Notes on minimal compactifications of the affine plane (Q731386)

The authors consider normal compact complex surfaces \(X\) which contain a closed irreducible analytic curve \(\Gamma\) such that \(X-\Gamma\) is biholomorphic to \(\mathbb{C}^2\). Such \(X\) can be obtained by taking a smooth projective compactification \(\widetilde X\) of \(\mathbb{C}^2\) and contracting suitable all but one irreducible components of \(\widetilde X- \mathbb{C}^2\) to finitely many normal singular points. With the assumption that \(X\) is not smooth and \(X\) has at most log canonical singular points, it is proved that \(X\) is a numerical del Pezzo surface of rank 1 and every singular point of \(X\) is rational. Letting \(\widetilde X-\mathbb{C}^2\) to be a minimal resolution of singularities, the authors classify the possible dual graphs of \(\widetilde X- \mathbb{C}^2\). C. P. Ramanujan has proved that any smooth projective compactification \(\widetilde X\) of \(\mathbb{C}^2\) has the property if \(\widetilde X-\mathbb{C}^2\) is not a linear chain then there is some \((-1)\)-curve in \(\widetilde X- \mathbb{C}^2\) such that after contracting it we still get a normal crossing divisor.