On the orbifold Euler characteristic of log del Pezzo surfaces of rank one (Q2877788)

A normal projective surface \(S/\mathbb{C}\) with at most quotient singular points is a log del Pezzo surface if \(-K_S\) is ample. If \(\text{Pic\,}S\) has rank 1 then \(S\) is said to have rank 1. There is a lot of work by many algebraic geometers about these surfaces. This paper deals with such surfaces.NEWLINENEWLINE The rational number \(X_{\text{top}}(S)-\sum(1-{1\over|G_p|})\), where the summation is over the singular points \(p\) of \(S\) with local fundamental group \(G_p\), denoted by \(e(S)\), is called the orbifold Euler characteristic of \(S\). It is easy to see that \(e(S)\leq 3\). \textit{S. Keel} and \textit{J. McKernan} [Mem. Am. Math. Soc. 669, 153 p. (1999; Zbl 0955.14031)] proved that \(0\leq e(S)\). In this paper the author has proved a useful improvement, viz. \(0<e(S)\). The proof uses the results of D. Q. Zhang, Belousov, Kojima-Takahashi, Hwang-Keum. Examples are given to show that this result is best possible.