• On Rees algebras of linearly presented ideals and modules (with A. Costantini and E. Price) (Collectanea Mathematica)

One of the essential conditions within the study of the Rees algebra of an ideal in a d-dimensional ring,  is that the number of generators of the ideal in question is bounded up to codimension d-1. In this paper, we proceed under the assumption that this value is bounded only up to codimension d-2, expanding on previous work of P.H.L Nguyen. Here we determine the generators of the defining ideal of the Rees algebra in a more general setting and study the Cohen-Macaulay property of both the Rees algebra and the fiber ring. Moreover, using generic Bourbaki ideals,  we then extend our results for perfect ideals of codimension two to Rees algebras of modules with projective dimension one. Lastly, we present some interesting phenomena which occurs under weaker assumptions, which will be the topic of future projects. 

• The equations on Rees algebras of height three Gorenstein ideals in hypersurface rings (Journal of Commutative Algebra)

In this paper,  we generalize a classic result within the study of defining equations of Rees algebras to the case of an ideal in a hypersurface ring. Gorenstein ideals of grade three are a particularly interesting class of ideals and their Rees rings are a source of many anomalies within the study of blow-up algebras. However, the study of these algebras is limited and  prior results insist that the ideal in question belong to a polynomial ring. Following the work below, we modify classical techniques to this new setting and introduce a recursive algorithm of gcd-iterations which produces a minimal generating set of the defining ideal.

• On Rees algebras of ideals and modules over hypersurface rings (Journal of Algebra)

Perfect ideals of grade two are an interesting class of ideals and their Rees algebras have been studied to great length. However, in much of the literature, it has been necessary that such an ideal lives in a polynomial ring. In this paper we consider a perfect ideal of grade two in hypersurface ring. Adapting classical techniques to this setting, the modified Jacobian dual is introduced along with a recursive algorithm of modified Jacobian dual iterations, which yields a minimal generating of the defining ideal of the Rees algebra. Through the use of generic Bourbaki ideals, this result is then extended to the case of Rees algebras of modules, having projective dimension one.