Cox rings
Cox rings of linear quotients
By a theorem of Arzhantsev and Gaifullin Ivan V. Arzhantsev, Sergei A. Gaǐfullin (2010), 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 $G$. This functionality is so far only available if the group does not contain any reflections.
cox_ring — Methodcox_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.
By a theorem of Arzhantsev–Gaifullin Ivan V. Arzhantsev, Sergei A. Gaǐfullin (2010) the Cox ring is graded isomorphic to the invariant ring of the derived subgroup of group(L). We use ideas from Maria Donten-Bury, Simon Keicher (2017) 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.
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 Ryo Yamagishi (2018).
cox_ring_of_qq_factorial_terminalization — Methodcox_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 Ryo Yamagishi (2018).