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 Algebraic Surfaces with Pencils of Curves of Genus 2
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
Introduction
Inspired by Kodaira's work on elliptic surfaces, several authors have studied pencils of curves of genus 2 on compact complex analytic surfaces. We understand that they have established a “local theory” on such pencils. We refer the reader to [7] for a brief account of results and references. We do not use the results of our predecessors in the following.
In this paper we shall study pencils of curves of genus 2 from a little more global point of view. We are more interested in surfaces S which carry these pencils rather than in the pencils themselves. We note that these surfaces are projective algebraic.
Our main results are as follows. Let g : S → Δ be a surjective holomorphic map onto a non-singular complete curve Δ whose general fibre C is an irreducible nonsingular curve of genus 2. We let K denote the canonical bundle of S. Then, for a sufficiently ample divisor m on Δ, the linear system ∣K + g*m∣ determines a rational map Φ : S → W‡ of degree 2 onto a surface W‡ which is a P1– bundle over Δ. Let ѽ : Š → S be a composition of quadric transformations such that Φ o ѽ is everywhere defined. We define the branch locus B‡ of Φ to be that of Φ o ѽ, which is independent of the choice of ѽ. The singularities of B‡ are classified into six types (see Lemma 6). We can calculate numerical characters of S in terms of BDagger (see Theorems 2 and 3).
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- Information
- Complex Analysis and Algebraic GeometryA Collection of Papers Dedicated to K. Kodaira, pp. 79 - 90Publisher: Cambridge University PressPrint publication year: 1977
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