Toric varieties as quotients: changing the status quo-tient

I’ve had a nice Saturday afternoon brushing up on something else toric that I half-learned once. Briefly, instead of the local + gluing construction of normal toric varieties from fans, one can construct them holistically as quotients by some reductive group (either a torus, or a product of a torus and a finite group). This global construction has certain benefits: one builds a ‘total coordinate ring’ that plays the same role as the (graded) polynomial ring does for projective space in allowing easy access to closed subvarieties (as homogeneous ideals) and quasicoherent sheaves (as graded modules).

Indeed, the grading here is by the class group, which for (weighted) projective spaces is just the familiar \mathbb{Z}-grading. Notice also that the construction doesn’t require the toric varieties produced to be projective.

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