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Altıntaş, Ayşe (2011) Multiple point spaces and finitely determined map-germs. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2533284~S1
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 (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Singularities (Mathematics), Mappings (Mathematics), Germs (Mathematics) | ||||
Official Date: | February 2011 | ||||
Dates: |
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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Publication Status: | Unpublished | ||||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 |
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