Hirzebruch surfaces and compactifications of \({\mathbb C^2}\) (Q2850553)

\textit{L. Brenton} [Math. Ann. 206, 303--310 (1973; Zbl 0267.32013)] constructed the following example: There exists a compactification \((\mathbb{F}_1,C^{(1)})\) of \(\mathbb{C}^2\) such that {\parindent=6mm\begin{itemize}\item[(1)] \(C^{(1)}=C_{11}\cup C_{12}\), where \(C_{11}\) is a smooth rational curve and \(C_{(12)}\) is a rational curve with a cusp singularity \(\{p\}=C_{11}\cap C_{12}\). In particular, one has the multiplicity \(\text{mult}_p C_{12}=2\). \item[(2)] \((C^2_{11})_{\mathbb{F}_1}=1,\,(C^2_{12})_{\mathbb{F}_1}=8\), and \((C_{11}\cdot C_{12})_{\mathbb{F}_1}=3\). NEWLINENEWLINE\end{itemize}} Using the Abhyankar-Moh-Suzuki theorem, the authors construct new compactifications \((\mathbb{F}_1,C^{(n)}),\,(n\geq 1)\) containing Brenton's construction. Moreover, the authors construct more general examples of the following kind:NEWLINENEWLINEThere exists a curve \(C^+=C_1^+\cup C_2^+\) in the Hirzebruch surface \(M^+=\mathbb{F}_m\,\,(m\geq 0)\) such that: {\parindent=8mm\begin{itemize}\item[(i)] \(M^+-C^+\cong\mathbb{C}^2\). \item[(ii)] \(C_1^+\cong\mathbb{P}^1\) and \(C_2^+\) is a rational curve with one cusp singularity \(p\) with \(\text{mult}_p C_2^+=n\). \item[(iii)] \(C_1^+\cap C_2^+=\{p\}\). \item[(iv)] There exists a fiber \(F_0\) such that \(C_1^+\cap C_2^+\cap F_0=\{p\}\). \item[(v)] \(C_1^+\sim\Sigma_m^0+mF\sim\Sigma_m^{\infty}\) and \(C_2^+\sim n\Sigma_m^0+(nm+1)F\) in \(\text{Pic}\mathbb{F}_m\), where \(\Sigma_m^0\) is the minimal section and \(F\) is a general fiber of \(\mathbb{F}_m\). NEWLINENEWLINE\end{itemize}} Brenton's example is the case where \(m=1\) and \(n=2\). The methods used by the authors clarify the relations between the Abhyankar-Moh-Suzuki's theorem and the Hirzebruch surface \(\mathbb{F}_m\) as a compactification of the affine plane.NEWLINENEWLINEFor the entire collection see [Zbl 1266.14004].