Cox rings

Cox rings of linear quotients

By a theorem of Arzhantsev and Gaifullin [AG10], the Cox ring of a linear quotient $V/G$ is graded isomorphic to the invariant ring $K[V]^{[G,G]}$, where $[G,G]$ is the derived subgroup of $G$.

cox_ringMethod
cox_ring(L::LinearQuotient)

Return the Cox ring of the linear quotient L in a presentation as a graded affine algebra (MPolyQuoRing) and an injective map from this ring into a polynomial ring.

Let G = group(L) and let H be the subgroup generated by the pseudo-reflections contained in G. By a theorem of Arzhantsev–Gaifullin [AG10], the Cox ring is graded isomorphic to the invariant ring of the group H[G,G], where [G,G] is the derived subgroup of G. We use ideas from [DK17] to find homogeneous generators of the invariant ring. To get a map from group(G) to the grading group of the returned ring, use class_group.

Experimental

This function is part of the experimental code in Oscar. Please read here for more details.

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Cox rings of $\mathbb Q$-factorial terminalizations

We provide an experimental algorithm to compute the Cox ring of a $\mathbb Q$-factorial terminalization $X\to V/G$ of a linear quotient due to [Yam18].

cox_ring_of_qq_factorial_terminalizationMethod
cox_ring_of_qq_factorial_terminalization(L::LinearQuotient)

Return the Cox ring of a QQ-factorial terminalization of the linear quotient L in a presentation as a graded affine algebra (MPolyQuoRing) and an injective map from this ring into a Laurent polynomial ring using the algorithm from [Yam18].

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