$\mathbb{A}^1$-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 $\mathbb{A}^2_{k} \setminus \left\{(0,0) \right\}$, then $X \cong \mathbb{A}^2_{k} \setminus \left\{(0,0) \right\}$ as $k$-varieties for any field $k$ of characteristic $0$.