## Research Articles.

The following is a list of papers that I have *completed and submitted *(it does not contain work in progress). After the list, you will find a brief presentation for each paper.

- 13.
*The projective dimension of three cubics is at most 5*, submitted. (with J. McCullough) *12. A tight bound on the projective dimension of 4 quadrics*, to appear on J. Pure and Applied Algebra. arXiv. (with C. Huneke, J. McCullough and A. Seceleanu)*11. A finite classification of (x,y)-primary ideals of low multiplicity*, Collect. Math. 69 (2018), 107-130. (with J. McCullough) This is the Macaulay2 file for the computations of section 4 of the paper.*10. Chudnovsky’s conjecture for very general points in P^N*, J. Algebra 498 (2018), 211-227. arXiv. (with L. Fouli and Y. Xie)*9. Hypergraphs with high projective dimension and 1-dimensional Hypergraphs*, Internat. J. Algebra and Comput. 27 (2017), 591-617. arXiv. (with K.-N. Lin)*8. Arithmetical Rank of strings and cycles*, J. Commut. Algebra 9 (2017), 89-106. arXiv. (with K. Kimura)*7. Multiple structures with arbitrarily large projective dimension supported on linear subspaces*, J. Algebra 447 (2016), 183-205. arXiv. (with C. Huneke, J. McCullough and A. Seceleanu)*6. Projective dimension of string and cycle hypergraphs*, Comm. Algebra 44 (2016), 1671-1694. arXiv. (with K.-N. Lin)-
*5. Generalized Stretched Ideals and Sally’s Conjecture*, J. Pure and Appl. Algebra 200 (2016), 1157-1177. arXiv. (with Y. Xie) - 4. A multiplicity bound and a criterion for the Cohen-Macaulay property, Proc. Amer. Math. Soc. 143 (2015), 2365-2377. Arxiv. (with C. Huneke, J. McCullough and A. Seceleanu)
*3. Ubiquity of liaison classes*, Illinois J. Math 58 (2014), 647-663. arXiv. (with M. Johnson)*2. The projective dimension of codimension two algebras presented by quadrics*, J. Algebra 393 (2013), 170-186. arXiv. (with C. Huneke, J. McCullough and A. Seceleanu)*1. On the Cohen-Macaulayness of the conormal module of an ideal*, J. Algebra 372, (2012), 35-55. arXiv.

(with Y. Xie)

Here is a brief introduction to each paper:

*1. On the Cohen-Macaulayness of the conormal module of an ideal*, J. Algebra 372, (2012), 35-55. arXiv.

(with Y. Xie)

**Summary:** my first paper! We worked on it at the MRC, where we received very good suggestions from C. Polini and C. Huneke. The counterexample was found by J. C. Migliore after discussing with him about the results. The main tool (Hilbert function) and the topic (conormal module) will appear in other papers.

**Abstract**: A long-standing question of W. Vasconcelos asks the following: given a generically complete intersection perfect ideal I in a regular local ring R, when does the Cohen–Macaulay property of the conormal module I / I^2 of I implies that R/I is Gorenstein? A positive answer is known for prime ideals of height 2 (work of Herzog), licci ideals (work of Huneke and Ulrich) and squarefree monomial ideals (work of Rinaldo, Terai and Yoshida). In the present paper we provide a positive answer when I is a monomial ideal, or R/I defines a stretched algebra, or R/I defines a short algebra with socle degree at least 3, or socle degree 2 provided the multiplicity satisfies some numerical conditions, or when the multiplicity of R/I is small (at most the embedding codimension + 4). We then show that for an ideal of 10 general points in P^5 Vasconcelos’ question has a negative answer; this example shows that the last two parts of our main result are sharp.

*2. The projective dimension of codimension two algebras presented by quadrics *, J. Algebra 393 (2013), 170-186. arXiv. (with C. Huneke, J. McCullough and A. Seceleanu)

**Summary:** during my semester-long membership at the MSRI in 2012, I had multiple mathematical discussions with C. Huneke, J. McCullough and A. Seceleanu regarding the difficulty of proving sharp bounds for Stillman’s conjecture, even in “small” cases. This paper is the first of a series of 4 manuscripts that we wrote as a result of our attempts to fill this gap.

