Automorphisms of cubic surfaces without points

We classify finite groups acting by birational transformations of a nontrivial Severi–Brauer surface over a field of characteristic zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field [Formula: see text] of characteristic zero that has no [Formula: see text]-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces.

Automorphism Groups of Compact Complex Surfaces

We study automorphism groups and birational automorphism groups of compact complex surfaces. We show that the automorphism group of such a surface $X$ is always Jordan, and the birational automorphism group is Jordan unless $X$ is birational to a product of an elliptic and a rational curve.

Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces

We study the birational geometry of irreducible holomorphic symplectic varieties arising as varieties of lines of general cubic fourfolds containing a cubic scroll. We compute the ample and moving cones, and exhibit a birational automorphism of infinite order explaining the chamber decomposition of the moving cone.

Canonical stability in terms of singularity index for algebraic threefolds

Throughout the ground field is always supposed to be algebraically closed of characteristic zero. Let X be a smooth projective threefold of general type, denote by ϕm the m-canonical map of X which is nothing but the rational map naturally associated with the complete linear system [mid ]mKX[mid ]. Since, once given such a 3-fold X, ϕm is birational whenever m [Gt ] 0, quite an interesting thing to find is the optimal bound for such an m. This bound is important because it is not only crucial to the classification theory, but also strongly related to other problems. For example, it can be applied to determine the order of the birational automorphism group of X [21, remark in section 1]. To fix the terminology we say that ϕm is stably birational if ϕt is birational onto its image for all t [ges ] m. It is well known that the parallel problem in the surface case was solved by Bombieri [1] and others. In the 3-dimensional case, many authors have studied the problem, in quite different ways. Because, in this paper, we are interested in the results obtained by Hanamura [7], we do not plan to mention more references here. According to 3-dimensional MMP, X has a minimal model which is a normal projective 3-fold with only ℚ-factorial terminal singularities. Though X may have many minimal models, the singularity index (namely the canonical index) of any of its minimal models is uniquely determined by X. Denote by r the canonical index of minimal models of X. When r = 1 we know that ϕ6 is stably birational by virtue of [3, 6, 13 and 14]. When r [ges ] 2, Hanamura proved the following theorem.