Although invariants for a curve with a single branch can be written in sequence, and calculated in terms of the Puiseux characteristic, for curves with several branches it is necessary to work on a tree. In this chapter we consider the dual graph of a tree produced from a plane by an arbitrary sequence of point blowings up. We will see that many invariants can be most conveniently expressed using the algebra of exceptional cycles on the surface T which is the result of the blowings up. This leads to many formulae; some of these complete the development of Chapter 4, others lead to a study of the ‘topological zeta function’.

We also prepare for the discussion of the topology of the Milnor fibration in Chapters 9 and 10. Indeed, on the boundary we have an isomorphism ∂T → ∂S, so ∂T includes the singularity link complement and allows a fairly explicit description of it, and of the Milnor fibration, which we will give in Chapter 9. Here we will introduce the invariants and notation in terms of which the later calculations will be expressed.

The homology of a blow-up

Let C be a curve defined near O ∈ 2, with branches Bj. We recall that by Theorem 3.4.4, C has a good resolution, which is a map π : T → S, where S is a (small enough) disc neighbourhood of the point O ∈ 2. The map π gives an isomorphism (T − π−1(O)) → (S − O), and the collection π−1(C) of curves has normal crossings. Moreover, π is constructed as a composite of maps πi : Ti+1 → Ti (with T0 = S), each obtained by blowing up a single point Oi, which thus gives rise to an exceptional curve Ei ⊂ Ti+1.

The study of singular points of algebraic curves in the complex plane is a meeting point for many different areas of mathematics. The beginnings of the study go back to Newton. During the nineteenth and early twentieth century algebraic geometers working on plane curves developed methods which allowed them to deal with singular curves: see e.g. [168], [84], and several articles in volume III of the mathematical encyclopaedia published 1906–1914. A notable achievement was the resolution of singularities of such curves. In the late 1920s results in the then new area of topology were applied to the knots and links in the 3-sphere obtained by looking at the neighbourhood of such a singularity. There was a resurgence of interest about 1970 due to the interaction with newly developing ideas from singularity theory in higher dimensions, most importantly, the fibration theorem which Milnor had just discovered, in the context of functions of several complex variables. There has been continuous development since then, a particular point of interest being the application of Thurston's (circa 1980) decomposition theorems for 3-manifolds and for homeomorphisms of 2-manifolds.

The interaction between ideas from these different sources makes the study of curve singularities particularly fruitful and exciting. Equisingularity is an equivalence relation which admits characterisations from numerous differing points of view. The development of the ideas leading up to this is the leitmotif of the first half of this book. I thus emphasise the equivalence of different approaches, and feel that many results gain in clarity from appearing in an integrated account.

In 1932, Mahler [376] introduced the first relevant classification of complex numbers into several classes. To this end, for given positive integers n and H and for any complex number ξ, he considered the minimum of the real numbers |P(ξ)|, where P(X) runs through the (finite) set of integer polynomials of degree at most n and height at most H, which do not vanish at ξ. Then, he let first H tend to infinity, and then n. This order is arbitrary, and we may as well do the converse, or let tend to infinity some given function of the height and the degree. The former suggestion has been proposed by Sprindžuk [532] in 1962, and the latter one by Mahler [393] in 1971. Both yield new classifications of complex numbers, to which Sections 8.1 and 8.2 are devoted, respectively. In Section 8.3, we present further results on the approximation by algebraic numbers, which, to some extend, refine Wirsing's Theorem 3.4.

Unlike in the previous chapters, we approximate complex numbers, and not only real numbers. The main reason for doing this is that the results obtained here are not sharp enough to ensure that, when we start with a real number, the approximants we construct are also real numbers.

In Chapters 1 to 7, we have exclusively considered approximation of real numbers. However, Mahler [376] and Koksma [333] defined their classifications for complex numbers as well, and Mahler [378] also introduced an analogous classification for the transcendental numbers in the field ℚp, the completion of ℚ with respect to the prime number p. Furthermore, approximation in the field of formal power series has also been investigated, for example, by Sprindžuk [534, 539]. In the present Chapter, we consider each of these settings, and we briefly describe the state of the art for the problems corresponding to those studied in Chapters 1 to 7. Roughly speaking, it is believed (and it often turns out to be true) that Diophantine approximation results in the real case have got their complex and p-adic analogues, the proofs of which are a (more or less) straightforward adaptation of those in the real case. This however does not hold true anymore for Diophantine approximation in fields of power series. For instance, the analogue of Roth's Theorem 2.1 does not exist when the ground field has positive characteristic, see, for example, the surveys by Lasjaunias [351] and by Schmidt [515] for additional information.

