Surfaces with K 2 7 and Pg 4
2001
American Mathematical Society (Verlag)
9780821826898 (ISBN)
American Mathematical Society (Verlag)
9780821826898 (ISBN)
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Intends to give the exact description of minimal smooth algebraic surfaces over the complex numbers with the invariants $K^2 = 7$ und $p_g = 4$. This title offers a very precise description of the smooth surfaces with $K^2 =7$ und $p_g =4$.
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. - 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 $/mathcal{M}_i$.; $/mathcal{M}_{36} /cap /mathcal{M}'_{36}$ is non empty. In particular, $/mathcal{M}_{K^2 = 7, p_g = 4}$ has at most two connected components; and $/mathcal{M}'_{36} /cap /mathcal{M}_{38}$ is empty.
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. - 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 $/mathcal{M}_i$.; $/mathcal{M}_{36} /cap /mathcal{M}'_{36}$ is non empty. In particular, $/mathcal{M}_{K^2 = 7, p_g = 4}$ has at most two connected components; and $/mathcal{M}'_{36} /cap /mathcal{M}_{38}$ is empty.
Introduction The canonical system Some known results Surfaces with $K^2=7, p_g=4$, such that the canonical system doesn't have a fixed part $/vert K/vert$ has a (non trivial) fixed part The moduli space Bibliography.
| Erscheint lt. Verlag | 30.6.2001 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Zusatzinfo | bibliography |
| Verlagsort | Providence |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-13 | 9780821826898 / 9780821826898 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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