Altıntaş, Ayşe (2011) Multiple point spaces and finitely determined map-germs. PhD thesis, University of Warwick.

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Abstract

This thesis is mainly based on the study of singularities of holomorphic mapgerms from n-space to p-space 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 map-germ with finite Ae-codimension, then there exists a resolution of ODk(f) over ODk-1(f) given by 0 -> Or-k+1 Dk(f) -> γ -> Or-k+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*λk-1k-1 . detf*λkk defines a free divisor in Dk(f) (Proposition 2.8.4). We investigate finitely A-determined map-germs 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 map-germs from C3 to C4 which satisfy the conjecture. For the dimensions (n; p) with n < p, we prove a criterion which yields finitely A-determined map-germs from the known ones (Theorem 5.1.2). We prove the existence of three series of finitely A-determined map-germs 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 2-jets of corank 2 map-germs from 3-space to 4-space.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Singularities (Mathematics), Mappings (Mathematics), Germs (Mathematics)
Date:	February 2011
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Mond, D. (David)
Sponsors:	University of Warwick
Extent:	vii, 203 leaves
Language:	eng
URI:	http://wrap.warwick.ac.uk/id/eprint/38086

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