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Surfaces with $K^2 = 7$ and $p_g = 4$
 
Ingrid C. Bauer University of Gottingen, Gottingen, Germany
Front Cover for Surfaces with K^2 = 7 and p_g = 4
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
Front Cover for Surfaces with K^2 = 7 and p_g = 4
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  • Front Cover for Surfaces with K^2 = 7 and p_g = 4
  • Back Cover for Surfaces with K^2 = 7 and p_g = 4
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.

    Readership

    Graduate students and research mathematicians interested in algebraic geometry.

  • Table of Contents
     
     
    • 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.

Readership

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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