The aim of this monography is the exact description of minimal smooth algebraic surfaces over the complex numbers with the invariants \(K^2 = 7\) und \(p_g = 4\). The interest in this fine classification of algebraic surfaces of general type goes back to F. Enriques, who dedicates a large part of his celebrated book Superficie algebriche to this problem. The cases \(p_g = 4\), \(K^2 \leq 6\) were treated in the past by several authors (among others M. Noether, F. Enriques, E. Horikawa) and it is worthwile to remark that already the case \(K^2 = 6\) is rather complicated and it is up to now not possible to decide whether the moduli space of these surfaces is connected or not.

We will give a very precise description of the smooth surfaces with \(K^2 =7\) und \(p_g =4\) which allows us to prove that the moduli space \(\mathcal{M}_{K^2 = 7, p_g = 4}\) has three irreducible components of respective dimensions \(36\), \(36\) and \(38\).

A very careful study of the deformations of these surfaces makes it possible to show that the two components of dimension \(36\) have nonempty intersection. Unfortunately it is not yet possible to decide whether the component of dimension \(38\) intersects the other two or not.

Therefore the main result will be the following:

Theorem 0.1. 1) The moduli space \(\mathcal{M}_{K^2 = 7, p_g = 4}\) has three irreducible components \(\mathcal{M}_{36}\), \(\mathcal{M}'_{36}\) and \(\mathcal{M}_{38}\), where \(i\) is the dimension of \(\mathcal{M}_i\).

2) \(\mathcal{M}_{36} \cap \mathcal{M}'_{36}\) is non empty. In particular, \(\mathcal{M}_{K^2 = 7, p_g = 4}\) has at most two connected components.

3) \(\mathcal{M}'_{36} \cap \mathcal{M}_{38}\) is empty.