UNL campus

Alexandra Seceleanu

Assistant Professor

Teaching | Research and Publications | Selected Talks | Software | Links

Curriculum vitae

Contact Information




338 Avery Hall


(402) 472-7253


Department of Mathematics
203 Avery Hall
Lincoln, NE 68588

About Me 

I am an assistant professor of mathematics at the University of Nebraska-Lincoln. My research focuses on commutative algebra with a geometric and computational flavor.

I have been awarded the 2018 Harold & Esther Edgerton Junior Faculty Award for creative research, extraordinary teaching abilities, and academic promise.

I co-organize the Commutative Algebra Seminar at UNL.

My Students

PhD students:

  • Ben DrabkinOutstanding First Year Student award (2015-2016), G.C. and W.H. Young award (2017-2018)
  • Andrew Connertwice SCIENCE SLAM finalist (2018 and 2019)
  • Erica Musgrave (co-advised with Mark Walker) - NPSC fellow (2016-2021), Outstanding First Year Student award (2017-2018)
  • Michael DeBellevue (co-advised with Mark Walker) – Outstanding Qualifying Exams award (2018-2019)

Undergraduate students:

  • Joey BeckerChair’s prize for best graduating math major (2015)
  • Diana (Xuehua) Zhong graduate student at North Carolina State University (since 2018)

Undergraduate Activities

I am a faculty advisor for the UNL Math Club and the organizer of the new lecture series Career Perspectives in Mathematics.

In April 2018 I organized the Central States Mathematics Undergraduate Research Conference (CeSMUR 2018) at UNL. CeSMUR 2019 will take place at Kansas State University.

Recent and upcoming conferences

Research and Publications

My research is in commutative algebra, with an interest in computational algebra, homological methods and connections to algebraic geometry.

1.    Connected sums of graded artinian Gorenstein algebras and Lefschetz properties (with A. Iarrobino and C. McDaniel)

A connected sum construction for local rings was introduced in a paper by H. Ananthnarayan, L. Avramov, and W.F. Moore. In the graded Artinian Gorenstein case, this can be viewed as an algebraic analogue of the topological construction of the same name. We give two alternative description of this algebraic connected sum: the first uses algebraic analogues of Thom classes of vector bundles and Gysin homomorphisms, the second is in terms of Macaulay dual generators. We also investigate the extent to which the connected sum construction preserves the weak or strong Lefschetz property.

2.    GMD functions for scheme-based linear codes and algebraic invariants of Geramita ideals (with S. Cooper, Ș. Tohăneanu, M. Vaz Pinto and R. Villarreal)

Motivated by notions from coding theory, we study the generalized minimum distance (GMD) function of a graded ideal I in a polynomial ring over an arbitrary field using commutative algebraic methods. It is shown that the GMD function is non-decreasing as a function of its first argument and non-increasing as a function of the second argument. For vanishing ideals over finite fields, we show that the GMD function is in fact strictly decreasing as a function of the second argument until it stabilizes. We also study algebraic invariants of Geramita ideals. Those ideals are graded, unmixed, 1-dimensional and their associated primes are generated by linear forms. We also examine GMD functions of complete intersections and show some special cases of two conjectures of Tohăneanu-Van Tuyl and Eisenbud-Green-Harris.

3.    Frieze varieties: A characterization of the finite-tame-wild trichotomy for acyclic quivers (with K. Li, L. Li, M. Mills and R. Schiffler)

We introduce a new class of algebraic varieties which we call frieze varieties. Each frieze variety is determined by an acyclic quiver. The frieze variety is defined in an elementary recursive way by constructing a set of points in affine space. From a more conceptual viewpoint, the coordinates of these points are specializations of cluster variables in the cluster algebra associated to the quiver. We give a new characterization of the finite-tame-wild trichotomy for acyclic quivers in terms of their frieze varieties by showing that an acyclic quiver is representation finite, tame, or wild, respectively, if and only if the dimension of its frieze variety is 0,1, or ≥2, respectively.

