Surfaces with K 2 7 and Pg 4
2001
American Mathematical Society (Verlag)
978-0-8218-2689-8 (ISBN)
American Mathematical Society (Verlag)
978-0-8218-2689-8 (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 |
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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-10 | 0-8218-2689-1 / 0821826891 |
ISBN-13 | 978-0-8218-2689-8 / 9780821826898 |
Zustand | Neuware |
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