Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
Surfaces with $K^2 = 7$ and $p_g = 4$
 
Ingrid C. Bauer University of Gottingen, Gottingen, Germany
Surfaces with K^2 = 7 and p_g = 4
eBook 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
Surfaces with K^2 = 7 and p_g = 4
Click above image for expanded view
Surfaces with $K^2 = 7$ and $p_g = 4$
Ingrid C. Bauer University of Gottingen, Gottingen, Germany
eBook 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.