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).