Due to the exponential growth of the dimension of the space of tensors V_(1)⊗• • •⊗V_(n), any naive method of representing these tensors is intractable on a computer. In practice, we consider feasible subspaces (subvarieties) which are defined to reduce the storage cost and the computational complexity. In this thesis, we study two such types of subvarieties: the third secant variety of the product of n projective spaces, and tensor network states. For the third secant variety of the product of n projective spaces, we determine set-theoretic defining equations, and give an upper bound of the degrees of these equations. For tensor network states, we answer a question of L. Grasedyck that arose in quantum information theory, showing that the limit of tensors in a space of tensor network states need not be a tensor network state. We also give geometric descriptions of spaces of tensor networks states corresponding to trees and loops.

We consider ideals of minors of a matrix, ideals of minors of a symmetric matrix, and ideals of Pfaffians of an alternating matrix. Assuming these ideals are of generic height, we characterize the condition $G_{s}$ for these ideals in terms of the heights of smaller ideals of minors or Pfaffians of the same matrix. We additionally obtain bounds on the generation and concentration degrees of the defining equations of Rees rings for a subclass of such ideals via specialization of the Rees rings in the generic case. We do this by proving that, given sufficient height conditions on ideals of minors or Pfaffians of the matrix, the specialization of a resolution of a graded component of the Rees ring in the generic case is an approximate resolution of the same component of the Rees ring in question. We end the paper by giving some examples of explicit generation and concentration degree bounds.

<p>The defining equations of Rees algebras provide a natural pathway to study these rings. However, information regarding these equations is often elusive and enigmatic. In this dissertation we study Rees algebras of particular classes of ideals and modules over hypersurface rings. We extend known results regarding Rees algebras of ideals and modules to this setting and explore the properties of these rings.</p> <p><br></p> <p>The majority of this thesis is spent studying Rees algebras of ideals in hypersurface rings, beginning with perfect ideals of grade two. After introducing certain constructions, we arrive in a setting similar to the one encountered by Boswell and Mukundan in [3]. We establish a similarity between Rees algebras of ideals with linear presentation in hypersurface rings and Rees algebras of ideals with <em>almost</em> linear presentation in polynomial rings. Hence we adapt the methods developed by Boswell and Mukundan in [3] to our setting and follow a path parallel to theirs. We introduce a recursive algorithm of <em>modified Jacobian dual iterations</em> which produces a minimal generating set for the defining ideal of the Rees algebra.</p> <p><br></p> <p>Once success has been achieved for perfect ideals of grade two, we consider perfect Gorenstein ideals of grade three in hypersurface rings and their Rees algebras. We follow a path similar to the one taken for the previous class of ideals. A recursive algorithm of <em>gcd-iterations</em> is introduced and it is shown that this method produces a minimal generating set of the defining ideal of the Rees algebra. </p> <p><br></p> <p>Lastly, we extend our techniques regarding Rees algebras of ideals to Rees algebras of modules. Using <em>generic Bourbaki ideals</em> we study Rees algebras of modules with projective dimension one over hypersurface rings. For such a module $E$, we show that there exists a generic Bourbaki ideal $I$, with respect to $E$, which is perfect of grade two in a hypersurface ring. We then adapt the techniques used by Costantini in [9] to our setting in order to relate the defining ideal of $\mathcal{R}(E)$ to the defining ideal of $\mathcal{R}(I)$, which is known from the earlier work mentioned above.</p> <p><br></p> <p>In all three situations above, once the defining equations have been determined, we investigate certain properties of the Rees algebra. The depth, Cohen-Macaulayness, relation type, and Castelnuovo-Mumford regularity of these rings are explored.</p>

Fundação de Apoio a Pesquisa e à Inovação Tecnológica do Estado de Sergipe - FAPITEC/SE / The Rees algebra of an ideal is an algebraic construction that takes place in commutative algebra and algebraic geometry. Currently, the study of arithmetic and homological properties of this object is cause for diverse research in commutative algebra. Our main goal in this work is to address aspects such as dimension and defining equations of the Rees algebra and other algebras that relate to it. / A álgebra de Rees de um ideal é uma construção algébrica que ocupa lugar de destaque na álgebra comutativa e na geometria algébrica. Atualmente, o estudo de propriedades aritméticas e homológicas desse objeto é motivo de diversas pesquisas em álgebra comutativa. Nosso principal objetivo nesse trabalho é tratar de aspectos como dimensão e equações de definição da álgebra de Rees e de outras álgebras que relacionam-se com ela.