Title:

Multiple point spaces and finitely determined mapgerms

This thesis is mainly based on the study of singularities of holomorphic mapgerms from nspace to pspace with n < p. We show that a minimal resolution of the kernel of the multiplication map μ : OCn,0 x OCn+1,0 OCn,0 > OCn,0, a x b > ab, is given by 0 > OrCn,0 > f*λ11 > OrCn,0 > ker(μ) > 0 where λ11 is the matrix obtained from λ, a symmetric matrix presenting OCn,0 over OCn+1,0, by deleting the first row and the first column (Proposition 2.5.2). We prove that if f is a corank 1 mapgerm with finite Aecodimension, then there exists a resolution of ODk(f) over ODk1(f) given by 0 > Ork+1 Dk(f) > γ > Ork+1 Dk(f) > ODk+1(f) > 0 in which γ is equal to f*λkk , the pullback of the matrix λkk obtained from λ by deleting the rows 1,...,k and the columns 1,...,k (Theorem 2.6.6). As a corollary, we show that detf*λk1k1 . detf*λkk defines a free divisor in Dk(f) (Proposition 2.8.4). We investigate finitely Adetermined mapgerms from Cn to Cn+1 (n ≥ 3) of corank ≥ 2 satisfying the Mond conjecture. We provide geometric criteria on finite determinacy for n = 3 (Theorem 4.4.1). We give two sets of examples of finitely A determined corank 2 mapgerms from C3 to C4 which satisfy the conjecture. For the dimensions (n; p) with n < p, we prove a criterion which yields finitely Adetermined mapgerms from the known ones (Theorem 5.1.2). We prove the existence of three series of finitely Adetermined mapgerms of corank 2 from C4 to C5 which also support the conjecture. We include a program code for a SINGULAR command that calculates Ae codimension, and a classification of 2jets of corank 2 mapgerms from 3space to 4space.