**Abstract:** Motivated by a celebrated conjecture of Stillman (proved in 2017 by Ananyan and Hocster), we prove that if I is an ideal in a polynomial ring R in an unspecified number of variables and I is generated by n quadrics (=homogeneous equations of degree 2), and the height of the ideal is 2, then the projective dimension of R/I is bounded above by 2n-2. We also prove that this bound is sharp. Moreover, we suggest possible more general tight upper bounds for ideals I generated by quadrics purely in terms of the number of quadrics and the height of I.

*3. Ubiquity of liaison classes*, Illinois J. Math 58 (2014), 647-663. arXiv.

(with M. Johnson)

**Summary:** in Fall 2012 at the MSRI, J.. C. Migliore and U. Nagel presented the following interesting result: if I is a Cohen-Macaulay ideal in R that is generically Gorenstein and X is a new variable, then I+(X) is glicci in R[x]. After discussing with C. Huneke and B. Ulrich about the lack of analogous results for CI-linkage, I worked with M. Johnson (with whom I share a strong interest for liaison) to introduce a new invariant that would allow us to fill this gap.

**Abstract:** While licci ideals (=ideals in the CI-liaison class of a complete intersection) have caught much attention in the last 30 years, very little is known about the (CI)-liaison classes of non-licci ideals. In the present paper we introduce a new invariant of a CI-liaison class, study its properties (especially with respect to localization, faithfully flat extensions, join and hyperplane sections) and employ it to distinguish many liaison classes. For instance, we show that there are at least as many CI-liaison classes of codimension (c+3) subschemes in (n+5)-dimensional projective space as there are generic complete intersections arithematically Cohen-Macaulay subschemes of codimension c in P^n. As another application, we show the following surprising result: let I be an unmixed ideal in a Gorenstein local ring (with infinite residue field), and X and Y are new variables. Then the ideals I+(X) and I+(Y) are in same CI-linkage classes if and only if I is licci (in stark contrast with the same situation under G-liaison, see the Summary above).

*4. A multiplicity bound and a criterion for the Cohen-Macaulay property*, Proc. Amer. Math. Soc. 143 (2015), 2365-2377. arXiv. (with C. Huneke, J. McCullough and A. Seceleanu)

**Summary:** this is the second paper by the same group of authors. The original purpose was to generalize an inequality proved by B. Engheta and employ it to reduce the number of cases to examine when proving sharp bounds for Stillman’s conjecture. We soon realized that ideals of maximal multiplicity have an interesting structure and high depth.

**Abstract**: Let R be a polynomial ring over a field, J a homogeneous ideal and F a homogeneous element not in J. It is well-known that the multiplicity e(R/J+(F)) equals e(R/J)e(R/(F)) provided that F is regular on R/J. What can one say when F is a zero-divisor on R/J? In the present paper we answer this question by proving that in this scenario e(R/J+(F)) is at most e(R/J) – max\\\\{1, s(J) – deg(F) + 1} where s(J) is an homological invariant of J. We show that this bound is sharp and the ideals for achieving this upper bound have high depth. As an application we provide a sufficient numerical condition (on the multiplicity) for a quasi-Gorenstein ring to be Gorenstein.

*5. Generalized Stretched Ideals and Sally’s Conjecture*, J. Pure and Appl. Algebra 200 (2016), 1157-1177. arXiv. (with Y. Xie)

**Summary:** my second paper with Y. Xie. Common interests on linkage, residual intersections multiplicities, Hilbert functions and Sally’s conjecture led us naturally to join forces to detect a class of ideals that would widely generalize several results about stretched ideals and ideals with almost minimal (j-)multiplicity. It is one of my most technical papers.

**Abstract**: Let R be a d-dimension Cohen-Macaulay ideal and I an ideal of R; let G be the associated graded ring of R with respect to I. We introduce the notion of j-stretched ideals which generalize stretched m-primary ideals (as defined by Rossi and Valla) and ideals of almost minimal j-multiplicity (as defined by Polini and Xie). In one of the main results of this papert we prove that if I is j-stretched and a general minimal reduction of I satisfies certain Valabrega-Valla type of assumptions and some numerical conditions, then G is an almost Cohen-Macaulay ring. This provides a proof of a generalization of Sally’s conjecture. The second main theorem states that G is Cohen-Macaulay if and only if a simple numerical condition (reduction number equals the index of nilpotency) is met. The theory we develop unifies work of Rossi-Valla and Polini-Xie, and generalizes results by a number of authors.

