Rank one log del Pezzo surfaces of index two (Q1890271)

The author considers normal projective surfaces \(X\) with \(-K_X\) ample and at worst quotient singularities. Such \(X\) is called a log del Pezzo surface. \textit{M. Miyanishi} and the reviewer [J. Algebra 118, 63--84 (1988; Zbl 0664.14019); ibid. 156, 183--193 (1993; Zbl 0801.14013)] classified del Pezzo surfaces of Cartier index 1 (i.e., Gorenstein) and Picard number 1 or 2 (and rel. minimal). \textit{V. A. Alekseev} and \textit{V. V. Nikulin} [Sov. Math., Dokl. 39, 507--511 (1989; Zbl 0705.14038)] announced the classification of del Pezzo surfaces \(Y\) of Cartier index \(\leq 2\) and Picard number 1. In the current paper, the author classifies these \(Y\) (with proof) in a different method (there are exactly \(18\) singularity types). He also calculates the fundamental group \(\pi_1(Y^0)\) for the smooth part \(Y^0\) of \(Y\). It turns out this group has order \(\leq 8\), and it is trivial if and only if it contains the affine plane. In general, a del Pezzo surface \(X\) has finite \(\pi_1(X^0)\) as proved by \textit{R. V. Gurjar} and the reviewer [J. Math. Sci., Tokyo 1, 137--180 (1994; Zbl 0841.14017); ibid. 2, 165--196 (1995; Zbl 0847.14021)]. Related work: {Isomorphism} classes of Gorenstein del Pezzo surfaces of Picard number \(\leq 2\) have been classified by \textit{Q. Ye} [Jap. J. Math. 28, 87--136 (2002; Zbl 1053.14044)].