4.    Computations involving symbolic powers (with B. Drabkin, E. Grifo, B. Stone) and the Macaulay2 package SymbolicPowers

J. Software for Algebra and Geometry

Symbolic powers are a classical commutative algebra topic that relates to primary decomposition, consisting, in some circumstances, of the functions that vanish up to a certain order on a given variety. However, these are notoriously difficult to compute, and there are seemingly simple questions related to symbolic powers that remain open even over polynomial rings. In this paper, we describe a Macaulay2 software package that allows for computations of symbolic powers of ideals and which can be used to study the equality and containment problems, among others.

5.    Negative curves on symmetric blowups of the projective plane, resurgences and Waldschmidt constants

(with Th. Bauer, S. Di Rocco, B. Harbourne, J. Huizenga, T. Szemberg) International Math. Research Notices

In this paper we study the surface X obtained by blowing up the projective plane in the singular points of one of two highly symmetric line configurations known as the Klein configuration and the Wiman configuration respectively. We study invariant curves on X in detail, with a particular emphasis on curves of negative self-intersection. We use the representation theory of the stabilizers of the singular points to discover several invariant curves of negative self-intersection on X, and use these curves to study Nagata-type questions for linear series on X.

6.    Symbolic powers of codimension two Cohen-Macaulay ideals (with S. Cooper, G. Fatabbi, E. Guardo, A. Lorenzini, J. Migliore, U. Nagel, J. Szpond, A. Van Tuyl)

Under the additional hypothesis that X is a local complete intersection, we classify when I(X)^(m) = I(X)^m for all m >= 1. The key tool to prove this classification is the ability to construct a graded minimal free resolution of I^m under these hypotheses. Among our applications are significantly simplified proofs for known results about symbolic powers of ideals of points in P^1 x P^1.

7.    The Waldschmidt constant for squarefree monomial ideals

(with C. Bocci, S. Cooper, E. Guardo, B. Harbourne, M. Janssen, U. Nagel, A. Van Tuyl, T. Vu) J. Algebraic Combinatorics 44 (2016) no. 4, 875-904                                  

Given a squarefree monomial ideal I, we show that the Waldschmidt constant of I can be expressed as the optimal solution to a linear program constructed from the primary decomposition of I. By applying results from fractional graph theory, we can then express the Waldschmidt constant in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of I. Moreover, we prove a Chudnovsky-like lower bound on this constant, thus verifying a conjecture of Cooper-Embree-Ha-Hoefel for monomial ideals in the squarefree case.

8.    A homological criterion for the failure of containment of the symbolic cube in the square of some ideals of points in P^2

                     J. Pure Applied Algebra 219 (2015) no.11, 4857-4871

We establish a criterion for the failure of the containment of the symbolic cube in the square for 3-generated ideals I defining reduced sets of points in P^2. Our criterion arises from studying the minimal free resolutions of the powers of I, specifically the minimal free resolutions for I^2 and I^3. We apply this criterion to two point configurations that have recently arisen as counterexamples to a question of B. Harbourne and C. Huneke: the Fermat configuration and the Klein configuration.

9.    Determinants of incidence and Hessian matrices arising from the vector space lattice (with S. Nasseh and J. Watanabe)

J. Commut. Algebra 11 (2019) no. 1, 131-154

We give explicit formulas for the determinants of the incidence and Hessian matrices arising from the interaction between the rank 1 and rank n−1 level sets of the subspace lattice of an n-dimensional finite vector space. Our exploration is motivated by the fact that both of these matrices arise naturally in the study of the combinatorial and algebraic Lefschetz properties.

10. Resurgences for ideals of special point configurations in P^N coming from hyperplane arrangements

(with M. Dumnicki, B. Harbourne, U. Nagel, T. Szemberg and H. Tutaj-Gasińska ) J. Algebra 443 (2015), 383-394

It had been expected for several years that I^(Nr−N+1) I^r should hold for the ideal I of any finite set of points in P^N for all r>0, but in the last year various counterexamples have now been constructed, all involving point sets coming from hyperplane arrangements. In the present work, we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal.

11. A tight bound on the projective dimension of four quadrics (with C. Huneke, P. Mantero and J. McCullough) with accompanying Macaulay 2 code.

J. Pure Appl. Algebra 222 (2018) no. 9, 2524-2551

Motivated by a question posed by Mike Stillman, we show that the projective dimension of an ideal generated by four quadric forms in a polynomial ring has projective dimension at most 6.

