This chapter is devoted to setting our general assumptions and conventions, to fixing notations and to recalling some basic notions and results in the form to be used throughout this book. The reader is assumed to be familiar with some very basic notions relating to analytic functions of several variables, such as germs of functions, varieties, manifolds, and analytic maps, including the inverse and implicit mapping theorems. For them he is referred to the first chapters of any book on analytic functions of several variables, such as for instance [41] or [40]. We will also make use of some rather elementary facts from ring and ideal theory: they can be found in general books such as [48] and, of course, also in those specifically devoted to commutative Algebra, [58], [8] or [32], for instance.

Throughout the book we will denote by ℤ the ring of integers, by ℕ the set of the natural or positive integers and by ℝ and ℂ the fields of the real and complex numbers. We will use the symbol ∞ with the usual algebraic rules and the total order of ℤ extended so that ∞ ≥ n for any n ∈ ℤ. Domain means connected non-empty open set and, unless otherwise stated, all neighbourhoods of points will be assumed to be open and connected.

Projective spaces

A projective space of dimension d over a field K is a set ℙ together with a exhaustive map π: F − {0} → ℙ, where F is a (d + 1)-dimensional K-vector space and π(v) = w(w) if and only if ν = aw for some a ∈ K.