Projections of Varieties

Meadows, Catherine Ann

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https://hdl.handle.net/2142/71199 Copy

Description

Title

Projections of Varieties

Author(s)

Meadows, Catherine Ann

Issue Date

1981

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Abstract

• This thesis is devoted to the study of varieties with the following property: if X is a smooth projectively normal variety in ('n), we say that X has the length one projection property if every isomorphic projection of X to ('n-1) has the property that the linear system cut out on it by the hypersurfaces of degree k is complete for k (GREATERTHEQ) 2.
• The main result of this thesis is that, if X is a smooth variety in ('n), then the d-uple embedding of X has the length one projection property for large enough d. The theorem is first proved for any d-uple embedding of ('n) by induction on r, and is then extended to the d-uple embedding of any smooth variety for large enough d.
• A theorem describing varieties with the length one projection property is also given. Suppose that X has the length one projection property non-trivially (i.e., that isomorphic projections of X do exist). Then for every variety Y such that X (L-HOOK) Y (L-HOOK) V(J), where J is the ideal generated by the quadratics in the defining ideal of X, we have Sec*(X) = Sec*(Y), where Sec*(Y) is defined to be the union of the secant lines through Y and the linear spaces tangent to Y. Thus in most cases we would expect the defining ideal of X to be minimal over its quadratic generators. An example is provided to show that this is not true of all cases.

Graduation Semester

1981

Type of Resource

text

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http://hdl.handle.net/2142/71199

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