Noether's inequality for non-complete algebraic surfaces of general type
Let \(V\) be a nonsingular projective surface. M. Noether's inequality says that \(\dim H^ 0 (V,K_ V) \leq 1/2 (K_ V^ 2) + 2\) provided that \(V\) is a minimal surface of general type. Let \(D\) be a reduced effective divisor on \(V\) with only simple normal crossings. In the paper, the authors extend Noether's inequality to the case of open algebraic surfaces \(V - D\) of general type. In this case, there is a Zariski decomposition for the adjoint divisor: \(K_ V + D = P + N\). The authors prove that \(\dim H^ 0 (V,K_ V + D) \leq 9/8 (P^ 2) + 2\) and several other inequalities. The pairs \((V,D)\) such that the inequality above becomes an equality are precisely described. The case that \(D\) is semistable has been treated by \textit{F. Sakai} [Math. Ann. 254, 89-120 (1980; Zbl 0437.14017)].
scientific article; zbMATH DE number 548920
scientific article