<div> This dissertation is based on joint work with Lindsey Hill. There are two main parts, which are linked by the common theme of the integral closure of the Rees algebra.</div><div> </div><div> In the first part of this dissertation, comprised of Chapter 3 and Chapter 4, we study the integral closure of the Rees algebra directly. In Chapter 3 we identify a bound for the multiplicity of the Rees algebra R[It] of a homogeneous ideal I generated in the same degree, and combine this result with theorems of Ulrich and Vasconcelos to obtain upper bounds on the number of generators of the integral closure of the Rees algebra as a module over R[It]. We also find various other upper bounds for this number, and compare them in the case of a monomial ideal generated in the same degree. In Chapter 4, inspired by the large depth assumption on the integral closure of R[It] in the results of Chapter 3, we obtain a lower bound for the depth of the associated graded ring and the Rees algebra of the integral closure filtration in terms of the dimension of the Cohen-Macaulay local ring R and the equimultiple ideal I. We finish the first part of this dissertation with a characterization of when the integral closure of R[It] is Cohen-Macaulay for height 2 ideals. </div><div> </div><div> In the second part of this dissertation, Chapter 5, we use the integral closure of the Rees algebra as a tool to discuss specialization of the integral closure of an ideal I. We prove that for ideals of height at least two in a large class of rings, the integral closure of I is compatible with specialization modulo general elements of I. This result is analogous to a result of Itoh and an extension by Hong and Ulrich which show that for ideals of height at least two in a large class of rings, the integral closure of I is compatible with specialization modulo generic elements of I. We then discuss specialization modulo a general element of the maximal ideal, rather than modulo a general element of the ideal I itself. In general it is not the case that the operations of integral closure and specialization modulo a general element of the maximal ideal are compatible, even under the assumptions of our main theorem. We prove that the two operations are compatible for local excellent algebras over fields of characteristic zero whenever R/I is reduced with depth at least 2, and conclude with a class of ideals for which the two operations appear to be compatible based on computations in Macaulay2.</div>

<div>This dissertation is based on joint work with Monte Cooper and is broken into two main parts, both of which study the defining ideals of the Rees rings of determinantal and Pfaffian ideals of generic height. In both parts, we attempt to place degree bounds on the defining equations.</div><div> </div><div> The first part of the dissertation consists of Chapters 3 to 5. Let $R = K[x_{1},\ldots,x_{d}]$ be a standard graded polynomial ring over a field $K$, and let $I$ be a homogeneous $R$-ideal generated by $s$ elements. Then there exists a polynomial ring $\mathcal{S} = R[T_{1},\ldots,T_{s}]$, which is also equal to $K[x_{1},\ldots,x_{d},T_{1},\ldots,T_{s}]$, of which the defining ideal of $\mathcal{R}(I)$ is an ideal. The polynomial ring $\mathcal{S}$ comes equipped with a natural bigrading given by $\deg x_{i} = (1,0)$ and $\deg T_{j} = (0,1)$. Here, we attempt to use specialization techniques to place bounds on the $x$-degrees (first component of the bidegrees) of the defining equations, i.e., the minimal generators of the defining ideal of $\mathcal{R}(I)$. We obtain degree bounds by using known results in the generic case and specializing. The key tool are the methods developed by Kustin, Polini, and Ulrich to obtain degree bounds from approximate resolutions. We recover known degree bounds for ideals of maximal minors and submaximal Pfaffians of an alternating matrix. Additionally, we obtain $x$-degree bounds for sufficiently large $T$-degrees in other cases of determinantal ideals of a matrix and Pfaffian ideals of an alternating matrix. We are unable to obtain degree bounds for determinantal ideals of symmetric matrices due to a lack of results in the generic case; however, we develop the tools necessary to obtain degree bounds once similar results are proven for generic symmetric matrices.</div><div> </div><div> The second part of this dissertation is Chapter 6, where we attempt to find a bound on the $T$-degrees of the defining equations of $\mathcal{R}(I)$ when $I$ is a nonlinearly presented homogeneous perfect Gorenstein ideal of grade three having second analytic deviation one that is of linear type on the punctured spectrum. We restrict to the case where $\mathcal{R}(I)$ is not Cohen-Macaulay. This is a natural next step following the work of Morey, Johnson, and Kustin-Polini-Ulrich. Based on extensive computation in Macaulay2, we give a conjecture for the relation type of $I$ and provide some evidence for the conjecture. In an attempt to prove the conjecture, we obtain results about the defining ideals of general fibers of rational maps, which may be of independent interest. We end with some examples where the bidegrees of the defining equations exhibit unusual behavior.</div>