Book contents
- Frontmatter
- Foreword
- List of Contributors
- Contents
- Introduction (by K. Ueno and T. Shioda)
- Part I
- Coverings of the Rational Double Points in Characteristic p
- Enriques' Classification of Surfaces in Char. p, II
- Classification of Hilbert Modular Surfaces
- On Algebraic Surfaces with Pencils of Curves of Genus 2
- New Surfaces with No Meromorphic Functions, II
- On the Deformation Types of Regular Elliptic Surfaces
- On Numerical Campedelli Surfaces
- On Singular K3 Surfaces
- On the Minimality of Certain Hilbert Modular Surfaces
- Part II
- Part III
- Index
On Numerical Campedelli Surfaces
Published online by Cambridge University Press: 03 May 2010
- Frontmatter
- Foreword
- List of Contributors
- Contents
- Introduction (by K. Ueno and T. Shioda)
- Part I
- Coverings of the Rational Double Points in Characteristic p
- Enriques' Classification of Surfaces in Char. p, II
- Classification of Hilbert Modular Surfaces
- On Algebraic Surfaces with Pencils of Curves of Genus 2
- New Surfaces with No Meromorphic Functions, II
- On the Deformation Types of Regular Elliptic Surfaces
- On Numerical Campedelli Surfaces
- On Singular K3 Surfaces
- On the Minimality of Certain Hilbert Modular Surfaces
- Part II
- Part III
- Index
Summary
Statement of Main Results
In this paper, a surface will mean a compact complex manifold of dimension 2. We denote by ∣mKs∣ (m ε N) the pluricanonical system on a surface S, and by ΦmKs the associated rational map (the pluricanonical map), assuming that ∣mKs∣ is not empty. A surface S is called of general type if, for a large number m, ΦmKs(S) is a variety of dimension 2 in the projective space PN (N = dim ∣mKs∣). We shall study a certain class of surfaces of general type.
Definition. (1) A minimal surface of general type is called a numerical Campedelli surface if S satisfies the following numerical conditions :
(2) A numerical Campedelli surface is called a Campedelli surface if its fundamental group π1(S) is isomorphic to Z/(2)⊕Z/(2)⊕Z/(2) (cf. [2]).
Then we have the following results.
Theorem A.The tricanonical system ∣3 Ks∣ for a numerical Campedelli surface S is free from base points and fixed components
Remark. Except for numerical Campedelli surfaces, we can enumerate all surfaces for which the tricanonical maps are not birational (see Bombieri [1] and Miyaoka [4]).
Theorem B. For a Campedelli surface S, the universal covering S of S is birational to a complete intersection of type (2, 2, 2, 2) in P6.
Remark. It is an interesting but, in general, a very difficult problem to determine the complex structures on a given underlying differentiate manifold.
- Type
- Chapter
- Information
- Complex Analysis and Algebraic GeometryA Collection of Papers Dedicated to K. Kodaira, pp. 113 - 118Publisher: Cambridge University PressPrint publication year: 1977
- 8
- Cited by