Embeddings of a family of Danielewski hypersurfaces
by
Lucy Moser-Jauslin
IMB, Université de Bourgogne, Dijon, France
Coauthors: P.-M. Poloni

This talk describes recent work in collaboration with P.-M. Poloni on embeddings of surfaces in complex three-dimensional space. We consider polynomials P_r in C[x, y, z] of the form P_r=x²y-z²+xr(z) where r(z) is a polynomial of one variable. We classify these polynomials up to equivalence. Two polynomials P and Q in R=C[x, y, z] are said to be (algebraically) equivalent if there is an algebraic automorphism f of R for which f(P)=Q. If there is a complex number c in C such that the zero set V(P-c) is not isomorphic to the zero set V(Q-c), then the two polynomials P and Q are not equivalent. We show, however, that the converse is not true. That is, there are examples of polynomials P and Q which are not equivalent, however the zero sets V(P-c) and V(Q-c) are isomorphic for all c. Moreover, we find families of polynomials which are analytically equivalent and also stably equivalent but not algebraically equivalent. We say that P and Q are analytically equivalent if there is an analytic isomorphism f of R for which f(P)=Q. We say they are stably equivalent if, when viewed as polynomials of 4 variables, they become equivalent.

These results use a theorem of L. Makar-Limanov describing the automorphism groups of hypersurfaces defined by an equation of the form x²y=p(z), where p is a polynomial of degree at least 2. The methods used are related to the techniques developed in an article by G. Freudenburg and L. Moser-Jauslin on embeddings of Danielewski surfaces.

Date received: March 30, 2005