Surfaces with $K^2 = 7$ and $p_g = 4$ (eBook)

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.