LAUSR

We show that a quotient of a non-trivial Severi–Brauer surface S over arbitrary field k of characteristic 0 by a finite group G ⊂ Aut(S) is k-rational, if and only if |G| is divisible by 3. Otherwise, the quotient is birationally equivalent to S. Let k be an arbitrary field of characteristic zero, and k be its algebraic closure. A d-dimensional variety X is called a Severi–Brauer variety if X = X ⊗ k is isomorphic to P k . If d = 1 then X is a conic, and if d = 2 then X is called a Severi–Brauer surface. A Severi–Brauer variety is called non-trivial if it is not isomorphic to P k . It is well known that a Severi–Brauer variety is trivial if and only if it has a k-point. Moreover, if X is a non-trivial d-dimensional Severi–Brauer variety and d + 1 is a prime number, then degree of any point on X is divisible by d+ 1 (see [Kol16, Theorem 53]). Complete classification of finite subgroups of automorphism groups of non-trivial Severi–Brauer surfaces is obtained in works of C. Shramov and V.Vologodsky [ShV20], [Sh20a], [Sh20b], [Sh21]. Let μn be a cyclic group of order n. Then we have the following. Theorem 1 (cf. [Sh21, Theorem 1.3(ii)]). Let S be a non-trivial Severi–Brauer surface over a field k of characteristic zero. Then any finite subgroup of Aut(S) is isomorphic to μn, μ3n, μn⋊μ3 or μ3×(μn ⋊ μ3), where n is a positive integer divisible only by primes congruent to 1 modulo 3 (including the case n = 1), and μn ⋊ μ3 is a semidirect product corresponding to an outer automorphism of μn of order 3 acting non-trivially on each non-trivial element of μn. Moreover, for any group G mentioned above there exists a field k of charectiristic zero and a non-trivial Severi–Brauer surface S over k, such that G ⊂ Aut(S). The aim of this paper is to obtain a birational classification of quotients of non-trivial Severi–Brauer surfaces by finite subgroups of automorphism groups. In particular, we want to answer a question, for which Severi–Brauer surfaces S and finite subgroups G ⊂ Aut(S) the quotient S/G is k-rational (i.e. birationally equivalent to P k ). Note that for an algebraically closed field k of characteristic zero quotients of k-rational surfaces by finite automorphism groups are always k-rational by Castelnuovo’s rationality criterion (see [Cast94]). Moreover, for arbitrary field k of characteristic zero quotients of P k by finite automorphism groups are always k-rational (see [Tr14, Theorem 1.3]). One can find more results about rationality of quotients of k-rational surfaces by finite automorphism groups in [Tr19]. The main result of this paper is the following theorem. Theorem 2. Let S be a non-trivial Severi–Brauer surface over a field k of characteristic zero, and G be a finite subgroup of Aut(S). Then the quotient S/G is k-rational, if and only if |G| is divisible by 3. Otherwise, the quotient S/G is birationally equivalent to S. This theorem is an analogue of the following proposition.