Statement of Main Results

In this paper, a surface will mean a compact complex manifold of dimension 2. We denote by ∣mKs∣ (m ε N) the pluricanonical system on a surface S, and by ΦmKs the associated rational map (the pluricanonical map), assuming that ∣mKs∣ is not empty. A surface S is called of general type if, for a large number m, ΦmKs(S) is a variety of dimension 2 in the projective space PN (N = dim ∣mKs∣). We shall study a certain class of surfaces of general type.

Definition. (1) A minimal surface of general type is called a numerical Campedelli surface if S satisfies the following numerical conditions :

(2) A numerical Campedelli surface is called a Campedelli surface if its fundamental group π1(S) is isomorphic to Z/(2)⊕Z/(2)⊕Z/(2) (cf. [2]).

Then we have the following results.

Theorem A.The tricanonical system ∣3 Ks∣ for a numerical Campedelli surface S is free from base points and fixed components

Remark. Except for numerical Campedelli surfaces, we can enumerate all surfaces for which the tricanonical maps are not birational (see Bombieri [1] and Miyaoka [4]).

Theorem B. For a Campedelli surface S, the universal covering S of S is birational to a complete intersection of type (2, 2, 2, 2) in P6.

Remark. It is an interesting but, in general, a very difficult problem to determine the complex structures on a given underlying differentiate manifold.