Introduction In this monograph we will study (smooth, projective, connected,) minimal sur- faces defined over the complex numbers with geometric genus pg = 4 and such that the canonical divisor has selfintersection 7. The study of these surfaces has a deep historical interest. In the book Le super fide algebriche (cf. [Enr]) F. Enriques summarized the results obtained in a life long research which took place in an arc of more than fifty years. First the so-called rough classification of surfaces is achieved, where algebraic surfaces defined over the complex numbers are divided into four classes according to their Kodaira dimension —oo, 0, 1, 2. While the first three classes are less wide and nowadays many problems which were still open at Enriques' time are quite well understood, the surfaces of Kodaira dimension 2, the so-called surfaces of general type, are a vast class and many problems considered by Enriques are not yet solved in a satisfactory way. In general the approach proposed by F. Enriques towards the surfaces of general type is the study of their geometry via the behaviour of the pluricanonical maps (or more special, of the canonical map) of a minimal surface S of general type with given selfintersection of the canonical divisor Ks- Before commenting more on these methods we first give some numerical crite- rion for an algebraic surface to be of general type as well as the basic inequalities which these surfaces have to fulfil. For a proof of these results we refer to the wellknown textbook [BPV]. THEOREM 0.2. 1) Let S be a surface with K2 0, then S is either of general type or rational In the first case we have for the second plurigenus P2 = X + K2 = l-q + pg + K2, in the second case 2) If S is a minimal surface, then S is of general type iff Kg 0 and P2 0. Moreover, we have the following theorem of Castelnuovo (for a proof in modern language we refer to [Bom]). THEOREM 0.3. If C2(S) 0, then S is a ruled surface. In particular, if S is a minimal surface of general type, then * i = c i ( S ) 2 l , x ( S ) l , (since 12* = K2 + c2). l

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