Research

Publication list:

Rees algebras of sparse determinantal ideals, with E. Celikbas, E. Dufresne, L. Fouli, E. Gorla, C. Polini, and I. Swanson. Transactions of the American Mathematical Society, (2024). https://doi.org/10.1090/tran/9101 https://arxiv.org/pdf/2101.03222.pdf
Abstract: We determine the defining equations of the Rees algebra and of the special fiber ring of the ideal of maximal minors of a $2\times n$ sparse matrix. We prove that their initial algebras are ladder determinantal rings. This allows us to show that the Rees algebra and the special fiber ring are Cohen-Macaulay domains, they are Koszul, they have rational singularities in characteristic zero, and are F-rational in positive characteristic.
Rees Algebras of unit interval Determinantal Facet Ideals, with A. Almousa and W. Liske, Journal of Pure and Applied Algebra Vol 228, No 2, (2024). https://doi.org/10.1016/j.jpaa.2023.107601 https://arxiv.org/pdf/2008.10950.pdf.
Abstract: Using SAGBI basis techniques, we find Gr\”obner bases for the presentation ideals of the Rees algebras and special fiber rings of unit interval determinantal facet ideals. In particular, we show that unit interval determinantal facet ideals are of fiber type and that their special fiber rings are Koszul. Moreover, their Rees algebras and special fiber rings are normal Cohen-Macaulay domains and have rational singularities.
Multi-Rees Algebras of Strongly Stable Ideals, with S. Kara and G. Sosa, to appear Collectanea Mathematica Vol 74 (2024). 213–246. https://link.springer.com/epdf/10.1007/s13348-022-00385-2. https://arxiv.org/pdf/2012.15447.pdf
Abstract: We prove that the multi-Rees algebra $\mathcal{R}(I_1 \oplus \cdots \oplus I_r)$ of a collection of strongly stable ideals $I_1, \ldots, I_r$ is of fiber type. In particular, we provide a Gr\”obner basis for its defining ideal as a union of a Gr\”obner basis for its special fiber and binomial syzygies. We also study the Koszulness of $\mathcal{R}(I_1 \oplus \cdots \oplus I_r)$ based on parameters associated to the collection. Furthermore, we establish a quadratic Gr\”obner basis of the defining ideal of $\mathcal{R}(I_1 \oplus I_2)$ where each of the strongly stable ideals has two quadric Borel generators. As a consequence, we conclude that this multi-Rees algebra is Koszul.
Regularity and Multiplicity of Toric Rings of Three-dimensional Ferrers Diagrams, with Y.-H. Shen, Journal of Algebraic Combinatorics Vol. 57 (2023), 1073-1101, https://link.springer.com/article/10.1007/s10801-023-01217-7. https://arxiv.org/pdf/1809.08351.pdf.
Abstract: We investigate the Castelnuovo–Mumford regularity and the multiplicity of the toric ring associated to a three-dimensional Ferrers diagram. In particular, in the rectangular case, we are able to provide direct formulas for these two important invariants. Then, we compare these invariants for an accompanied pair of Ferrers diagrams under some mild conditions, and bound the Castelnuovo–Mumford regularity for more general cases.
Symbolic Powers and Free Resolutions of Generalized Star Configurations of Hypersurfaces, with Y.-H. Shen, Michigan Mathematical Journal Vol. 73 (2023), 33–66. https://arxiv.org/abs/1912.04448
Abstract. As a generalization of the ideals of star configurations of hypersurfaces, we consider the a-fold product ideal I_a(f_1^{m_1} ···f__s^{m_s}) when f_1,…,f_s is a sequence of generic forms and 1 ≤ a ≤ m_1 +· · ·+m_s. Firstly, we show that this ideal has complete intersection quotients when these forms are of the same degree and essentially linear. Then we study its symbolic powers while focusing on the uniform case with m_1 = ··· = m_s. For large a, we describe its resurgence and symbolic defect. And for general a, we also investigate the corresponding invariants for meeting-at-the-minimal-components version of symbolic powers.
Blow-up algebras of secant varieties of rational normal scrolls, with Y.-H. Shen, Collectanea Mathematica Vol. 74 (2023), 247-278. https://doi.org/10.1007/s13348-021-00345-2 https://arxiv.org/pdf/2107.04168.pdf
Abstract: In this paper, we are mainly concerned with the blow-up algebras of the secant varieties of balanced rational normal scrolls. In the first part, we give implicit defining equations of their associated Rees algebras and fiber cones. Consequently, we can tell that the fiber cones are Cohen–Macaulay normal domains. Meanwhile, these fiber cones have rational singularities in characteristic zero, and are F -rational in positive characteristic. The Gorensteinness of the fiber cones can also be characterized. In the second part, we compute the Castelnuovo–Mumford regularities and a-invariants of the fiber cones. We also present the reduction numbers of the ideals defined by the secant varieties.
Lattices and Hypergraphs Associated to Squarefree Monomial Ideals, with S. Mapes, Communications in Algebra, Vol. 50 (2022), 4710-4724.

DOI: 10.1080/00927872.2022.2072853 https://www.tandfonline.com/doi/epub/10.1080/00927872.2022.2072853?needAccess=true

https://arxiv.org/pdf/1804.05919.pdf

Abstract: Given a square-free monomial ideal I in a polynomial ring R over a field K, one can associate it with its LCM-lattice and its hypergraph. In this short note, we establish the connection between the LCM-lattice and the hypergraph, and in doing so we provide a sufficient condition for removing higher dimension edges of the hypergraph without impacting the projective dimension of the square-free monomial ideal. We also offer algorithms to compute the projective dimension of a class of square-free monomial ideals built using the new result and previous results of Lin-Mantero.
Fiber cones of rational normal scrolls are Cohen–Macaulay, with Y.-H. Shen, Journal of Algebraic Combinatorics, 56, pages 547–563 (2022). https://doi.org/10.1007/s10801-022-01123-4

https://arxiv.org/pdf/2009.03484.pdf.

Abstract: In this paper, we are mainly concerned with the blow-up algebras of the secant varieties of balanced rational normal scrolls. In the first part, we give implicit defining equations of their associated Rees algebras and fiber cones. Consequently, we can tell that the fiber cones are Cohen–Macaulay normal domains. Meanwhile, these fiber cones have rational singularities in characteristic zero, and are F -rational in positive characteristic. The Gorensteinness of the fiber cones can also be characterized. In the second part, we compute the Castelnuovo–Mumford regularities and a-invariants of the fiber cones. We also present the reduction numbers of the ideals defined by the secant varieties.
Toric Ideals of Weighted Oriented Graphs, with J. Biermann, S. Kara, and A. O’Keefe, International Journal of Algebra and Computation, Vol. 32, No. 02, (2022) 307-325. https://doi.org/10.1142/S0218196722500151. https://arxiv.org/abs/2107.04524
Abstract: Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which depends on the combinatorial structure and weights of the graph. We provide a result analogous to the unweighted, unoriented graph case, to show that when the associated simple graph has only trivial even closed walks, the toric ideal is the zero ideal. Moreover, we give necessary and sufficient conditions for the toric ideal of a weighted oriented graph to be generated by a single binomial and we describe the binomial in terms of the structure of the graph.
Projective Dimension of Hypergraphs, with S. Mapes, Women in
Commutative Algebra Proceedings of the 2019 WICA Workshop, (2022) 369-398. https://arxiv.org/pdf/1910.01053.pdf.
Abstract: Given a square-free monomial ideal $I$, satisfying certain hypotheses, in a polynomial ring $R$ over a field $\mathbb{K}$, we compute the projective dimension of $I$. Specifically, we focus on the cases where the 1-skeleton of an associated hypergraph is either a string or a cycle. We investigate the impact on the projective dimension when higher dimensional edges are removed. We prove that the higher dimensional edge either has no effect on the projective dimension or the projective dimension only goes up by one with the extra higher dimensional edge.
Algebraic Invariants of Weighted Oriented Graphs, with S. Beyarslan, J. Biermann, and A. O’Keefe, Journal of Algebraic Combinatorics 55, (2022) 461-491. 10.1007/s10801-021-01058-2, https://arxiv.org/pdf/1910.11773.pdf.
Abstract: Let D be a weighted oriented graph and let I(D) be its edge ideal in a polynomial ring R. We give the formula of Castelnuovo-Mumford regularity of R/I(D) when D is a weighted oriented path or cycle such that edges of D are oriented in one direction. Additionally, we compute the projective dimension for this class of graphs.
Symbolic powers of generalized star configurations of hypersurfaces, with Y.-H. Shen, Journal of Algebra, Vol. 593 (2022) 193–216. https://arxiv.org/pdf/2106.02955.pdf
Abstract: We introduce the class of sparse symmetric shifted monomial ideals. These ideals have linear quotients and their Betti numbers are computed. Using this, we prove that the symbolic powers of the generalized star configuration ideal are sequentially Cohen–Macaulay under some mild genericness assumption. With respect to these symbolic powers, we also consider the Harbourne–Huneke containment problem and establish the Demailly-like bound.
On the Conjecture of Vasconcelos for Artinian Almost Complete Intersection Monomial Ideals, with Y.-H. Shen, Nagoya Mathematical Journal. Vol 243 (2021) 263-277. https://arxiv.org/pdf/1902.03068.pdf.
Abstract: In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen–Macaulay.
