Introduction

In this article we shall define an analytic Néron model, i.e., an analytic counterpart of Néron's minimal model [11], and prove the minimality of it among principal homogeneous spaces (§ 2, 3). This portion deals with a partial generalization of the notion of analytic fiber system of groups and related results in [4].

Secondly we shall relatively compactify an analytic Néron model by applying the theory of torus embeddings [3], [5]. One should recall that the usefulness of a relative compactification of Néron model has been conjectured in [11]. The original construction in [6] was given in a slightly different, elementary, however rather complicated form. To facilitate better understanding, we employ here the notations of torus embeddings.

Kodaira has made a deep investigation of elliptic fibrations of surfaces [4]. Iitaka [2] and Ogg [12] gave a numerical classification of singular fibers in a pencil of curves of genus two. Namikawa and Ueno [8], [9] gave their complete classification and made a systematic study of them, following in principle Kodaira. The final objective of this article is to construct a family of reduced singular fibers of genus two in a systematic and geometric way by making use of a relative compactification of the Neron model (§ 5, 6, 7). In this respect, the present article should be viewed as a continuation of [9], which its authors had intended to write at that time.