Normal Gorenstein del Pezzo surfaces with quasi-lines (Q2472506)

A normal projective Gorenstein surface \(X\) over the complex field \(\mathbb C\) is called del Pezzo surface if the anti-canonical divisor \(-K_X\) is ample. Let us assume that Sing(\(X\)) \(\neq\) \(\emptyset\). Let \(\varphi: M \rightarrow X\) be the minimal desingularization of \(X\). An irreducible curve on \(X\) is called a quasi-line if its proper transform on \(M\) is a (\(-1\))-curve. Let \(N_X\) be the number of quasi-lines on \(X\). The aim of the paper under review is to give a complete classification of del Pezzo surfaces \(X\) with quasi-lines and determine the geometric structure of the complement of \(N\) quasi-lines on \(X\), where \(N \leq N_X\), under the hypothesis that \(1 \leq N_X \leq 3\). In addition, a complete list of compactifications \(X\) of \(\mathbb C^2\) with quasi-lines as boundaries is given. One of the key points of the paper is the following result, proved by \textit{L. Brenton} [Math. Ann. 148, No. 2, 117--124 (1980; Zbl 0407.14013)] and \textit{F. Hidaka} and \textit{K. Watanabe} [Tokyo J. Math. 4, No. 2, 319--330 (1981; Zbl 0496.14023)]: If \(X\) contains quasi-lines, then \(M\) is a rational surface and Sing(\(X\)) consists of rational double points.