sch, MapSch

    SetVerbose("IsoToSub", n):          Maximum: 1

As described above, this function projects the scheme X isomorphically down to a linear subspace of its ambient space P of dimension 2d + 1, if 2d + 1 < n. In the case of finite base fields or infinite ones of small characteristic, the process may stop before reaching this dimension but this should be very rare (difficulties finding random points outside the secant/tangent varieties). The return values consist of the image scheme, with the subspace as its ambient space, and the explicit map taking X to this image.

EmbedPlaneCurveInP3(C) : Crv -> Sch, MapSch

    SetVerbose("EmbCrv", n):            Maximum: 1

This function embeds a plane curve C as a non-singular projective curve in ordinary 2- or 3-dimensional projective space over the base field. This is effected by first using the function field machinery to give a non-singular projective embedding and then projecting down to a 3-dimensional subspace if necessary. In rare cases with small finite base fields or in small characteristic, the final embedding may still be into a higher-dimensional ambient space. The image scheme is returned along with the mapping of C to it.

> P<x,y,z> := ProjectiveSpace(RationalField(),2);
> f := x^5+x^2*y^3+y^2*z^3+x^2*z^3-y*z^4-z^5;
> C := Curve(P,f);
> Genus(C);
5
> SetVerbose("IsoToSub",1);
> SetVerbose("EmbCrv",1);
> C1, mp := EmbedPlaneCurveInP3(C);
Curve of genus 5.
Mapping to 4-space by canonical divisor map
Map was an embedding (non-hyperelliptic curve).
Beginning projection to 3-space.
Projection from dimension 4 to dimension 3:
Finding good projection point...
Performing projection...
Time: 0.002
> P1 := AmbientSpace(C1);
> AssignNames(~P1,["a","b","c","d"]);
> C1;
Scheme over Rational Field defined by
a^2*d^3 + a*b^2*c*d - a*b*c^2*d + a*b*d^3 + a*c^2*d^2 - a*d^4 + b^5 + b^2*c^3 - 
    b^2*c*d^2 - b^2*d^3 + 2*b*d^4 - c^2*d^3 - d^5,
a^2*c*d + a*b^3 + a*c^3 - a*c*d^2 + b*c*d^2 - c*d^3,
a*b^2*d - a*b*d^2 + a*c^2*d + b^3*c + c^4 - c^2*d^2,
a^3*d + a^2*c^2 - a^2*d^2 + a*b*d^2 - a*d^3 + b^2*c*d,
a^2*b - c^2*d
> Dimension(C1);
1
> ArithmeticGenus(C1);
5
> IsNonsingular(C1);
true