Research

My research is in commutative algebra, and I am especially interested in its interactions with algebraic geometry and combinatorics. The goal of commutative algebra is to study commutative rings, and ideals and modules over them.  An ideal in a polynomial ring can be interpreted as the vanishing locus of an algebraic variety, or can be associated with various combinatorial objects, such as a graph, a simplicial complex or a convex polytope. In this way, problems in commutative algebra, algebraic geometry and combinatorics are deeply interconnected with each other, and one can often solve problems in one of these areas borrowing techniques from another one. Furthermore, one can sometimes use representation-theoretic methods to study the algebraic properties of ideals which are stable under permutation of the variables. 

In this context, my work revolves around the study of Rees algebras of ideals and modules, with geometric motivations coming from the study of singularities, multiplicity theory, and toric algebraic geometry.  Algebraically, I am interested in understanding  integral dependence of ideals and modules, as well as in studying the asymptotic behavior of ordinary and symbolic powers of ideals for large exponents. Rees algebras have proved to be extremely useful tools for this, especially when considering ideals with a rich combinatorial structure. 

Publications and preprints

Published or accepted for publication

1. (with Edward Price III and Matthew Weaver) On Rees algebras of linearly presented ideals and modules, accepted in Collectanea Mathematica,  arxiv:2308.16010 

Rees algebras of height-two perfect ideals with a linear presentation have been extensively studied under the assumption that the the minimal number of generators of the given ideal is locally bounded on the punctured spectrum. In this paper, we treat the case that this number of generators is locally bounded up to codimension d-2, if d is the dimension of the ring. We then extend the result to modules of projective dimension one.  slides

2. (with Kyle Maddox and Lance E. Miller)  Rees algebras and generalized depth-like conditions in prime characteristic,  Math. Nachr. (2023) 1-19,  arxiv:2212.11374. 

We introduce a new depth-like invariant to study certain singularities in prime characteristic, defined in terms of nilpotent Frobenius actions on local cohomology. Extending a well-known result of Huneke, we prove that these singularities behave similarly to Cohen-Macaulay singularities under blowups. slides

3.  (with Louiza Fouli and Jooyoun Hong)  Residual intersections and core of modules, J. Algebra 629 (2023), 227-246,  arxiv:2205.13721 

The core is an important object in the study of integral dependence, but  is difficult to determine, being a priori an infinite intersection of ideals or modules. We  introduce the notion of residual intersection of modules and describe the core of modules satisfying certain homological conditions. Our results generalize work of Corso, Polini and Ulrich on the core of ideals with good residual intersection properties.  slides 

4. (with Ben Drabkin and Lorenzo Guerrieri)  Rees algebras of ideals of star configurations, Linear Algebra and Appl.  645 (2022), 91-122, arxiv:2107.12260 

A generalized star configuration is a union of complete intersection subschemes in projective space, obtained by intersecting hypersurfaces meeting properly.  This includes the case of 10 points located at pairwise intersections of 5 lines in the projective plane, with the lines forming a star shape.  We study the defining ideal of blow ups of generalized star configurations, using methods from linear algebra.   slides 

5. (with Tan Dang)  On the Cohen-Macaulay property of the Rees algebra of the module of differentials,  Proc. Amer. Math. Soc. 150 (2022), 941-950, arxiv:2007.15772 

The module of Kähler differentials of a variety X captures the singularities of X. In the case when X is a complete intersection, we give sufficient conditions for the Rees algebra of its modules of differentials to be Cohen-Macaulay.  slides 

6. Cohen-Macaulay fiber cones and defining ideal of Rees algebras of modules, Women in Commutative Algebra - Proceedings of the 2019 WICA Workshop, Association for Women in Mathematics Series, vol. 29, Springer (2022), arxiv:2011.08453 

Extending previous work of Simis, Ulrich and Vasconcelos, we introduce methods to study the Cohen-Macaulay property of fiber cones of modules and to determine the defining ideal of Rees algebras of modules.  notes

7. Residual intersections and modules with Cohen-Macaulay Rees algebras, J. Algebra 587 (2021), 36-63,  arxiv:1811.08402 

We identify classes of modules with Cohen-Macaulay Rees algebras, generalizing a well-known result of Johnson and Ulrich on Rees algebras of ideals with good residual intersection properties.  slides 

Submitted for publication

8. (with Alexandra Seceleanu)  The combinatorial structure of symmetric strongly shifted ideals, arxiv:2208.10476.

Symmetric strongly shifted ideals are a class of monomial ideals which are invariant under permutation of the variables. We study the combinatorial and algebraic properties of ideals of this kind, highlighting similarities with the classes of strongly stable and polymatroidal ideals. We also prove that ideals in this class are associated with polytopes which are the convex hull of unions of permutohedra. slides

Ph.D. Thesis 

Rees algebras and fiber cones of modules doi.org/10.25394/PGS.9107936.v1