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 $G$.
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.
Let G = group(L) and let H be the subgroup generated by the pseudo-reflections contained in G. 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 group H[G,G], where [G,G] is the derived subgroup of G. 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).