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Title:  Projections of Varieties 
Author(s):  Meadows, Catherine Ann 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(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 ('n1) 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 duple embedding of X has the length one projection property for large enough d. The theorem is first proved for any duple embedding of ('n) by induction on r, and is then extended to the duple 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 nontrivially (i.e., that isomorphic projections of X do exist). Then for every variety Y such that X (LHOOK) Y (LHOOK) 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 
Type:  Text 
Description:  149 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1981. 
URI:  http://hdl.handle.net/2142/71199 
Other Identifier(s):  (UMI)AAI8203532 
Date Available in IDEALS:  20141216 
Date Deposited:  1981 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois