Noether's inequality for non-complete algebraic surfaces of general type. II
[For part I see \textit{S. Tsunoda} and the author, ibid. No. 1, 21-38 (1992; see the preceding review).] Let \(V\) be a non-ruled non-singular projective surface. Let \(D\) be a reduced, effective non-zero divisor on \(V\) with only simple normal crossings. The author considers here the case where the open surface \(V - D\) is ``minimal'' and of general type. On the one hand, \textit{F. Sakai} [Math. Ann. 254, 89-120 (1980; Zbl 0437.14017)] proved the Miyaoka-Yau type inequality: \((c_ 1^ 2) : = (K_ V + D)^ 2 \leq 3 c_ 2 : = 3c_ 2 (V) - 3e (D)\). Here the author will give a lower bound of \(c_ 1^ 2\) in terms of \(c_ 2\): \(c_ 1^ 2 \geq 1/9 c_ 2^ 2-2\) under the assumption that the rational map associated with the linear system \(| K_ V + D |\) maps \(V\) to a surface. When the above inequality becomes an equality, then \(p_ g (V - D) = 1/2c_ 1^ 2 + 2 = 3\), \(D\) is a smooth elliptic curve and \(V\) is Horikawa's canonical resolution of a double covering surface over \(\mathbb{P}^ 2\).
scientific article; zbMATH DE number 597470