CHAPTER 2 Some known results In this section we will recall a couple of results on surfaces with K2 7, pg = 4, which have already been established in the past. We do not want to enter in the details of proof and refer to the original papers. The cases (0) and (1.3) of theorem (1.6) have already been settled by F. Catanese in [Catl] and by F. Zucconi in [Zuc] respectively. They obtain the following results: THEOREM 2.1. (F. Catanese) Surfaces withpg = 4 and K2 = 7 such that \Ks\ is without base points form an irreducible, unirational open set of their moduli space 9RX=5,K2=7 of dimension 36. THEOREM 2.2. (F. Zucconi) The surfaces with K2 = 7 and pg = 4 such that the canonical map has degree three form an irreducible, unirational set of dimension 35 in the moduli space of surfaces with K2 = 7 and pg = 4. In the remaining part of this section we will discuss an example of F. Enriques. In his famous book v Superficie algebriche", chapter 8, ([Enr]), he gives ideas and concrete examples how to classify smooth algebraic surfaces with K2 = 7 and pg 4. Unfortunately, there are also some mistakes. Among other things he gives the following construction: EXAMPLE 2.3. (F. Enriques) We consider two different planes a, 7 of P3. Let C C 7 = P2 be a (reduced) conic. Furthermore, let E C P3 be a surface of degree six having as only singularities (apart from rational double points) C as double curve and having an (ordinary) tacnode o(£ C) contained in the plane 7, whose tacnodal plane is a. We recall that an ordinary tacnode is given by the following analytic equation: z2 = xy(x + y)(x-\y). Before showing that E is the canonical image of a minimal surface S with K2 7 and pg = 4 such that the canonical system \Ks\ of S has exactly one (simple) base point, i.e. S is of type (LI) (cf. prop. (1.5)), we give another description of the hypersurfaces E C P3 in Enriques' example (2.3). First we give some definition generalizing the notion of a tacnode: A generalized tacnode is a two dimensional elliptic hypersurface singularity (X, 0), such that the fundamental cycle has self intersection (—2). In particular, (X, 0) is Gorenstein and by [Lau], theorem (1.3), (X, 0) is a double point singularity, hence the local analytic equation is given by z2 = f(x,y), where / vanishes of order four in 0. I.e., a generalized tacnode (X, 0) is the singularity obtained as the double cover 10
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