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Title:Projections of Varieties
Author(s):Meadows, Catherine Ann
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
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.
Issue Date:1981
Description:149 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981.
Other Identifier(s):(UMI)AAI8203532
Date Available in IDEALS:2014-12-16
Date Deposited:1981

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