Generalized Newton Complementary Duals of Monomial Ideals, with K. Ansaldi and Y.-H. Shen, Journal of Algebra and its Applications Vol. 20, No. 2 (2021). https://arxiv.org/pdf/1702.00519.pdf.
Abstract: Given a monomial ideal in a polynomial ring over a field, we define the generalized Newton complementary dual of the given ideal. We show good properties of such duals including linear quotients and isomorphisms between the special fiber rings. We construct the cellular free resolutions of duals of strongly stable ideals generated in the same degree. When the base ideal is generated in degree two, we provide an explicit description of cellular free resolution of the dual of a compatible generalized stable ideal.
Multi-Rees Algebras and Toric Dynamical Systems, with D. Cox and G. Sosa, Proceedings of the American Mathematical Society, Vol. 147, No. 11 (2019) 4605-4616. http://www.ams.org/journals/.
Abstract: This paper explores the relation between multi-Rees algebras and ideals that arise in the study of toric dynamical systems from the theory of chemical reaction networks.
Edge Ideals of Oriented Graphs, with T. Hà, S. Morey, E. Reyes, and R. Villarreal, International Journal of Algebra and Computation Vol. 29, No. 3 (2019) 535–559. https://arxiv.org/pdf/1805.04167.pdf.
Abstract: Let D be a weighted oriented graph and let I(D) be its edge ideal. Under a natural condition that the underlying (undirected) graph of D contains a perfect matching consisting of leaves, we provide several equivalent conditions for the Cohen-Macaulayness of I(D). We also completely characterize the Cohen-Macaulayness of I(D) when the underlying graph of D is a bipartite graph. When I(D) fails to be Cohen-Macaulay, we give an instance where I(D) is shown to be sequentially Cohen-Macaulay.
Koszul Blow-up Algebras Associated to Three-dimensional Ferrers Diagrams, with Y.-H. Shen, Journal of Algebra, Vol. 514 (2018) 219-253. https://arxiv.org/pdf/1709.03251.pdf.
Abstract: We investigate the Rees algebra and the toric ring of the squarefree monomial ideal associated to the three-dimensional Ferrers diagram. Under the projection property condition, we describe explicitly the presentation ideals of the Rees algebra and the toric ring. We show that the toric ring is a Koszul Cohen–Macaulay normal domain, while the Rees algebra is Koszul and the defining ideal is of fiber type.
Hypergraphs with High Projective Dimension and 1-dimensional Hypergraphs, with P. Mantero, International Journal of Algebra and Computation, Vol. 27, No. 6 (2017) 591-617. https://arxiv.org/pdf/1603.01331.pdf.
Abstract: We prove a sufficient and a necessary condition for a square-free monomial ideal J associated to a (dual) hypergraph to have projective dimension equal to the minimal number of generators of J minus 2. We also provide an effective explicit procedure to compute the projective dimension of 1-dimensional hypergraphs H when each connected component contains at most one cycle. An algorithm to compute the projective dimension is also included. Applications of these results are given; they include, for instance, computing the projective dimension of monomial ideals whose associated hypergraph has a spanning Ferrers graph.
Cohen-Macaulayness of Rees Algebra of Modules, Communications in Algebra, Vol. 44 (2016) 3673-3682. https://arxiv.org/pdf/1502.06584.pdf.
Abstract: We provide the sufficient conditions for Rees algebras of modules to be Cohen-Macaulay, which has been proven in the case of Rees algebras of ideals in [10] and [4]. As it turns out the generalization from ideals to modules is not just a routine generalization, but requires a great deal of technical development. We use the technique of generic Bourbaki ideals introduced by Simis, Ulrich and Vasconcelos [12] to obtain the Cohen-Macaulayness of Rees Algebras of modules.
Projective Dimension of String and Cycle Hypergraphs, with P. Mantero, Communications in Algebra, Vol. 44 (2016) 1671-1694. https://arxiv.org/pdf/1309.7948.pdf.
Abstract: We present a closed formula and a simple algorithmic procedure to compute the projective dimension of square-free monomial ideals associated to string or cycle hypergraphs. As an application, among these ideals we characterize all the Cohen-Macaulay ones.
Normal 0-1 Polytopes, with T. Hà, SIAM Journal on Discrete Mathematics, Vol. 1 No. 1 (2015) 210-223. https://arxiv.org/pdf/1309.4807.pdf.
Abstract: We study the question of when 0-1 polytopes are normal or, equivalently, having the integer decomposition property. In particular, we shall associate to each 0-1 polytope a labeled hypergraph, and examine the equality between its Ehrhart and polytopal rings via the combinatorial structures of the labeled hypergraph.