*6. Projective dimension of string and cycle hypergraphs*, Comm. Algebra 44 (2016), 1671-1694. arXiv. (with K.-N. Lin)

**Summary:** My first paper in Combinatorial Commutative Algebra!

**Abstract:** Rinaldo, Terai and Yoshida in 2009 introduced a construction associated a (dual separated) hypergraph H to any squarefree monomial ideal J and used this construction to classify combinatorially the ideals J with high projective dimension, namely pd(R/J) being at least b(J) – 1 where b(J) is the minimal number of generators of J. Already when the hypergraph H is 1-dimensional the computation of pd(R/J) for the associated monomial ideal J is complicated. In this paper, we provide a combinatorial formula to compute pd(R/J) when the associated hypergraph is either a string or a cycle. The key point is the definition of a combinatorial invariant detecting the presence of certain particular configurations of open/closed vertices in H.

*7. Multiple structures with arbitrarily large projective dimension supported on linear subspaces*, J. Algebra 447 (2016), 183-205. arXiv. (with C. Huneke, J. McCullough and A. Seceleanu)

**Summary:** this is the third paper by the same group of authors. In this paper we employ linkage effectively to construct families of ideals with prescribed multiplicity, projective dimension and primary decomposition. The purpose is to show that a natural attempt to construct tight bounds for Stillman’s conjecture fail because of the lack of a finite a classification of certain primary ideals. Our results are even stronger. Fortunately, in paper 11. below with J. McCullough we manage to overcome this difficulty for (x,y)-primary ideals of low multiplicity.

** ****Abstract:** Let R be a polynomial ring over a field and J an unmixed ideal. If e(R/J) =1, then Samuel proved that J is a linear prime. If e(R/J)=2 and ht(J) =2, Engheta proved that there are precisely 5 possible normal forms for J. One may hope that there is also a finite classification when ht(J)=3 and e(R/J)=2, or ht(J)=2 and e(R/J)=3. However, we prove in this paper that this is not the case, even under the stronger assumption that J is (x,y,z)-primary (or (x,y)-primary) in a very strong sense.

*8. Arithmetical Rank of strings and cycles*, J. Commut. Algebra 9 (2017), 89-106. arXiv. (with K. Kimura)

**Summary:** An evolution of paper 6. above arising from email communications with Prof. Kimura.

**Abstract:** It is known that the arithmetical rank ara(J) of a squarefree monomial ideal J is bounded above by its projective dimension pd(R/J). In the present paper we prove that equality holds for the ideals J associated to the hypergraphs introduced in paper 6. above. The proof is constructive and in fact shows that J can be generated up to radicals by trinomials.

*9. Hypergraphs with high projective dimension and 1-dimensional Hypergraphs*, Internat. J. Algebra and Comput 27 (2017), 591-617. arXiv.(with K.-N. Lin)

**Summary:** The natural continuation of paper 6. above.

**Abstract:** Rinaldo, Terai and Yoshida in 2009 classified combinatorially the ideals J whose projective dimension pd(R/J) is > b(J) -2, where b(J) is the minimal number of generators of J. Understanding precisely when pd(R/J) is >b(J) -3 appears pretty challenging; in the present paper we prove sufficient and necessary conditions for the inequality pd(R/J) > b(J) -3. As an application, we prove that pd(R/J) = b(J) -2 when the 1-dimensional skeleton of the hypergraph associated to H has a spanning Ferrer subgraph. Moreover, we provide a non-trivial “divide-and-conquer” method allowing to compute pd(R/J) when the hypergraph H associated to J is 1-dimensional and belongs to a wide class, namely the class of hypergraphs that are disjoint union of trees and 1-dimensional hypergraphs containing at most 1 cycle. To achieve it, we introduce refined combinatorial invariants generalizing the one introduced in paper 6 above.

