Ben Drabkin, Eloísa Grifo, Alexandra Seceleanu, Branden Stone
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.
Marcin Dumnicki, Brian Harbourne, Uwe Nagel, Alexandra Seceleanu, Tomasz Szemberg, Halszka Tutaj-Gasińska
Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence. There have been exciting new developments in this area recently. It had been expected for several years that $I^{Nr-N+1}\subseteq I^r$ should hold for the ideal $I$ of any finite set of points in ${\bf P}^N$ 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.
Brian Harbourne, Alexandra Seceleanu
When $I$ is the radical homogeneous ideal of a finite set of points in projective $N$-space, ${\bf P}^N$, over a field $K$, it has been conjectured that $I^{(rN-N+1)}$ should be contained in $I^r$ for all $r\geq 1$. Recent counterexamples show that this can fail when N=r=2. We study properties of the resulting ideals. We also show that failures occur for infinitely many $r$ in every characteristic $p>2$ when N=2, and we find additional positive characteristic failures when $N>2$.
Daniel R. Grayson, Alexandra Seceleanu, Michael E. Stillman
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 Groebner bases of the ideals of relations and give applications to computation of intersections, as implemented in Macaulay2.
Alexandra Seceleanu, Liana Şega
We investigate the structure and properties of symmetric ideals generated by general forms in the polynomial ring under the natural action of the symmetric group. This work significantly broadens the framework established in our earlier collaboration with Harada on principal symmetric ideals. A novel aspect of our approach is the construction of a bijective parametrization of general symmetric ideals using Macaulay-Matlis duality, which is asymptotically independent of the number of variables of the ambient ring. We establish that general symmetric ideals exhibit extremal behavior in terms of Hilbert functions and Betti numbers, and satisfy the Weak Lefschetz Property. We also demonstrate explicit asymptotic stability in their algebraic and homological invariants under increasing numbers of variables, showing that such ideals form well-behaved $\mathfrak{S}_\infty$-invariant chains.
Brian Harbourne, Hal Schenck, Alexandra Seceleanu
Migliore-Miró-Roig-Nagel [Trans. A.M.S. 2011, arXiv: 0811.1023] show that the weak Lefschetz property (WLP) can fail for an ideal I in K[x_1,x_2,x_3,x_4] generated by powers of linear forms. This is in contrast to the analogous situation in K[x_1,x_2,x_3], where WLP always holds [H.Schenck, A.Seceleanu, Proc. A.M.S. 2010, arXiv:0911.0876]. We use the inverse system dictionary to connect I to an ideal of fat points and show that failure of WLP for powers of linear forms is connected to the geometry of the associated fat point scheme. Recent results of Sturmfels-Xu in [J. Eur. Math. Soc. 2010, arXiv:0803.0892] allow us to relate WLP to Gelfand-Tsetlin patterns. See the paper "On the weak Lefschetz property for powers of linear forms" by Migliore-Miró-Roig-Nagel [arXiv:1008.2149] for related results.
Nasrin Altafi, Roberta Di Gennaro, Federico Galetto, Sean Grate, Rosa M. Miro-Roig, Uwe Nagel, Alexandra Seceleanu, Junzo Watanabe
The connected sum construction, which takes as input Gorenstein rings and produces new Gorenstein rings, can be considered as an algebraic analogue for the topological construction having the same name. We determine the graded Betti numbers for connected sums of graded Artinian Gorenstein algebras. Along the way, we find the graded Betti numbers for fiber products of graded rings; an analogous result was obtained in the local case by Geller. We relate the connected sum construction to the doubling construction, which also produces Gorenstein rings. Specifically, we show that a connected sum of doublings is the doubling of a fiber product ring.
Saeed Nasseh, Alexandra Seceleanu, Junzo Watanabe
Let $\mathcal{V}=\bigsqcup_{i=0}^n\mathcal{V}_i$ be the lattice of subspaces of the $n$-dimensional vector space over the finite field $\mathbb{F}_q$ and let $\mathcal{A}$ be the graded Gorenstein algebra defined over $\mathbb{Q}$ which has $\mathcal{V}$ as a $\mathbb{Q}$ basis. Let $F$ be the Macaulay dual generator for $\mathcal{A}$. We compute explicitly the Hessian determinant $|\frac{\partial ^2F}{\partial X_i \partial X_j}|$ evaluated at the point $X_1 = X_2 = \cdots = X_N=1$ and relate it to the determinant of the incidence matrix between $\mathcal{V}_1$ and $\mathcal{V}_{n-1}$. Our exploration is motivated by the fact that both of these matrices arise naturally in the study of the Sperner property of the lattice and the Lefschetz property for the graded Artinian Gorenstein algebra associated to it.
Alexandra Seceleanu
We establish a criterion for the (failure of) the containment $I^{(m)}\subset I^r$ for 3-generated ideals $I$ defining reduced sets of points in $\mathbb{P}^2$. Our criterion arises from studying the minimal free resolutions of the powers of $I$, specifically the minimal free resolutions for $I^m$ and $I^r$. 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.
