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.
Readership
Graduate students and research mathematicians interested in
algebraic geometry.