12. A multiplicity bound for graded rings and a criterion for the Cohen-Macaulay property (with C. Huneke, P. Mantero and J. McCullough)

 Proc. Amer. Math. Soc. 143 (2015) no.6, 2365-2377

We prove an upper bound for the multiplicity of R/I, where I is a homogeneous ideal of the form I=J+(F) and J is a Cohen-Macaulay ideal. The bound is given in terms of invariants of R/J and the degree of F. We show that ideals achieving this bound have high depth and deduce a numerical criterion for the Cohen-Macaulay property. Applications to quasi-Gorenstein rings and almost complete intersection ideals are given.

13. Containment counterexamples for ideals of various configurations of points in P^N (with B. Harbourne) J. Pure Appl. Algebra 219 (2015), 1062-1072

We provide counterexamples to a conjecture of Harbourne and Huneke regarding containments between regular powers and symbolic powers of ideals of points in projective space P^N. We show that the conjecture fails in every prime characteristic p>2 when N=2 and we provide additional counterexamples for higher dimensional projective spaces.

14. The projective dimension of codimension two algebra presented by quadrics (with C. Huneke, P. Mantero and J. McCullough) J. Algebra 393 (2013), 170-186

 We prove a sharp upper bound for the projective dimension of ideals of height two generated by quadrics in a polynomial ring with arbitrary large number of variables.

15. Multiple structures with arbitrarily large projective dimension on linear subspaces (with C. Huneke, P. Mantero and J. McCullough)

J. Algebra 447 (2016), 183-205

We show that no finite characterization of multiple structures on linear spaces is possible if one only assumes Serre’s S_1 property holds by by constructing structures with arbitrarily large projective dimension. Our methods build upon a family of ideals with large projective dimension using linkage. The result is in stark contrast to Manolache's characterization of Cohen-Macaulay multiple structures in codimension 2 and multiplicity at most 4 and also to Engheta's characterization of unmixed ideals of height 2 and multiplicity 2.

16. Computations in intersection rings of flag bundles (with D. Grayson and M. Stillman)

This paper arose out of an observation that was made while I was working on the “Symmetric polynomials” package for Macaulay2. Intersection rings of flag varieties and of isotropic flag varieties are generated by Chern classes of the tautological bundles modulo the relations coming from multiplicativity of total Chern classes. In this paper we describe the Gröbner bases of the ideals of relations and give applications to computation of intersections, as implemented in Macaulay2.

17. Bounding projective dimension (with J. McCullough) a chapter in the book Commutative Algebra, Springer-Verlag London, 2013

The celebrated Hilbert Syzygy Theorem states that the projective dimension of any ideal in a polynomial ring on n variables is at most n-1. This paper surveys recent progress on Stillman’s question, asking whether the degrees of a set of homogeneous polynomials suffice in order to bound the projective dimension of the ideal they generate, without prior knowledge of the ambient polynomial ring (hence without using the number of variables).

18. Syzygies and singularities of tensor product surfaces of bidegree (2,1) (with H. Schenck and J. Validashti), Math. Comp. 83 (2014), 1337-1372

We study the associated ideal of a bigraded parametrization of a surface in P^3 from the standpoint of commutative algebra, proving that there are exactly six numerical types of possible bigraded minimal free resolution. These resolutions play a key role in determining the implicit equation of the image, via work of Buse-Jouanolou, Buse-Chardin, Botbol and Botbol-Dickenstein-Dohm on the approximation complex. In particular this allows us to completely describe the implicit equation and singular locus of the image.

19.  Syzygy theorems via comparison of order ideals on a hypersurface (with P. A. Griffith) J. Pure Appl. Algebra 216 (2012), no. 2, 468-479

We introduce a weak order ideal property that suffices for establishing the Evans-Griffith Syzygy Theorem. We study this weak order ideal property in settings that allow for comparison between homological algebra over a local ring R versus a hypersurface ring R/(x^n). Consequently we solve some relevant cases of the Evans-Griffith syzygy conjecture over local rings of unramified mixed characteristic p, with the case of syzygies of prime ideals of Cohen-Macaulay local rings of unramified mixed characteristic being noted.