*10. Chudnovsky’s conjecture for very general points in P^N*, J. Algebra 498 (2018), 211-227. arXiv. (with L. Fouli and Y. Xie)

**Summary:** Together with Yu and Louiza, we worked for quite some time to develop the ideas and techniques of this paper, but the result was very rewarding! This is my first paper entirely dedicated to points in projective space and symbolic powers, two research topics to which I dedicated many energies very recently.

**Abstract:** A long-standing conjecture raised by G. V. Chudnovsky in 1981 predicts a sharp lower bound on the smallest degree D of an hypersurface passing through a finite set X of points at least m times in the n-dimensional projective space. The bound depends on n and the smallest degree of an hypersurface through X (at least once). When n=2 the conjecture was proved by Chudnovsky and re-proved by Harbourne and Huneke (2011). When n=3 the conjecture was proved by Dumnicki for *general* points in 2012. For arbitrary n>3, very little was known. In the present paper we prove Chudnovsky’s conjecture for *very general* points, or generic points, or any configuration of points lying on a quadric in any n-dimensional projective space. We also suggest a generalization of the conjecture to any homogeneous ideal J in polynomial rings. To this end, we provide some evidence by proving Chudnovsky’s conjecture for sufficiently large symbolic powers of any given ideal J.

*11. A finite classification of (x,y)-primary ideals of low multiplicity*, to appear in Collect. Math. (with J. McCullough) – This is the Macaulay2 file for the computations of section 4 of the paper.

**Summary:** for a few years Jason and I worked to prove the tight version of Stillman’s conjecture for 3 cubics, i.e. pd(R/I) is at most 5 when I is generated by three homogeneous equations of degree 3. One of the key ingredients would be a finite classification of (x,y)-primary ideals of low multiplicity…except that we proved in paper 7. above that there is no such classification. In this paper we show that the fact that something is impossible does not mean you should give up; by adding a restriction on the degrees of the generators, we prove a finite classification!

**Abstract: **Let R be a polynomial ring over a field in an unspecified number of variables and J an (x,y)-primary ideal. We prove that if e(R/J) = 3, then there are 8 possible “normal forms” for the equations of J of degree at most 3. If e(R/J) = 4, we prove that there are 23 possible “normal forms” for the equations of J of degree at most 3. We mploy these classification theorems to prove under additional assumptions that if I is generated by 3 cubics, then pd(R/I) is at most 5 (see also paper 13. below).

*12. A tight bound on the projective dimension of 4 quadrics*, to appear on J. Pure and Applied Algebra. arXiv. (with C. Huneke, J. McCullough and A. Seceleanu)

**Summary:** this is the fourth paper by the same group of authors written to prove tight bounds on the projective dimension in special cases of Stillman’s conjecture.

**Abstract:** Let R be a polynomial ring over a field in an unspecified number of variables. We prove that if I is an ideal generated by four quadratic homogeneous equations, then pd(R/I) is at most 6 and this bound is sharp. A number of techniques, classification results, “normal forms” for certain ideals, and linkage statements are employed to achieve this result.

*13. The projective dimension of three cubics is at most 5*, submitted. (with J. McCullough)

**Summary:** after a few years of working on it, Jason and I prove the conjectured tight bound pd(R/I) is at most 5 when I is generated by 3 cubics. I am very proud of this result! When we started, we could prove the non-sharp bound pd(R/I) is at most 12 in about 5 pages (thanks to previous work developed by Engheta and in our previous papers). Proving that pd(R/I) is at most 7 would require about 30 pages; and improving it to pd(R/I) at most 6 requires us 40 pages. Proving the actual sharp bound required us about 55 pages of math (paper 11. above + this paper).

**Abstract:** Let R be a polynomial ring over a field. Engheta in a series of 3 papers proved that if I is generated by 3 cubic equations, then pd(R/I) is at most 36. However, he conjectured the sharp bound to be 5. In the present paper we prove this 10-year old conjecture. Our techniques rely heavily on the previous work of Engheta, the classification theorems proved in paper 11. above, a few techniques from the papers on the quadrics, a number of linkage arguments, normal forms for ideals, Hilbert functions and their “local” versions, and projective dimension considerations of various forms.