On the Defining Ideals of Rees Rings for Determinantal and Pfaffian Ideals of Generic Height

Edward F Price, Purdue University

Abstract

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. The first part of the dissertation consists of Chapters 3 to 5. Let R = K[x1, . . . , xd] be a standard graded polynomial ring over a field K, and let I be a homogeneous Rideal generated by s elements. Then there exists a polynomial ring S = R[T1, . . . , Ts], which is also equal to K[x1, . . . , xd, T1, . . . , Ts], of which the defining ideal of R(I) is an ideal. The polynomial ring S comes equipped with a natural bigrading given by deg xi = (1, 0) and deg Tj = (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 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 in [45] 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.

Degree

Ph.D.

Advisors

Ulrich, Purdue University.

Subject Area

Mathematics

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