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Surfaces with $K^2 = 7$ and $p_g = 4$

Ingrid C. Bauer University of Gottingen, Gottingen, Germany
Available Formats:
Electronic ISBN: 978-1-4704-0314-0
Product Code: MEMO/152/721.E
List Price: $52.00 MAA Member Price:$46.80
AMS Member Price: $31.20 Click above image for expanded view Surfaces with$K^2 = 7$and$p_g = 4$Ingrid C. Bauer University of Gottingen, Gottingen, Germany Available Formats:  Electronic ISBN: 978-1-4704-0314-0 Product Code: MEMO/152/721.E  List Price:$52.00 MAA Member Price: $46.80 AMS Member Price:$31.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 1522001; 79 pp
MSC: Primary 14; 32;

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.

Graduate students and research mathematicians interested in algebraic geometry.

• Chapters
• Introduction
• 1. The canonical system
• 2. Some known results
• 3. Surfaces with $K^2 = 7$, $p_g = 4$, such that the canonical system doesn’t have a fixed part
• 4. $|K|$ has a (non trivial) fixed part
• 5. The moduli space
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Volume: 1522001; 79 pp
MSC: Primary 14; 32;

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.

• 3. Surfaces with $K^2 = 7$, $p_g = 4$, such that the canonical system doesn’t have a fixed part
• 4. $|K|$ has a (non trivial) fixed part