20.  Inverse systems, fat points and the weak Lefschetz property (with B. Harbourne and H. Schenck) J. Lond. Math. Soc. (2) 84 (2011), no. 3, 712-730

We use the inverse system dictionary to connect ideals generated by powers of linear forms to ideals of fat points and show that failure of WLP for powers of linear forms in at least four variables is connected to the geometry of the associated fat point scheme. This is in stark contrast with the situation described in the paper “The Weak Lefschetz Property and powers of linear forms in K[x, y, z]” below. Closely related results can be found in the paper See the paper "On the weak Lefschetz property for powers of linear forms" by Migliore-Mirò-Roig-Nagel.

21.  Ideals with Large(r) Projective Dimension and Regularity (with J. Beder, J. McCullough, L. Nunez-Betancourt, B. Snapp, B. Stone)

J. Symbolic Comput. 46 (2011), no. 10, 1105-1113

This paper is an outcome of the Mathematical Research Communities program. I am grateful to AMS and the organizers for this wonderful opportunity. We define a family of homogeneous ideals with large projective dimension and regularity relative to the number of generators and their common degree. This family subsumes and improves upon constructions given by Caviglia and McCullough In particular, we describe a family of homogeneous ideals with three generators of degree d in arbitrary characteristic whose projective dimension grows asymptotically as.

Here is the Macaulay 2 code mentioned in the paper.

22.  The Weak Lefschetz Property and powers of linear forms in K[x, y, z]  (with H. Schenck) Proc. Amer. Math. Soc. 138 (2010), no. 7, 2335-2339

We show that any artinian quotient of K[x, y, z] by an ideal I generated by powers of linear forms has the Weak Lefschetz Property.

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Slides and Videos from Selected Talks

  • Resolutions for powers of ideals and applications to symbolic powers (slides) SIAM Conference on Applied Algebraic Geometry in Atlanta, Aug 2017
  • Combinatorial methods for symbolic powers (video) CMO Workshop on Symbolic and Ordinary Powers of Ideals in Oaxaca, May 2017
  • A hands-on approach to tensor product surfaces of bidegree (2,1) (video) CMO Workshop on Computational Algebra and geometric Modeling in Oaxaca, Aug 2016
  • Ordinary and symbolic Rees algebras for Fermat configurations of points (slides) AMS Sectional Meeting in Fargo, Apr 2016
  • The Waldschmidt constant for squarefree monomial ideals (slides) AMS Sectional Meeting in Athens, Mar 2016 and AMS Sectional Meeting in Salt Lake City, Apr 2016
  • Polynomial growth for Betti numbers (slides) AMS Sectional Meeting in Athens, Mar 2016
  • Using syzygies to test containments between ordinary and symbolic powers (slides) AMS Joint Mathematics Meeting in San Antonio, Jan 2015
  • Configurations of points and lines with interesting algebraic properties (slides) AMS Sectional Meeting in Eau Claire, Sept 2014
  • Regular vs symbolic powers for ideals of points (slides) AMS Sectional Meeting in Louisville, Oct 2013
  • Tight bounds on projective dimension: the case of quadrics of height two (slides) AMS Sectional Meeting in Louisville, Oct 2013
  • The complexity of bounding projective dimension (slides) SIAM conference on Applied Algebraic Geometry in Fort Collins, Aug 2013
  • Syzygies and singularities of tensor product surfaces (slides) SIAM conference on Applied Algebraic Geometry in Fort Collins, Aug 2013
  • A hands-on approach to tensor product surfaces (slides) AMS Sectional Meeting in Lawrence, Apr 2012
  • Bounding projective dimension and regularity (slides) AMS Sectional Meeting in Lincoln, Oct 2011
  • Fat points and the weak Lefschetz property (slides) International Congress of Romanian Mathematicians in Brasov, Jul 2011
  • From syzygies to the weak Lefschetz property and back (slides) Resolutions Day at Cornell, May 2011
  • Inverse systems, fat points and the weak Lefschetz property (slides) AMS Joint Mathematics Meeting in New Orleans, Jan 2011
  • Syzygy theorem via comparison of order ideals (slides) AMS Joint Mathematics Meeting in San Francisco, Jan 2010
  • Weak Lefschetz Property and powers of linear forms (slides) AMS Sectional Meeting in Boca Raton, Nov 2009
  • Weak Lefschetz property for ideals generated by powers of linear forms (poster), PASI conference in Olinda, Aug 2009

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I have authored or co-authored several packages for the computer algebra program Macaulay2:

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