This note presents the study of the conjugacy classes of elements of some given finite order in the Cremona group of the plane. In particular, it is shown that the number of conjugacy classes is infinite if is even, or , and that it is equal to (respectively ) if (respectively if ) and to for all remaining odd orders. Some precise representative elements of the classes are given.
We describe the set of points over which a dominant polynomial map is not a local analytic covering. We show that this set is either empty or it is a uniruled hypersurface of degree bounded by .
We investigate an approach of Bass to study the Jacobian Conjecture via the degree of the inverse of a polynomial automorphism over an arbitrary ℚ-algebra.
We study the group of tame automorphisms of a smooth affine -dimensional quadric, which we can view as the underlying variety of . We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in is linearizable, and that satisfies the Tits alternative.