My research interests lie in commutative algebra and its connections to algebraic geometry. Specifically, I am interested in the study of Rees algebras of ideals and modules, the algebraic structure of these rings, and their interactions with algebraic geometry and other aspects of commutative algebra. Moreover, I am interested in the applications of Rees algebras, and related syzygy techniques, to implicitization problems arising in the applied sciences.
Within algebraic geometry, a natural problem that arises is how to obtain the implicit equations of the closed image of a rational map between projective spaces. We study this so-called implicitization problem for the higher dimension analog of the classic plane de Jonquières transformations, as given in previous work of Hassanzadeh and Simis. We pass from the image of this map and study its graph, with coordinate ring the Rees algebra of the ideal of forms defining this transformation. When the underlying Cremona support of a de Jonquières transformation is tame, we show that its Rees ring is defined by the ideal of a downgraded sequence associated to a particular syzygy, and moreover we characterize the Cohen-Macaulayness of this algebra. We then show that the techniques developed here extend to the setting of generalized de Jonquières transformations, and answer a conjecture of Ramos and Simis.
Blowups of hypersurfaces (submitted)
Within the study of Rees rings, one often considers ideals with accessible syzygies, so as to leverage homological data to describe properties of this algebra. In the case that an ideal is generated by a regular sequence, i.e. its syzygies are Koszul and hence as simple as possible, a classical result of Micali asserts that its Rees ring is defined by a particular ideal of minors of generic height. In particular, this describes the Rees ring of the maximal ideal of any regular local ring and moreover, the converse holds as well. Namely, a Noetherian local ring is regular if and only if the Rees algebra of its maximal ideal has this particular shape. In this paper, we ask how far the shape of the Rees algebra differs from this form in the case the ring is non-regular, and answer this question for hypersurface rings. We show that the defining ideal differs from the aforementioned determinantal ideal only by a downgraded sequence of polynomials associated to the hypersurface equation. Once it is show that this recursive algorithm yields a minimal generating set of the defining ideal, we explore the Cohen-Macaulayness of the Rees ring and several other invariants.
Tensor product surfaces and quadratic syzygies (Linear Algebra and its Applications)
Tensor product surfaces are defined as the closed image of a rational map defined by a bihomogeneous ideal over the product of projective lines. These surfaces arise frequently in the field of geometric modeling for their applications to computer-aided geometric design (CAGD). In this context, these surfaces are bi-parameterized and knowledge of the implicit equation is particular desirable, as it allows for more efficient computation and geometric rendering. In this paper, we address this implicitization problem through the lens of commutative algebra, via the syzygies of the bihomogeneous ideal. Following previous work of Duarte and Schenck when this ideal has a linear syzygy, we address the case when it has a quadratic syzygy. From this given syzygy, an additional subset of syzygies is constructed which completely determines the implicit equation of such a surface.
(with A. Costantini and E. Price) On Rees algebras of ideals and modules with weak residual conditions (submitted)
Within the study of Rees rings, one must often impose certain residual conditions, namely bounds on the generation of the ideal or module in question, locally up to a certain codimension. Typically, this assumption is precisely the Artin-Nagata condition G_d where d is the dimension of the ring R. In previous work, the present authors studied the case when this is weakened to the condition G_(d-1). In the current paper, this condition is further weakened to the condition G_s for any s<d. This is accomplished by introducing a method of successive approximations of Rees algebras, allowing one to produce a sequence of algebras that are simpler than the Rees ring, but still carry information on it. This method appears quite general, and further study of some of its other applications will be the topic of a future paper.
In this paper, we study the Rees algebra of a module of projective dimension one, expressing this ring as a quotient of the symmetric algebra of the module. When the symmetric algebra is a complete intersection ring, we employ a particular duality between the symmetric algebra and the ideal defining this quotient. With this, we study the Rees ring in a handful of different settings. In particular, when the ground ring is a complete intersection defined by quadrics, we consider its module of Kähler differentials and the tangent algebras associated to this module.
(with A. Costantini and E. Price) On Rees algebras of linearly presented ideals and modules (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 of 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.