Thesis (doctoral)--Christian-Albrechts-Universität zu Kiel, 1917. / Vita. Includes bibliographical references.

While there is considerable literature about algebras satisfying a polynomial identity, there are only scant results about varieties of algebras. For such an algebra we can introduce the notions of bimodule, birepresentation and universal enveloping algebra as an extension of the notions of module and representation for associative algebras. Moreover, it is possible to define injective hulls for these restricted representations. We derive a rather concrete structure theorem of I-bimodules M for a finite dimensional algebra in a certain variety by studying a universal enveloping algebra and injective hulls. / Science, Faculty of / Mathematics, Department of / Graduate

The thesis consists of four chapters. First chapter is introductory. In Chapter 2, we recall some basic facts from the singularity theory of algebraic varieties (see Section 2.2) and the theory of minimal models (see Section 2.3), which will be used throughout the rest of the thesis. We also make some conventions on the notions and notation used in the thesis (see Section 2.1). Each Chapter 3 and 4 starts with some preliminary results (see Sections 3.1 and 4.1, respectively). Each Chapter 3 and 4 ends with some corollaries and conclusive remarks (see Sections 3.7 and 4.4, respectively). In Chapter 3, we prove Theorem 1.2.7, providing the complete description of Halphen pencils on a smooth projective quartic threefold X in P4. Let M be such a pencil. Firstly, we show that M ⊂ | − nKX | for some n ∈ N, and the pair (X,1n M) is canonical but not terminal. Further, if the set of not terminal centers CS(X, 1 ) (see Remark 2.2.8) does not contain points, we show that n = 1 (see Section 3.2). Finally, if there is a point P ∈ CS(X, n M), in Section 3.1 we show first that a general M ∈ M has multiplicity 2n at P (cf. Example 1.2.3). After that, analyzing the shape of the Hessian of the equation of X at the point P , we prove that n = 2 and M coincides with the exceptional Halphen pencil from Example 1.2.6 (see Sections 3.3-3.6). In Chapter 4, we prove Theorem 1.2.11, which shows, in particular, that a general smooth K3 surfaces of type R is an anticanonical section of the Fano threefold X with canonical Gorenstein singularities and genus 36. In Section 4.2, we prove that X is unique up to an isomorphism and has a unique singular point, providing the geometric quotient construction of the moduli space F in Section 4.3 (cf. Remark 1.2.12). Finally, in Section 4.3 we prove that the forgetful map F −→ KR is generically surjective.

Thesis (Ph. D.)--Northeastern University, 2008. / Title from title page (viewed March 26, 2009). Graduate School of Arts and Sciences, Dept. of Mathematics. Includes bibliographical references (p.80-82).

Let <em>X</em>(<em>K</em>) &sub; <strong>P</strong>^<em>n</em> (<em>K</em>) be a projective algebraic variety over <em>K</em>, and let <em>D</em> be a subset of <strong>P</strong>^<em>n</em>_<em>OK</em> such that the codimension of <em>D</em> with respect to <em>X</em> &sub; <strong>P</strong>^<em>n</em>_<em>OK</em> is two. We are interested in points <em>P</em> on <em>X</em>(<em>K</em>) with the property that the intersection of the closure of <em>P</em> and <em>D</em> is empty in <strong>P</strong>^<em>n</em>_<em>OK</em>, we call such points <em>D</em>-integral points on <em>X</em>(<em>K</em>). First we prove that certain algebraic varieties have infinitely many <em>D</em>-integral points. Then we find an explicit description of the complete set of all <em>D</em>-integral points in projective n-space over Q for several types of <em>D</em>.

Let <em>X</em>(<em>K</em>) &sub; <strong>P</strong>^<em>n</em> (<em>K</em>) be a projective algebraic variety over <em>K</em>, and let <em>D</em> be a subset of <strong>P</strong>^<em>n</em>_<em>OK</em> such that the codimension of <em>D</em> with respect to <em>X</em> &sub; <strong>P</strong>^<em>n</em>_<em>OK</em> is two. We are interested in points <em>P</em> on <em>X</em>(<em>K</em>) with the property that the intersection of the closure of <em>P</em> and <em>D</em> is empty in <strong>P</strong>^<em>n</em>_<em>OK</em>, we call such points <em>D</em>-integral points on <em>X</em>(<em>K</em>). First we prove that certain algebraic varieties have infinitely many <em>D</em>-integral points. Then we find an explicit description of the complete set of all <em>D</em>-integral points in projective n-space over Q for several types of <em>D</em>.

No description available.

Thesis (Ph. D.)--University of Washington. / Bibliography: l. 73-74.

Inaug.-Diss.--Rheinsche Friedrich-Wilhelms-Universität, 1986. / Bibliography: p. 101-104.

Thesis (Doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1990. / Includes bibliographical references.