Nasrin Altafi, Rodica Dinu, Sara Faridi, Shreedevi K. Masuti, Rosa M. Miró-Roig, Alexandra Seceleanu, Nelly Villamizar
This paper initiates a systematic study for key properties of Artinian Gorenstein \(K\)-algebras having binomial Macaulay dual generators. In codimension 3, we demonstrate that all such algebras satisfy the strong Lefschetz property, can be constructed as a doubling of an appropriate 0-dimensional scheme in \(\mathbb{P}^2\), and we provide an explicit characterization of when they form a complete intersection. For arbitrary codimension, we establish sufficient conditions under which the weak Lefschetz property holds and show that these conditions are optimal.
Eliana Duarte, Alexandra Seceleanu
We derive the implicit equations for certain parametric surfaces in three-dimensional projective space termed tensor product surfaces. Our method computes the implicit equation for such a surface based on the knowledge of the syzygies of the base point locus of the parametrization by means of constructing an explicit virtual projective resolution.
Craig Huneke, Paolo Mantero, Jason McCullough, Alexandra Seceleanu
Motivated by Stillman's question, we show that the projective dimension of an ideal generated by four quadric forms in a polynomial ring is at most 6; moreover, this bound is tight. We achieve this bound, in part, by giving a characterization of the low degree generators of ideals primary to height three primes of multiplicities one and two.
Hal Schenck, Alexandra Seceleanu, Javid Validashti
Let U be a basepoint free four-dimensional subspace of the space of sections of O(2,1) on P^1 x P^1. The sections corresponding to U determine a regular map p_U: P^1 x P^1 --> P^3. We study the associated bigraded ideal I_U in k[s,t;u,v] 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 p_U(P^1 x P^1), via work of Buse-Jouanolou, Buse-Chardin, Botbol and Botbol-Dickenstein-Dohm on the approximation complex. In four of the six cases I_U has a linear first syzygy; remarkably from this we obtain all differentials in the minimal free resolution. In particular this allows us to describe the implicit equation and singular locus of the image.
Jesse Beder, Jason McCullough, Luis Nunez-Betancourt, Alexandra Seceleanu, Bart Snapp, Branden Stone
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 in [Cav04] and [McC]. In particular, we describe a family of three-generated homogeneous ideals in arbitrary characteristic whose projective dimension grows asymptotically as sqrt{d}^(sqrt(d) - 1).
Eleonore Faber, Martina Juhnke-Kubitzke, Haydee Lindo, Claudia Miller, Rebecca R. G., Alexandra Seceleanu
We generalize Buchsbaum and Eisenbud's resolutions for the powers of the maximal ideal of a polynomial ring to resolve powers of the homogeneous maximal ideal over graded Koszul algebras. Our approach has the advantage of producing resolutions that are both more explicit and minimal compared to those previously discovered by Green and Martínez-Villa \cite{GreenMartinezVilla} or Martínez-Villa and Zacharia \cite{MartinezVillaZacharia}.
Penelope Beall, Erenay Boyali, Nancy Chen, Ellen Chlachidze, Trong Toan Dao, Frederic Garvey, Mitchell Johnson, Yu Olivier Li, Nikola Kuzmanovski, Kelvin Ma, Treanungkur Mal, Rukshan Marasinghe Mudiyanselage, Quinlan Mayo, Nava Minsky-Primus, Alexandra Seceleanu, Sriram Veerapaneni
A poset is Macaulay if its partial order and an additional total order interact well. Analogously, a ring is Macaulay if the partial order defined on its monomials by division interacts nicely with any total monomial order. We investigate methods of obtaining new structures through combining Macaulay rings and posets by means of certain operations inspired by topology. We examine whether these new structures retain the Macaulay property, identifying new classes of posets and rings for which the operations preserve the Macaulay property.
Hrishikesh Bodas, Benjamin Drabkin, Caleb Fong, Su Jin, Justin Kim, Wenxuan Li, Alexandra Seceleanu, Tingting Tang, Brendan Williams
We study several consequences of the packing problem, a conjecture from combinatorial optimization, using algebraic invariants of square-free monomial ideals. While the packing problem is currently unresolved, we successfully settle the validity of its consequences. Our work prompts additional questions and conjectures, which are presented together with their motivation.
Jennifer Biermann, Hernán De Alba, Federico Galetto, Satoshi Murai, Uwe Nagel, Augustine O'Keefe, Tim Römer, Alexandra Seceleanu
We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered as an analogue of stable monomial ideals within the class of monomial ideals. We show that a symmetric shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we obtain several consequences for graded Betti numbers of symbolic powers of defining ideals of star configurations.
Kyungyong Lee, Li Li, Matthew Mills, Ralf Schiffler, Alexandra Seceleanu
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. We show 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 $\ge2$, respectively.
Craig Huneke, Paolo Mantero, Jason McCullough, Alexandra Seceleanu
Let $R$ be a polynomial ring over a field. We prove an upper bound for the multiplicity of $R/I$ when $I$ is a homogeneous ideal of the form $I=J+(F)$, where $J$ is a Cohen-Macaulay ideal and $F\notin J$. The bound is given in terms of two invariants of $R/J$ and the degree of $F$. We show that ideals achieving this upper bound have high depth, and provide a purely numerical criterion for the Cohen-Macaulay property. Applications to quasi-Gorenstein rings and almost complete intersections are given.