The polynomials of the Cox ring of a toric variety X provide homogeneous coordinates on X that can be used to define subschemes of X. These subschemes are true Magma schemes, and so the usual scheme machinery works for them. However, there is a substantial caveat to this for the first version of the toric geometry package: affine patches have not been installed systematically, and so scheme machinery that uses affine patches of schemes will not work.
The subscheme of the toric variety X defined by the polynomial f from the Cox ring of X.
The subscheme of the toric variety X defined by the sequence Q of polynomials from the Cox ring of X.
Takes the binomial equations in the ideal of the scheme Z and constructs the toric variety given by the normalisation of the closure of the subtorus described by those binomials. Returns the pullback of Z and the normalisation map into the ambient of Z.
First make a curve. (This curve is in fact trigonal---it admits a 3-to-1 cover of the projective line. Once you've had that thought, it's actually pretty clear: the defining equation is a cubic in y. But there's more to it than just being trigonal, as we will see.)
> P<x,y,z> := ProjectiveSpace(Rationals(),2); > C := Curve(P,x^8 + x^4*y^3*z + z^8); > Genus(C); 8This curve is of general type (that is, its genus is at least 2), so we can consider the canonical map: that will either be an embedding or a 2-to-1 map to a projective line.
We make the canonical map take its image in a toric variety.
> eqns := Sections(CanonicalLinearSystem(C)); > X<[a]> := ProjectiveSpace(Rationals(),7); > f := map< P -> X | eqns >; > V := f(C); > V; Curve over Rational Field defined by a[1]^3 + a[2]^2*a[4] + a[1]*a[8]^2, a[1]^2*a[3] + a[2]^2*a[6] + a[3]*a[8]^2, a[1]^2*a[5] + a[2]*a[4]*a[6] + a[5]*a[8]^2, a[1]*a[4]*a[6] - a[2]^2*a[7], a[1]*a[6]^2 - a[2]^2*a[8], a[2]*a[6]^2 + a[1]^2*a[7] + a[7]*a[8]^2, a[4]*a[6]^2 + a[1]^2*a[8] + a[8]^3, a[2]*a[3] - a[1]*a[4], a[3]^2 - a[1]*a[5], a[3]*a[4] - a[1]*a[6], a[4]^2 - a[2]*a[6], a[2]*a[5] - a[1]*a[6], a[3]*a[5] - a[1]*a[7], a[4]*a[5] - a[2]*a[7], a[5]^2 - a[1]*a[8], a[3]*a[6] - a[2]*a[7], a[5]*a[6] - a[2]*a[8], a[3]*a[7] - a[1]*a[8], a[4]*a[7] - a[2]*a[8], a[5]*a[7] - a[3]*a[8], a[6]*a[7] - a[4]*a[8], a[7]^2 - a[5]*a[8]All those binomial equations suggest that V lies on a toric variety embedded in X=P7. We can recover this toric variety and its map to X.
> W,g := BinomialToricEmbedding(V);
> Y<[b]> := Domain(g);
> Y;
Toric variety of dimension 2
Variables: b[1], b[2], b[3], b[4]
The components of the irrelevant ideal are:
(b[3], b[2]), (b[4], b[1])
The 2 gradings are:
0, 1, 1, 0,
1, 0, 2, 1
It is a well-known consequence of (geometric) Riemann--Roch that
trigonal curves lie on scrolls in their canonical embeddings.
Exactly which scroll is an intrinsic property of the
particular curve: the Maroni invariant of a trigonal curve
can be realised as the twist that occurs in the scroll, in this
case 2 (visible in the last line of output above).
This makes good sense: the scroll Y has a natural map to P1, and the equation of the curve W is a cubic in the fibre variables b[2], b[3] so defines a 3-to-1 cover of the base.
> I := Saturation(DefiningIdeal(W),IrrelevantIdeal(Y));
> Basis(I);
[
b[1]^8*b[2]^3 + b[1]*b[3]^3*b[4] + b[2]^3*b[4]^8
]
The need for saturation is already visible in the
equations of V: all those cubics are really
multiples of a single cubic on the scroll by irrelevant
ideals, but written in the coordinates of the projective space.