# Surfaces with \(K^{2} = 7\) and \(p_{g} = 4\)

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*Ingrid C. Bauer*

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:

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.

#### Table of Contents

# Table of Contents

## Surfaces with $K^{2} = 7$ and $p_{g} = 4$

- Contents vii8 free
- Introduction 110 free
- Chapter 1. The canonical system 716 free
- Chapter 2. Some known results 1019
- Chapter 3. Surfaces with K[sup(2)] = 7, p[sub(g)] = 4, such that the canonical system doesn't have a fixed part 1423
- Chapter 4. |K| has a (non trivial) fixed part 4352
- Chapter 5. The moduli space 5160
- Bibliography 7887