Log terminal singularities, platonic tuples and iteration of Cox rings

Abstract

Looking at the well understood case of log terminal surface singularities, one observes that each of them is the quotient of a factorial one by a finite solvable group. The derived series of this group reflects an iteration of Cox rings of surface singularities. We extend this picture to log terminal singularities in any dimension coming with a torus action of complexity one. In this setting, the previously finite groups become solvable torus extensions.As explicit examples, we investigate compound du Val threefold singularities. We give a complete classification and exhibit all the possible chains of iterated Cox rings.Theorem 2. Let X 1 be a rational, normal, affine variety with a torus action of complexity one of Type 2 and at most log terminal singularities. Then there is a unique chain of quotients

Polynomial curves on trinomial hypersurfaces

“…So one may try to construct a horizontal polynomial curve on X and then to project it to a horizontal polynomial curve on X via the quotient morphism X → X. The difficulty with this approach is that the total coordinate space need not be rational, see [3,Example 5.12] and the following example.…”

“…Let us recall that a triple of positive integers (p, q, r) is called platonic if we have 1/p + 1/q + 1/r > 1. It is well known that the platonic triples up to renumbering are the following ones (5,3,2), (4,3,2), (3,3,2), (p, 2, 2), (p, q, 1), p, q ∈ Z >0 .…”

Divisor class groups of rational trinomial varieties

“…Note that due to [ABHW18,Cor. 5.8] one can easily decide if a given trinomial variety is rational or factorial just in terms of the numbers l i , see also Remark 2.2.…”

“…We call this triple the basic platonic triple of X. Note that these varieties comprise all total coordinate spaces of affine log terminal varieties of complexity one; see [ABHW18] for the precise statement. Due to [HW18, Thm.…”

Polyhedral Divisors of Affine Trinomial Hypersurfaces

“…[4, Theorem 5.3] Let Y be the curve in P(d 12 , d 02 , d 01 ) given by the equationwd The genus of this curve is equal tog = d 2 d − (d 01 + d 02 + d 12 ) + 1.Corollary 5.2. [4, Lemma 5.6] Let Y be the curve from Theorem 3.2.…”