A1-homotopy type of A2∖{(0,0)}

Abstract

In this article we prove that any $A^1$-connected smooth k-variety is $A^1$-uniruled for any algebraically closed field k. We establish that if a non-empty open subscheme X of a smooth affine k-scheme is $A^1$-weakly equivalent to $A_k^2\setminus \left\{\right(0,0\left)\right\}$, then $X\cong A_k^2\setminus \left\{\right(0,0\left)\right\}$ as k-varieties for any field k of characteristic 0.