Dr. Stéphane Vénéreau Post-Doc Assistant, Dep. Math. University of Basel Affine Algebraic Geometry

stephane.venereau(at)unibas.ch Büro 23, 2. UG Telefon +41 (0)61 267 26 94 Fax +41 (0)61 267 26 95 (Sprechstunde nach Vereinbarung)

Mathematisches Institut Universität Basel Rheinsprung 21 CH-4051 Basel Schweiz

Lehre im FS 2011

Research

 
Links: Automorphismes des espaces affines, Seminar Algebra & Topologie.

A few words on my research (more details in my CV in english or french )

I work in Affine Algebraic Geometry and mainly around Polynomial Automorphisms . If you have never seen one, the following should give you an idea of what we are talking about: (x,y)|--->(x,y+x²). Here are some of the problems I deal with: construction of "new" automorphisms, recognition of so called variables i.e. components of automorphisms, generation of the automorphisms group, tame and non-tame automorphisms, group actions, Locally Nilpotent Derivations (LNDs).
The tools I am using are various: commutative algebra, topology (of complex varieties), LNDs (again!), degree functions...
 
I am a member of the workgroup Automorphismes des espaces affines and an organizer of the Seminar Algebra & Topologie.
 
The Keywords and MSC-Classification are below .
 

Publications

Published:

• A parachute for the degree of a polynomial in algebraically independent ones, Mathematische Annalen 349, No. 3, 589-598 2011 ( journal version , arXiv version) .

• Jointly with P.Bonnet, Relations between the leading terms of a polynomial automorphism, Journal of Algebra 322, No. 2, 579-599, 2009 ( journal , arXiv ).

• Jointly with A. van den Essen and Stefan Maubach, The Special Automorphism Group of R[t]/(t^m)[X_1,...,X_n] and Coordinates in a Subring of R[t][X_1,...X_n], Journal of Pure and Applied Algebra 210(1),  141-146, 2007, doi:10.1016/j.jpaa.2006.09.013 ( journal , preprint version).

• New bad lines in R[x,y] and optimization of the Epimorphism Theorem, Journal of Algebra 302, No. 2, 729-749, 2006 ( journal , preprint ).

• Hyperplanes of the Form f_1(x,y)z_1+...+f_k(x,y)z_k+g(x,y) are variables, Bull. Can. Math. 48, No.4, 622-635, 2005 (journal, preprint ).

• Jointly with S. Kaliman and M. Zaidenberg, Simple birational extensions of the polynomial algebra C^[3], Trans. Am. Math. Soc. 356, No.2, 509-555, 2004 (journal , arXiv).

• Jointly with S. Kaliman and M. Zaidenberg, Extensions birationnelles simples de l'anneau de polynomes C^[3], C.R. Acad. Sci. Paris Ser. I Math, 333(4) :319-322, 2001 (journal ).

• jointly with E.Edo, Lenght 2 Variables of A[x,y] and Transfer, Ann. Pol. Math. 76, No.1-2, 67-76, 2001 ( journal, preprint ).

• Is y+x[xz+y(yu+z^2)] an x-variable of C[x][y,z,u]?, (Problem no. 13 in:) Open problems in affine algebraic geometry from the workshop on group actions on rational varieties, CRM 2002 (G. Freudenburg and P. Russell, editors), Contemporary Mathematics 369, AMS, 2005 (preprint ).

Submitted:

• Jointly with S. Lamy, The tame and the wild automorphims of an affine quadric threefold, arXiv:1103.4291v1 .

Unpublished:

• Degree Semigroup and the Abhyankar-Moh-Suzuki Theorem, 11p.

  
   

  PhD Thesis

• Automorphismes et variables de l'anneau de polynômes A[y_1,...,y_n] (in PDF)
   

  Keywords and MSC

  
   
  Keywords: automorphism; variable; Nagata's automorphism; Abhyankar-Sathaye's embedding problem; Vénéreau's polynomial; tame automorphism; wild automorphism; residual variable; Locally Nilpotent Derivation; polynomial ring; affine space; birational extension; affine modification; affine fibration; acyclic variety; seminormality; p-seminormality.
   
  MSC 2010:

  14R Affine geometry
  14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
  14E25 Embeddings
  14R25 Affine fibrations
  13B10 Morphisms
  13B25 Polynomials over commutative rings
  13F45 Seminormal rings
  13P10 Polynomial ideals, Gröbner bases

  
   

  My CV in french or english