Collop, Michael J. (1973) Projective fibres schemes. PhD thesis, University of Warwick.

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Abstract

This thesis takes the construction (due to Grothendieck) of the projective fibred scheme of a coherent sheaf and iinvestigates certain aspects of its geometry, (and, in Chapter IV, topology).

In Chapter I, after quoting the basic definitions, of the projective fibred scheme of ϵ with its projection π : ρϵ·→X and fundamental invertible sheaf ϴ(1) on ρϵ, and giving some simple illustrative examples, we turn (§§ 2,3) to some particular features of the geometry, notably the Fitting subschemes and the dominating component (and interactions between the two). Here and throughout most of the work we keep close to geometrical intuition by considering only coherent modules on locally-neotherian, reduced schemes. In an appendix we introduce some sheaves that are in: a sense universal for coherent sheaves on projective varieties. (These do not play any essential role in the rest of the thesis).

Chapter II is concerned with the canonical homomorphism X, ϵ→π*ϴ (1) which is known to be an isomorphism when ϵ is locally-free. We extend this result to a larger class of sheaves and show, for example, that X is an isomorphism if ρϵ normal. In view of this result, and for general reasons, it is of interest to look for examples of smooth projective fibred schemes. This we do in Chapter III and show that a "generic" Module ϵ. of the type that locally has a resolution 0→0pu→θqu→∈u→0, (U, smooth where p(x) =0, has smooth ρϵ in a neighbourhood of ρϵ(x).

Chapter IV considers the (singular) cohomology ring of PE when ϵ is a sheaf on a complex variety (with the classical topology). We include a discussion of the effect on cohomology of blowing-up a smooth variety with centre of smooth subvariety.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Geometry, Projective, Differential Geometry, Homology theory, Topology
Official Date:	1973
Dates:	Date Event 1973 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Schwarzenberger, R. L. E.
Sponsors:	Science Research Council (Great Britain)
Extent:	75 leaves
Language:	eng

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