Non-complete rationalT-varieties of complexity one

Abstract

We consider rational varieties with a torus action of complexity one and extend the combinatorial approach via the Cox ring developed for the complete case in earlier work to the non‐complete, e.g. affine, case. This includes in particular a description of all factorially graded affine algebras of complexity one with only constant homogeneous invertible elements in terms of canonical generators and relations.

Divisor class groups of rational trinomial varieties

“…We work in the notation of [HH13,HW17], where the Cox rings of rational Tvarieties of complexity one are described. Note that the trinomial varieties defined in the introduction arise as the spectrum of these rings.…”

On Torus Actions of Higher Complexity

“…Beyond toric geometry, there are the T-varieties X of higher complexity, where one requires X to be normal, the T-action to be effective, meaning that only the neutral element acts trivially, and the complexity is the difference dim(X)−dim(T). Torus actions of complexity one are studied as well since the 1970s; we mention the combinatorial approaches [48,62], the geometric work [29][30][31][55][56][57][58] on K * -surfaces and the more algebraic point of view based on trinomial relations [38,39,44,53]. In arbitrary complexity, we have the general approach via polyhedral divisors [1,2], unifying in particular aspects of [31,62].…”

“…Let us say a few words about what kind of T-varieties we obtain. First, Construction 3.5 generalizes the Cox ring based approach to rational T-varieties of complexity one developed in [38,44]; see Remark 5.14. For a general statement, recall the Mori dream spaces introduced by Hu and Keel [45]: these varieties behave perfectly with respect to the minimal model programme and are characterized as the Q-factorial, projective varieties with finitely generated Cox ring.…”

On homogeneous locally nilpotent derivations of trinomial algebras

“…[18,Chapter 4]. At the same time, Cox rings establish a close relation between torus actions of complexity one and trinomials, see [10,9,8,2,7]. In particular, any trinomial hypersurface admits a torus action of complexity one.…”