${\mathbb{A}}^1$-homotopy type of ${\mathbb{A}}^2\setminus \left\{(0,0)\right\}$

Abstract

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