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

1989, … of the mini-symposium on algebraic …

Abstract

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

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References (16)

  1. // (K v , A)^-l then h\V , A)^(A 2 }+2-g(A).
  2. Proof. (1) Suppose (K v , A)^Q. Then conditions of Lemma 2.6 with X:=A and L:=A\ A are satisfied. Hence we have c(A\ A }=(A Z }+2(1-h\A, O A (A)))^Q.
  3. This, together with the equality h\V, A)^l + h°(A, O A (A)\ implies the asser- tion (1).
  4. Suppose (K v , ,4)<0. Then H\A, O A (A))^H\A, O A (K V ))=0. Hence we have h°(V, A)^l+h°(A, O A (A))=(A z )+2-g(A) by the Riemann-Roch theorem. The assertion (2) is proved. Now we can prove the following : Theorem 2.8. Assume that A satisfies the condition (*) given at the beginn- ing of § 2 and assume further that V is not a rational surface. Then the follow- ing assertions hold true, where A is replaced by a general member of \A\ if necessary.
  5. Suppose h\V, A)^(A*)+1. Then h\V , A)=(A 2 )+l=3, g(A)=p a (A)^2
  6. and l+p g (V)^h\V 9 A)+TL(Ov)= h\V , K v -A)+g(A). Moreover, 0 lAl : V->P Z is a morphism of degree two.
  7. Suppose h°(V, A)-(A 2 ). Then one of the following cases takes place. (2-1) V is an elliptic ruled surface with general members of \A\ as cross- sections of the P l -fibration. @u\ is a morphism of degree ^3 onto a surface. (See the precise description of V and A in Proposition 2.5). (2-2) We have g(A)=p a (A)=2 and (K v , A)^-l. V is a ruled surface satisfying q(V)=h l (V, A)^2. 0 lAl is a morphism of degree ^3 onto a surface. (2-3) We have g(A)=p a (A)^2, l+p g (W)^hW, f'(A))+X(O w )=h°(W, K w - f'(A))+g(A)-l and (A 2 )=3 or 4. Here f :=id if Bs\A\= and f: W-+V is the blowing-up of a base point P of A if Bs\A\^$. If (A 2 )=3 and \A\ is base point free, then g(A)^3 and 0 iA \ : V->P Z is a morphism of degree three. If (A z )=3 and \A\ has base points, then A\ contains only one base point P and &if<.A->\'. W-*P Z is a morphism of degree two. If (A 2 )=4, then ^(^1)^3 and $\A\ : V-^P Z is a morphism of degree one (resp. or two) onto a quartic surface (resp. or a quardratic surface}.
  8. Suppose V is not a ruled surface. Then h\V, A)^(l/2)(A z )+2. Suppose further h°(V, ^)-(l/2)(A 2 )+2. Then g(A)=p a (A)^(l/2)(A z )+l, l + p g (V)<h\V, A)+-l(O v }=h\V,K v -A) + g(A) + l-(l/2)(A z ), and either 0 lAl : V->P N (N:= h°(V, A)-1) is a birational morphism onto a surface of degree 2(N-1) or 0\ A \ '.
  9. V->P N is a morphism of degree two onto a normal rational surface W of degree N-l in P N . (See Lemma 2.2 for the precise description of W.) Proof. (1) Suppose h°(V, ^)^(^4 2 )+l. Since V is not a rational surface, we have then h\V , A)=(A 2 )+l by Lemmas 2.1 and 2.2. By Lemma 2.4, \A\ is base point free. Hence A is a nonsingular curve and g(A)=p a (A). Since V References
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  15. Sakai, F., Semi-stable curves on algebraic surfaces and logarithmic pluricanonical maps, Math. Ann., 254 (1980), 89-120.
  16. Zhang, D.-Q., Noether's inequality for non-complete algebraic surfaces of general type, II, forthcoming.