3.00 - 3.50 & 4.00 - 4.50
On Zariski's Cancellation Problem
Neena Gupta
ISI, Kolkata.
25-09-13

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Abstract

Let $k$ be an algebraically closed field. The Zariski Cancellation Problem asks whether the affine space $\A^n_k$ is cancellative, i.e., for any affine variety $\V$ whether $\V \times \A^1_k \cong \A^{n+1}_k$ implies $\V \cong \A_k^n$; equivalently, whether, for any ring $A$, $A[T]\cong k[X_1, \dots, X_{n+1}]$, implies $A\cong k[X_1, \dots, X_n]$. It was known that $\A^n_k$ is cancellative for $n=1$ (Abhyankar-Eakin-Heinzer 1972) and $n=2$ (Fujita 1979, Miyanishi-Sugie 1980 for ch $k=0$ and Russell 1981 for ch $k >0$). Recently, I have shown that the affine 3-space $\A^3_k$ is not cancellative in positive characteristic.

In this talk we shall give a brief survey of problems in Affine Algebraic Geometry related to the above Cancellation Problem and discuss the recent developments.