By Alexandru Dimca

The physique of arithmetic constructed within the final 40 years or so that are positioned less than the heading Singularity concept is sort of huge. And the superb introductions to this colossal sub ject that are already to be had (for example [AGVJ, [BGJ, [GiJ, [GGJ, [LmJ, [Mr], [WsJ or the extra complicated [Ln]) disguise unavoidably merely aside of even the main simple subject matters. the purpose of the current booklet is to introduce the reader to some vital themes from ZoaaZ Singularity thought. a few of these themes have already been handled in different introductory books (e.g. correct and make contact with finite determinacy of functionality germs) whereas others were thought of purely in papers (e.g. Mather's Lemma, class of straightforward O-dimensional whole intersection singularities, singularities of hyperplane sections and of twin mappings of projective hypersurfaces). Even within the first case, we believe that our therapy isn't the same as the introductions pointed out above - the overall cause being that we supply detailed cognizance to the aompZex anaZytia state of affairs and to the connections with AZgebraia Geometry. we provide now an in depth description of the contents, aspect ing out certain facets and new fabric (i.e. formerly un released, although for the main half definitely identified to the ts!). bankruptcy 1 is a quick advent for the newbie. We keep in mind the following simple effects (the Submersion Theorem and Morse Lemma) and make a couple of reviews on what's intended through the neighborhood behaviour of a functionality or of a aircraft algebraic curve.

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7). 8) CONTACT EQUIVALENCE (K-EQUIVALENCE) Let G be the semidireat produat D mUltiplication rule n ~M n,p given by the where composition with h~I refers to all the entries of the matrix A2 • This group G will be also denoted by Kn,p or simply by K. 19 The K-equivalence is associated to the action k:GxE o n,p where the germs fand foh- 1 should be considered as column vectors wi th entries in the maximal ideal mn C En • The mul tiplicati on in the right-hand side is then matrix multiplication. A geometrie explanation for the use of the word contaat here can be found in [GG], p.

To the element x). One assurnes here Gx~Gy. Notation: x + y, Gx + Gy. 25 that x and only if there is areal analytic path c:[O,€) + + y if M for some e>O such that y=c(O) and c«O,e))CG·x. Hence y can be regarded as a limit of points xt=c(t)EG'x for t + O. 9). e. 12) as action a for d=l. ; i=l, ... ,k; j=l, ... ,n>+ J 1. 2) Note that dirn S=(n-k) (p-k) and hence G,u k is a smooth manifold of dimension np-dirn S. g. [GG], p. 3) PROPOSITION The specialization relations among the normal forms u k are the following other words: Guk =GU k U GU k _ 1 U •• ,U GU 1 U {O}.

The space Jk(n,p) is called the spaoe of k-jets of type (n,p). t i. e. J 0 J =J • If we analyze now aga in the results in Chapter 1, we see that essentially the Submersion Theorem (resp. Morse Lemma) says that the germ of the ma9ping f at the origin is equal (up to a coordinate change) to the germ associated to the first jet j1 f (resp. second jet j2 f ) of f. The atternpt to un- der stand and generalize these special cases will be our main concern in the sequel. Finally, we remark that for two manifolds (smooth or complex analytic) X and Y, there is a k-jet space Jk(X,y) which is a fiber bundle over XxY with typical fiber Jk(n,p), where n=dim X, p=dim Y, and which plays a basic role in studying the global mappings f:X [GGJ.