Noether's Inequality for Non-complete Algebraic Surfaces of General type, II

Abstract

Let V be a nonsingular projective surface of Kodaira dimension κ(V)≥0. Let D be a reduced, effective, nonzero divisor on V with only simple normal crossings. In the present article, a pair (V, D) is said to be a minimal logarithmic surface of general type, if, by definition, K_V+D is a numerically effective divisor of self intersection number (K_V+D)²>0 and if K_V+D has positive intersection with every exceptional curve of the first kind on V. Here K_V is the canonical divisor of V. In the case, on the one hand, Sakai [8; Theorem 7.6] proved a Miyaoka—Yau type inequality (\bar{c}₁²):=(K_V+D)²≤ 3\bar{c}₂:=3c₂(V)−3e(D). On the other hand, we can easily obtain (\bar{c}₁²)≥\frac{1}{15}\bar{c}₂−\frac{8}{5} by making use of [8; Theorem 5.5]. In the present article, we shall prove that (\bar{c}₁²)≥\frac{1}{9}\bar{c}₂−2 provided that the rational map Φ_|KV+D| defined by the complete linear system |K_V+D| has a surface as the image of V. Moreover, if the equality holds, then the logarithmic geometric genus \bar{p}_g:=h⁰(V, K_V+D)=½(\bar{c}₁²)+2=3, D is an elliptic curve and V is the canonical resolution in the sense of Horikawa associated with a double covering h: Y→P². In addition, the branch locus B of h is a reduced curve of degree eight and the singular locus Sing B consists of points of multiplicity ≤ 3 except for at most one “simple quadruple point”.