CHAPTER 1 The canonical system Let S be a minimal (smooth, connected, projective) surface over the complex numbers with K2 7 and pg = 4. REMARK 1.1. 1) S is of general type. 2) (cf. [Deb]) S is a regular surface, i.e. q(S) := H^S, Os) = 0. Moreover, we have the following result: THEOREM 1.2. Let S be a minimal surface with K2 = 1, pg 4. Then the canonical system \Ks\ of S is not composed with a pencil. Proof Follows immediately from [Deb], q.e.d First of all we treat the case, where \Ks\ has a (non-trivial) fixed part. We will show that if \Ks\ has a non-trivial fixed part then either F2 = —2, Ks F 0 or we have Ks F = 1, F2 = - 1 and H F - 2. PROPOSITION 1.3. Let S be a minimal surface with K2 = 7 andpg 4. More- over, Ze£ F be the fixed part of the canonical system \Ks\ of S. Then either (1) F = 0, or (2) Ks-F = 0 and F2 = -2, or (3) Ks-F = 1 andF2 = - 1 . In case (2) as we/Z as in case (3) we Ziave H F = 2. Proof Let |iJ| be the movable part of |i^s-|, in particular \Ks\ = |if | + F. Then we have: (*) 7 = K2S = H2 + H-F + KS-F. By the 2-connectedness of the canonical divisor of 5 (cf.[BPV],VII, prop. (6.1)) we have H F 2 and therefore H2 + Ks F 5. Using the fact that H2 degtp\H\ deg^\H\{S) 4, we immediately see from (*) that Ks F 1. Ks F = 1 implies that H-F = 2 and F 2 = - 1 . Since F^- is numerically effective it follows that the only other possibility is Ks F = 0 and from (*) we see that we have the following two possibilities: (i) # 2 = 5, H -F = 2 (F 2 = -2), (ii) H2 = 4 , HF = 3 (F 2 - - 3 ) , 7
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