The Equations Defining Rees Algebras of Ideals and Modules over Hypersurface Rings

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.

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 almost 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 modified Jacobian dual iterations which produces a minimal generating set for the defining ideal of the Rees algebra.

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 gcd-iterations is introduced and it is shown that this method produces a minimal generating set of the defining ideal of the Rees algebra. 

Lastly, we extend our techniques regarding Rees algebras of ideals to Rees algebras of modules. Using generic Bourbaki ideals 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.

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.