Noether's inequality for non-complete algebraic surfaces of general type, the part 2 (original) (raw)

1989, … of the mini-symposium on algebraic …

Let V be a nonsingular projective surface. M. Noether proved that dim H°(V, K v } where K v is the canonical divisor of V, provided V is a minimal surface of general type. Let D be a reduced, effective divisor on V with only simple normal crossings. An open surface V-D is said to be of general type if the Kodaira dimension K (V, Kv + D}=2. In this case, K V +D has the Zariski decomposition and we denote by P, which is a Q-divisor, the numerically effective part of the decomposition. We have (P 2) = (c 1 (y) 2) if D=Q and if V is a minimal surface of general type. In the present article, we shall verify that dim H\V, #,/ + £)^9/8(-P 2)+2 and several other inequalities. Such pairs (V, D) that the above inequality becomes an equality are precisely described. The case that D is semi-stable has been treated by Sakai [Math. Ann. 254, 89-120 (1980)]. We intend, in the present article, to extend this inequality to a logarithmic surface of general type, which is to be defined below. Let V be a nonsingular projective surface defined over k and let D be a reduced effective divisor on V with only simple normal crossings. Denote by K v the canonical divisor of V. Let D-^Di be the decomposition into irreducible components. We call a pair (V, D} a logarithmic surface (log surface, for short). It is called minimal if K V +D has a decomposition into a sum of Qdivisors: K v +D=(K v 4-^a i D l)+^l(l-a i)D l , where