Factorially graded rings of complexity one

Abstract

Abstract. We consider finitely generated normal algebras over an algebraically closed field of characteristic zero that come with a complexity one grading by a finitely generated abelian group such that the conditions of a UFD are satisfied for homogeneous elements. Our main results describe these algebras in terms of generators and relations. We apply this to write down explicitly the possible Cox rings of normal complete rational varieties with a complexity one torus action. Statement of the resultsThe subject of this note are finitely generated normal algebras R = ⊕ K R w over some algebraically closed field K of characteristic zero graded by a finitely generated abelian group K. We are interested in the following homogeneous version of a unique factorization domain: R is called factorially (K-)graded if every homogeneous nonzero nonunit is a product of K-primes, where a K-prime element is a homogeneous nonzero nonunit f ∈ R with the property that whenever f divides a product of homogeneous elements, then it divides one of the factors. For free K, the properties factorial and factorially graded are equivalent [3], but for a K with torsion the latter is more general. Our motivation to study factorially graded algebras is that the Cox rings of algebraic varieties are of this type, see for example [2].We focus on effective K-gradings of complexity one, i.e., the w ∈ K with R w = 0 generate K and K is of rank dim(R) − 1. Moreover, we suppose that the grading is pointed in the sense that R 0 = K holds. The case of a free grading group K and hence factorial R was treated in [5, Section 1]. Here we settle the more general case of factorial gradings allowing torsion. Our results enable us to write down explicitly the possible Cox rings of normal complete rational varieties with a complexity one torus action. This complements [6], where the Cox ring of a given variety was computed in terms of the torus action.In order to state our results, let us fix the notation. For r ≥ 1, let A = (a 0 , . . . , a r ) be a sequence of vectors a i = (b i , c i ) in K 2 such that any pair (a i , a k ) with k = i is linearly independent, n = (n 0 , . . . , n r ) a sequence of positive integers and L = (l ij ) a family of positive integers, where 0 ≤ i ≤ r and 1 ≤ j ≤ n i . For every 0 ≤ i ≤ r, define a monomial

The automorphism group of a rigid affine variety

“…By [7], the Cox ring of a toric variety is a polynomial algebra. In turn, it is shown in [14], [15], [16], [17] that the Cox ring of a rational variety with a torus action of complexity one is a factor ring of a polynomial ring by an ideal generated by trinomials. The study of homogeneous locally nilpotent derivations on such rings lead to a description of the automorphism group of a complete rational variety with a complexity one torus action [3].…”

Non-complete rationalT-varieties of complexity one

“…The first results in this direction concern the case R 0 = K in dimension two, see [14], [17]. The case R 0 = K in arbitrary dimension was settled in [10] and occurs as part of Type 2 in our subsequent considerations. A simple example of Type 1 presented below is the coordinate algebra of the special linear group SL (2):…”

“…The purpose of this article is to extend the toolkit developed in [10], [11], [12] for normal complete rational varieties with a torus action of complexity one also to non-complete varieties, for example affine ones. Recall that the complexity of a variety X with an effective action of an algebraic torus T equals dim(X ) − dim(T ).…”

Homogeneous Locally Nilpotent Derivations of Nonfactorial Trinomial Algebras

“…[16,Chapter 4]. At the same time, Cox rings establish a close relation between torus actions of complexity one and trinomials, see [11,10,9,3,8]. In particular, any trinomial hypersurface admits a torus action of complexity one.…”