On C-fibrations over projective curves

The goal of this paper is to present a modified version (GML) of ML invariant which should take into account rulings over a projective base and allow further stratification of smooth affine rational surfaces. We provide a non-trivial example where GML invariant is computed for a smooth affine rational surface admitting no C-actions. We apply GML invariant to computation of ML invariant of some threefolds.Comment: 21 pages, LaTe

Centralizers of rank one in the first Weyl algebra

Centralizers of rank one in the first Weyl algebra have genus zero. <br

The Freiheitssatz for generic Poisson algebras

We prove the Freiheitssatz for the variety of generic Poisson algebras

Automorphisms and derivations of free Poisson algebras in two variables

AbstractLet P be a free Poisson algebra in two variables over a field of characteristic zero. We prove that the automorphisms of P are tame and that the locally nilpotent derivations of P are triangulable

On the lifting of the Nagata automorphism

It is proved that the Nagata automorphism (Nagata coordinates, respectively) of the polynomial algebra $F[x,y,z]$ over a field $F$ cannot be lifted to a $z$-automorphism ($z$-coordinate, respectively) of the free associative algebra $K$. The proof is based on the following two new results which have their own interests: degree estimate of ${Q*_FF}$ and tameness of the automorphism group ${\text{Aut}_Q(Q*_FF)}$.Comment: 15 page

Holomorphic automorphisms of Danielewski surfaces II -- structure of the overshear group

We apply Nevanlinna theory for algebraic varieties to Danielewski surfaces and investigate their group of holomorphic automorphisms. Our main result states that the overshear group which is known to be dense in the identity component of the holomorphic automorphism group, is a free amalgamated product.Comment: 24 page

Endomorphisms of quantized Weyl algebras

Belov-Kanel and Kontsevich conjectured that the group of automorphisms of the n'th Weyl algebra and the group of polynomial symplectomorphisms of C^2 are canonically isomorphic. We discuss how this conjecture can be approached by means of (second) quantized Weyl algebras at roots of unity