A note on Sakai's theorem concerning polarized normal surfaces (Q1401611)

Let \(Y\) be a normal projective surface and \(Y\) an ample Cartier divisor on \(X\). The author generalizes earlier classification results of such polarized normal surfaces in terms of the logarithmic Kodaira dimension to the case of algebraically closed fields of arbitrary characteristic. More precisely, \textit{F. Sakai} [J. Reine Angew. Math. 366, 121--128 (1986; Zbl 0582.14011)] completed this classification in the case of characteristic zero. While most of his arguments are independent of the characteristic, a vanishing theorem used there does not hold if \(\text{char}(k)>0\), and the classification theory of normal surfaces in characteristic zero is used. In the paper under review, the author avoids this problems by using a result of \textit{H. Terakawa} [Pac. J. Math. 187, No.1, 187--199 (1999; Zbl 0967.14008)] and himself [Arch. Math. 77, No.6, 517--521 (2001; Zbl 1063.14046)].