Noether's Inequality for Non-complete Algebraic Surfaces of General Type

Abstract

Let V be a nonsingular projective surface of Kodaira dimension K(V) > 0. Let D be a reduced, effective, nonzero divisor on V with only normal crossings. In the present article, a pair (F, D) is said to be a minimal logarithmic surface of general type, if, by definition, Kv + D is a numerically effective divisor of self intersection number (Kv + D) 2 > 0 and if Kv + D has positive intersection with every exceptional curve of the first kind on V. Here Kv is the canonical divisor of V. In the case, on the one hand, Sakai [8; Theorem 7.6] proved a Miyaoka — Yau type inequality (cf ) := (Kv + D) 2 — c2 -by making use of [8; Theorem 5.5]. In the present article, we shall prove that (cf ) > -c2 — 2 provided that the rational map &\Ky+D defined by the complete linear system \KV + D| has a surface as the image of V. Moreover, if the equality holds, then the logarithmic geometric genus pg := h°(V, Kv + D) = -(cf ) + 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 -> P. 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. Introduction This is a succession of the previous paper [9]. We work over an algebraically closed field fe of characteristic zero. Let V be a nonsingular projective surface defined over fe. If V is a minimal surface of general type, we have the following inequality due to M. Noether: This inequality, together with the Noether formula 12%((9V) = c±(V) 2 + c2(V Communicated by K. Saito, May 18, 1990. 1991 Mathematics Subject Classification: 14J29 Department of Mathematics, the National University of Singapore, Singapore.