Rees Algebras of Square-Free Monomial Ideals, with L. Fouli, Journal of Commutative Algebra, Vol. 7, No. 1 (2015) 25-54. https://arxiv.org/pdf/1205.3127.pdf.
Abstract: We study the defining equations of the Rees algebras of square-free monomial ideals in a polynomial ring over a field. We propose the construction of a graph, namely the generator graph of a monomial ideal, where the monomial generators serve as vertices for the graph. When I is a square-free monomial ideal such that each connected component of the generator graph of I has at most 5 vertices then I has relation type at most 3. In general, we establish the defining equations of the Rees algebra in this case and give a combinatorial interpretation of them. Furthermore, we provide new classes of ideals of linear type. We show that when I is a square-free monomial ideal and the generator graph of I is the graph of a disjoint union of trees and graphs with a unique odd cycle, then I is an ideal of linear type.
Cohen-Macaulayness of Rees Algebras of Diagonal Ideals, Journal of Commutative Algebra, Vol. 6, No. 4 (2014) 3561-586. https://arxiv.org/pdf/1106.0741.pdf.
Abstract: Given two determinantal rings over a field k. We consider the Rees algebra of the diagonal ideal, the kernel of the multiplication map. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the join variety. When the Rees algebra and the Symmetric algebra coincide, we show that the Rees algebra is Cohen-Macaulay.
Rees Algebras of Truncations of Complete Intersections, with C. Polini, Journal of Algebra, Vol. 410 (2014) 36-52. https://reader.elsevier.com.
Abstract: In this paper we describe the defining equations of theRees algebra and the special fiber ring of a truncation of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal m. To describe explicitly the Rees algebra R(I) in terms of generators and relations we map another Rees ring R(M) on to it, where M is the direct sum of powers of m. We compute a Gröbner basis of the ideal defining R(M). It turns out that the normal domain R(M) is a Koszul algebra and from this we deduce that in many instances R(I) is a Koszul algebra as well.
Hypergraphs and Regularity of Square-Free Monomial Ideals, with J. McCullough, International Journal of Algebra and Computation, Vol. 23, No. 7 (2013) 1573-1590. https://arxiv.org/pdf/1211.4301.pdf.
Abstract: We define a new combinatorial object, which we call a labeled hypergraph, uniquely associated to any square-free monomial ideal. We prove several upper bounds on the regularity of a square-free monomial ideal in terms of simple combinatorial properties of its labeled hypergraph. We also give specific formulas for the regularity of square-free monomial ideals with certain labeled hypergraphs. Furthermore, we prove results in the case of one-dimensional labeled hypergraphs.
Rees Algebras of Diagonal Ideals, Journal of Commutative Algebra, Vol. 5, No. 3 (2013) 329-475. https://projecteuclid.org/.
Abstract: There is a natural epimorphism from the symmetric algebra to the Rees algebra of an ideal. When this epimorphism is an isomorphism, we say that the ideal is of linear type. Given two determinantal rings over a field, we consider the diagonal ideal, kernel of the multiplication map. We prove in many cases that the diagonal ideal is of linear type and recover the defining ideal of the Rees algebra. In our cases, the special fiber rings of the diagonal ideals are the homogeneous coordinate rings of the join varieties.
Regularities and multiplicities of Veronese type algebras, with Y.-H. Shen, submitted, 2023. https://arxiv.org/pdf/2305.01859.pdf
Abstract: In this paper, we study the algebra of Veronese type. We show that the presentation ideal of this algebra has an initial ideal whose Alexander dual has linear quotients. As an application, we explicitly obtain the Castelnuovo-Mumford regularity of the Veronese type algebra. Furthermore, we give an effective upper bound on the multiplicity of this algebra.
On Cohen–Macaulay modules over the Weyl algebra, with J.-C. Hsiao, submitted, 2023. https://arxiv.org/pdf/2309.05864.pdf
Abstract: We propose a definition of Cohen–Macaulay modules over the Weyl algebra $D$ and give a sufficient condition for a GKZ $A$-hypergeometric $D$-module to be Cohen–Macaulay.

Works in Progress

Normality of Weighted Oriented Graphs, with J. Biermann, S. Kara, and A. O’Keefe.
Rees Algebras of Almost Complete Intersection Determinantal Facet Ideals, with A.
Almousa and W. Liske.
Multi-Rees Algebras of Lexsegment Ideals, with A. Costantini, and G. Sosa.

Kuei-Nuan Lin

Research

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