Approximation in the field of complex numbers

Let ξ be a complex non-real number and let n be a positive integer.

In this chapter we begin the deeper study of the topology attached to the Milnor fibration. One key problem is to obtain an understanding of the monodromy. A major tool for this is a canonical decomposition of the Milnor fibre. Because the decomposition is intrinsic, it gives a better picture of the topology than we attained in Chapter 5, particularly when the curve has several branches. We discuss the decomposition theorems in this chapter, leaving the application to monodromy to Chapter 10. Although we present an introductory account of these matters, we will necessarily assume a higher level of mathematical sophistication than was the case in earlier chapters.

We may use the carousel of Section 5.3 or the resolution tree of Section 3.6 to obtain a decomposition. We will see directly that the same is obtained from each approach, but this fact is underpinned by major theorems of great generality. Although we do not need these results, we describe them to set our discussions in a wider context. We thus begin with a section stating the general decomposition theorems in 2- and 3-dimensional topology which underlie the constructions.

We now explain what we mean by ‘decomposition’. A decomposition of a connected manifold M is effected by cutting along submanifolds of codimension 1. If T is a connected submanifold which separates M into two pieces, then if M1,M2 are the closures of the two complementary regions, the result of cutting is defined to be the disjoint union of M1 and M2.

In this book, we study various questions related to classifications of real numbers and we mainly focus our attention on the approximation of real transcendental numbers by algebraic numbers. In the present Chapter however, we briefly review the most important results which have followed Liouville's Theorem 1.2 and deal with algebraic approximation to algebraic numbers. Since a broad literature is available on this topic, we omit most of the proofs and refer the reader to, for example, the monographs of Mahler [388], Schmidt [510, 512], A. Baker [44], and Feldman and Nesterenko [244] for further information.

Completeness is not the only reason for making this survey. Indeed, some results of the present Chapter will be used in subsequent parts of the book. For instance, Theorem 2.7 (or Theorem 2.6) is crucial for proving the main result of Chapter 7, namely the existence of T -numbers. These real transcendental numbers with very specific properties of approximation by algebraic numbers are defined in Chapter 3.

We divide our exposition into four main Sections, dealing respectively with rational approximation, effective rational approximation, algebraic approximation to algebraic numbers, and effective algebraic approximation to algebraic numbers. A broad variety of methods are needed for the proofs of the results below. In a fifth Section, we briefly mention various applications to irrationality and transcendence statements.

Rational approximation

Let ξ be a real algebraic number.

Theorem 5.4, due to A. Baker and Schmidt [45], asserts that for any integer n ≥ 1 and any real number τ ≥ 1 the Hausdorff dimension of the set W*n (τ) of real numbers ξ with w*n (ξ) = τ(n + 1) – 1 is equal to 1/τ. In the present Chapter, we are concerned with various refinements, including the determination of the Hausdorff measure of W*n (τ) at the critical exponent (Corollary 6.3 below).

There are essentially two new ingredients. On the one hand, we need an improvement of Proposition 5.4, which is due to Beresnevich [61] and asserts that real algebraic numbers of bounded degree are distributed ‘as evenly as they could be’. On the other hand, we present a refined analysis of the Hausdorff measure of sets of real numbers close to infinitely many points in a given real sequence.

One essential tool, introduced in Section 6.1, is the notion of ‘optimal regular systems’ (also termed ‘best possible regular systems’ by Beresnevich, Bernik, and Dodson [67]). We state four general results on sets of real numbers close to infinitely many points in an optimal regular system. We establish the first one in Section 6.2, which allows us to give an alternative proof of (a slightly stronger form of) Khintchine's Theorem 1.10. The second one, stated in Section 6.3, provides the Hausdorff dimension of general exceptional sets.

For curves in the projective plane, the most basic invariant is the degree of the defining equation. This gives a qualitative bound for the possible complexities of the curve, and also of its singularities. We will see that this may be turned into precise quantitative bounds.

