New bad lines in R[x, y] and optimization of the Epimorphism Theorem
by
Stéphane Vénéreau
University of Basel, Switzerland

This work is about one of the main problem in Affine Algebraic Geometry: the Abhankar-Sathaye (Embedding) Problem. Here I study the special case (in terms of dimension and codimension) of so called "lines" in R[x, y], that is, polynomials f such that R[x, y]/(f) is isomorphic to R[z]. The question is: are lines variables (= coordinates)? Quite a lot is known about this problem:
1. If R is a field a positive char. there exists bad lines (lines that are not variables) found by Nagata;
2. If R is a field of char. 0 then lines are variables; this is the Abhankar-Moh-Suzuki theorem.
Part 2. was generalized by Russell-Sathaye and by Bhatwadekar and part 1 has also some easy generalizations to rings of positive char. or having integers dividing 0. The main thing here is the quite unexpected discovery of new bad lines for rings R which are not of this latter type. This allows me to answer completely the question, that is to say, which rings are "good" and which are "bad" in this context.

Date received: March 31, 2005