On the classification and geometry of finite map-germs
Summary of Chapter I: 1. We use Samuel's theory of multiplicity to describe the structure of (f)/T [cal Af in terms of the multiplicity e((f)/T cal Af) and the dimension of the instability locus (this dimension is also the Krull dimension of (f)/T cal Af). 2. We extend the theory of trivial unfoldings of map-germs to the theory of k-trivial unfoldings of k-jets. 3. We develop a method for obtaining normal forms for the (k+1)-jets having a given k-jet f by inspection of certain submodules of T cal Af. We give a test for sufficiency of a normal form and a method for constructing a (k+1)-trivial unfolding of the normal form. 4. We show that if f is weighted homogeneous and the Krull dimension and multiplicity of (f)/T cal Af are both 1 then f is a weak stem and we can find an integer k and a complete list of normal forms for finitely determined map-germs whose k-jet is f. The list has many similarities with the series found by Arnold and Mond. Summary of Chapter II: We extend Mond's classification of map-germs f:(cal C^2,O)(cal C^3?O) under cal A-equivalence. This chapter also demonstrates the use of the classification theory developed in I.2 and provides a large number of examples for use in the rest of the thesis. Summary of Chapter III: 1. We prove that f is a geometric stem if and only if is irreducible and the localised module ((f)/T cal Af) has length char61 1 (localise with respect to the prime ideal defining ). 2. We also prove that if has transversal type A_2n or A_2n+1 for some n1 then char61 n. This makes it easier to determine n if has transversal type of A_2n or A_2n+1 since we shall calculate anyway. 3. If f is a stem we show that is irreducible and give a list of transversal types that may have (although having one of these transversal types does not necessarily indicate that f is a stem). 4. We look at how the numbers C, T and (D_2(f)/cal Z_2) behave for the families of map-germs associated with some weak stems. We observe that for a given family (f+p_s) the integer C(f+p_s)+T(f+p_s)+(D2(f+ps)/cal Z2)-cod(cal A,f+ps) appears to be a constant. Appendices A and B contain supplementary calculations. Appendix C is a description of the computer programs written to calculate the modules used in classifying the map-germs of Chapter II.