Schemes in Toric Varieties#
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.
Construction of Subschemes#
- Scheme(X, f): TorVar, RngMPolElt -> Sch#
The subscheme of the toric variety \(X\) defined by the polynomial \(f\) from the Cox ring of \(X\).
- Scheme(X, Q): TorVar, [RngMPolElt] -> Sch#
The subscheme of the toric variety \(X\) defined by the sequence \(Q\) of polynomials from the Cox ring of \(X\).
- BinomialToricEmbedding(Z): Sch -> Sch, TorMap#
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 \(Z\).
- Example: Toric Mmp Example1 (ex-641627)#
Toric varieties are the natural ambient space for many varieties. Here we review the example of a trigonal curve from the Schemes chapter (it is self-contained here).
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); 8
This 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 \(X={\mathbb{P}}^7\). 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 \({\mathbb{P}}^1\), 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.