First, however, we discuss two classical topics. The first is a formula for the genus of a curve in terms of its degree and its singularities. The second is a corresponding formula for its class: the number of tangents to the curve from a generic point. This can be reinterpreted. The tangents to a curve are lines in the plane, which can also be interpreted as points in a dual plane. The locus of these points, or rather its closure, is the dual curve, and the formulae yield relations between the original curve and its dual. The oldest version of these formulae consists of the celebrated Plücker equations.

Our proofs of these results use the technique involving the Euler characteristic which we developed in Theorems 6.4.1 and 6.4.2. A further refinement of this technique consists of the calculus of so-called constructible functions. We describe this, and give a further application, to Klein's formula concerning the singularities of a curve in the real projective plane.

These formulae do not exhaust the relations between the singularities of a curve and those of its dual, and we give a full discussion of the relation between the two.

We conclude with a survey of some of the known results concerning the possible configurations of singularities of curves of a given degree.

We begin this chapter by explaining what a fibration is, and giving a method of establishing that certain maps are fibrations. Then we show that two maps defined explicitly in terms of an isolated curve singularity give equivalent fibrations: each of these is termed ‘the Milnor fibration’. All the more delicate topology of C is encoded in the Milnor fibration, and studying its geometry gives a very close insight into the topology and geometry associated to the singularity. In this chapter, we give some elementary properties, leading to various calculations of the Betti numbers of the fibre. A detailed study will be made in Chapter 10.

Fibrations

A fibration is a sort of twisted product. More precisely, a map π : E → B is (the projection of) a fibration with fibre F if each point b ∈ B has a neighbourhood U such that there is a homeomorphism φ of π-1(U) onto F × U whose second component is the restriction of π. Thus to construct the homeomorphism one needs only the first component, a map onto F, which may well be defined via a map onto Fb = π-1(b).

Properties of fibrations are derived in full in textbooks of algebraic topology, e.g. [169], and we content ourselves here with citing those we shall require.

If U is any contractible subset of B, then a homeomorphism φ as above may be constructed over U. In particular, if B = S1, we may take U to be either the upper or the lower semicircle: U+, U- say.

The set of real numbers splits into algebraic and transcendental numbers, but these two subsets do not have the same size, the former being countable, while the latter has the power of continuum. Such a crude classification of real numbers seems to be rather unsatisfactory, and one aims to find some way to classify the transcendental numbers. First, we have to ask which requirements should satisfy a ‘good’ classification. Ideally, for a given real number ξ, we would like to have a simple criterion to determine the class to which ξ belongs. Furthermore, two algebraically dependent real numbers should belong to the same class. The first classification of transcendental real numbers has been proposed by Maillet [403, 404], and others were subsequently described by Perna [453] and by Morduchai-Boltovskoj [430], but none of these has proved to be relevant. For instance, Maillet's classification depends on the size of the partial quotients of the real numbers and, clearly, does not satisfy the second requirement.

An attempt towards a ‘reasonable’ classification was made in 1932 by Mahler [376], who proposed to subdivide the set of real numbers into four classes (including the class of algebraic numbers) according to, roughly speaking, their properties of approximation by algebraic numbers. Mahler's classification satisfies our second requirement: two algebraically dependent real numbers belong to the same class.

In order to give a full discussion of the case of curves with several branches at the singular point it is necessary to discuss the type of contact that two such branches can have. This also allows us to fill in several details in our discussion of the geometry of a single branch.

We will express our result in terms of the exponent of contact of two branches. In fact, we obtain a more flexible concept by introducing the notion of pro-branch and exponents of contact of pro-branches.

The most important result is a formula relating exponent of contact to intersection multiplicity. This is the key to numerous later developments. The basic formula relates to the case when each curve has just one branch. We then develop a notation to express the type of contact of curves with several branches. It takes the form of a tree with numerical information attached, which seems best suited to describe the numerical invariants of curves with several branches.

We use our main formula to give a complete description of the semigroup of a branch. The intersection multiplicity can also be expressed in terms of the calculus of infinitely near points, and establish the essential equivalence of these two approaches, which is formalised by the notion of equisingularity.

A further section gives an application of these techniques to give a proof of a recent theorem on the decomposition of polar curves. As this is somewhat outside the main line of development of the first half of this book, it may be omitted on a first reading.