Coordinate ring of `ToricVariety`

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Gareth Ma

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Jan 23, 2024, 1:29:17 PMJan 23
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Hi all,

I am working with `ToricVariety` within Sage, and I have a question. Say I define a toric variety as follows:

sage: PP = toric_varieties.WP([1, 3, 1]); PP
2-d toric variety covered by 3 affine patches
sage: PP.linear_equivalence_ideal()
Ideal (z0 - z2, z1 - 3*z2) of Multivariate Polynomial Ring in z0, z1, z2 over Rational Field

(The repr output of toric varieties is too simplified, but that's another issue.) As seen in the second output, the "3" is definitely "somewhere within the object". Thus I would expect the polynomial ring / coordinate ring to also be a weighted one. However, that's not the case:

sage: PP.coordinate_ring().term_order()
Degree reverse lexicographic term order
sage: [g.degree() for g in PP.coordinate_ring().gens()]
[1, 1, 1]

Is this intended or should it return a weighted polynomial ring, something like this?

sage: to = TermOrder("wdegrevlex", [1, 3, 1])
sage: R = PolynomialRing(QQ, 3, names="z", order=to)
sage: [g.degree() for g in R.gens()]
[1, 3, 1]

Dima Pasechnik

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Jan 25, 2024, 12:06:37 PMJan 25
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Yes, this seems to be a bug. The Cox ring of a weighted projective
space must be weighted.
Here is what Macaulay2 says about your example:

$ M2
Macaulay2, version 1.22.0.1
with packages: ConwayPolynomials, Elimination, IntegralClosure,
InverseSystems, Isomorphism, LLLBases, MinimalPrimes, OnlineLookup,
PrimaryDecomposition, ReesAlgebra, Saturation, TangentCone

i1 : loadPackage("NormalToricVarieties");

i2 : PP = weightedProjectiveSpace {1,3,1}

o2 = PP

o2 : NormalToricVariety

i3 : S = ring PP

o3 = S

o3 : PolynomialRing

i4 : degrees S

o4 = {{1}, {3}, {1}}

I've opened https://github.com/sagemath/sage/issues/37167 to track this

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