# On the Zariski Cancellation Problem

Seminar
Speaker
Prof. Mikhail Zaidenberg, Fourier Institute, Grenoble, France
Date
28/05/2015 - 15:00 - 14:00Add to Calendar 2015-05-28 14:00:00 2015-05-28 15:00:00 On the Zariski Cancellation Problem Given complex affine algebraic varieties $X$ and $Y$, the general Zariski Cancellation Problem asks whether the existence  of an isomorphism $X\times\mathbb{C}^n\cong Y\times\mathbb{C}^n$ implies that $X\cong Y$. Or, in other words, whether varieties with isomorphic cylinders should be isomorphic. This occurs to be true for affine  curves (Abhyankar, Eakin, and Heinzer $'72$)  and false for affine surfaces (Danielewski $'89$). The special Zariski Cancellation Problem asks the same question provided that $Y=\mathbb{C}^k$. In this case, the answer  is "yes" in dimension $k=2$ (Miyanishi-Sugie $'80$ and Fujita $'79$), and unknown in higher dimensions, where the situation  occurs to be quite mysterious (indeed, over a field of positive characteristic, there is a recent counter-example due to Neena Gupta $'14$). The birational counterpart of the special Zariski Cancellation Problem asks whether stable rationality implies rationality.  The answer  occurs to be negative; the first counter-example was constructed by Beauville, Colliot-Th\'el\ene, Sansuc, and Swinnerton-Dyer $'85$.  We will survey on the subject, both on some classical results and on a very recent development, reporting in particular on a joint  work with Hubert Flenner and Shulim Kaliman. 2nd floor Colloquium Room, Building 216 אוניברסיטת בר-אילן - המחלקה למתמטיקה mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
2nd floor Colloquium Room, Building 216
Abstract
Given complex affine algebraic varieties $X$ and $Y$, the general Zariski Cancellation Problem asks whether the existence
of an isomorphism $X\times\mathbb{C}^n\cong Y\times\mathbb{C}^n$ implies that $X\cong Y$.
Or, in other words, whether varieties with isomorphic cylinders should be isomorphic. This occurs to be true for affine
curves (Abhyankar, Eakin, and Heinzer $'72$)  and false for affine surfaces (Danielewski $'89$).
The special Zariski Cancellation Problem asks the same question provided that $Y=\mathbb{C}^k$. In this case, the answer
is "yes" in dimension $k=2$ (Miyanishi-Sugie $'80$ and Fujita $'79$), and unknown in higher dimensions, where the situation
occurs to be quite mysterious (indeed, over a field of positive characteristic, there is a recent counter-example due to Neena Gupta $'14$).
The birational counterpart of the special Zariski Cancellation Problem asks whether stable rationality implies rationality.  The answer
occurs to be negative; the first counter-example was constructed by Beauville, Colliot-Th\'el\ene, Sansuc, and Swinnerton-Dyer $'85$.
We will survey on the subject, both on some classical results and on a very recent development, reporting in particular on a joint
work with Hubert Flenner and Shulim Kaliman.

תאריך עדכון אחרון : 25/05/2015