ﬁbered spaces in 3 -manifolds

MR0414561 (54 #2662) 14H20; 14H45 Zariski, Oscar Le problème des modules pour les branches planes. (French) Avec un Appendice de Bernard Teissier et une réimpression de “Characterization of plane algebroid curves whose module of differentials has maxuimum torsion” (Proc. Nat. Acad. Sci. U.S.A. 56 (1966), no. 3, 781–786), par Oscar Zariski. École Polytechnique, Paris , 1973, iii+199 pp. This book consists of notes from a course of Zariski at the École Polytechnique, plus an appendix by B. Teissier. The main problem is to study the space of isomorphism classes of plane branches (analytically irreducible curve germs) of given equisingularity type, i.e., with the same valuation semigroup Γ. The first three chapters of the course review the basic material and introduce several important notions, such as short parametrization: every plane branch has a Puiseux parametrization of the form (t, t + ∑q 1 aνit i), where ν1, · · · , νq are the integers in {m+1, · · · , c}−Γ (c = conductor). If the characteristic is (n;β1, · · · , βg), then the aβi = 0; and certain of the aνi may be eliminated after appropriate coordinate change. Thus, one can associate (in a non-unique way) to each curve of semigroup Γ a point in C − ⋃g 2 Hi, using the aνi as coordinates; here, p is some integer ≤ q, and the hyperplanes are removed because aβi = 0 (i = 2, · · · , g). The moduli space M(Γ) is then the quotient of C − ⋃ Hi by the equivalence relative of analytic isomorphism of the corresponding branches. M(Γ) is not separated in general; it is compact if and only if g = 1 or g = 2, n = 4, m = 6 (in the last case, M(Γ) is a point). Further, if g = 1 (i.e., (n,m) = 1), the curve y = x gives a point in the closure of every other point. That M(Γ) is non-compact in other cases is established by finding analytic invariants of the parametrization (i.e., aβ3 β2−β1 , if g ≥ 3), which provide surjective maps of M(Γ) onto C − {0}. The basic technique is to change the parametrization via appropriate automorphisms, to see how the coefficients can change. In Chapter V, the complete structure of M(Γ) is worked out for (n,m) ∈ {(2,m), (3,m), (4, 5), (5, 6), (6, 7)}. The last case is the most interesting; M(Γ) consists, of six components (there are “six types of curves”), the one of largest dimension being an algebraic surface. Thus, the “generic” component has dimension 2. Chapter VI examines the generic component for g = 1 via equisingular deformations. The main result here is that the generic dimension is the dimension of the space of equisingular deformations for a “general” curve of type (n,m); this is computed if m = n + 1. At the end of the course an article of the author’s is reproduced [Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 781–786; MR0202716]. Teissier’s appendix is based on the very pretty idea that every plane branch C of semi-group Γ appears as the general fibre in a one-parameter deformation of the monomial curve C (defined by (t0 , t1 , · · · , tg ), where a0, · · · , ag are minimal generators for Γ). The idea is that O = OC can be filtered using the natural filtration on the integral closure, and the associated graded is the local ring OΓ of C.