Powers of binomial edge ideals with quadratic gröbner bases

Using SAGBI basis techniques, we find Grobner bases for the presentation ideals of the Rees algebra and special fiber ring of a closed determinantal facet ideal. In particular, we show that closed determinantal facet ideals are of fiber type and their special fiber rings are Koszul. Moreover, their Rees algebras and special fiber rings are normal Cohen-Macaulay domains, and have rational singularities.

Rees Algebras of Closed Determinantal Facet Ideals

A BSTRACT . We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to inﬁnity in certain directions. Such an arrangement deﬁnes a decomposition of Euclidean space where a cell is determined by its ‘type’ data, analogous to the covectors of an oriented matroid. By work of Develin-Sturmfels and Fink-Rinc´on, these ‘tropical complexes’ are dual to (regular) subdivisions of root polytopes, which in turn are in bijection with mixed subdivisions of certain generalized permutohedra. Extending previous work with Joswig-Sanyal, we show how a natural monomial labeling of these complexes describes polynomial relations (syzygies) among ‘type ideals’ which arise naturally from the combinatorial data of the arrangement. In particular, we show that the cotype ideal is Alexander dual to a corresponding initial ideal of the lattice ideal of the underlying root polytope. This leads to novel ways of studying algebraic properties of various monomial and toric ideals, as well as relating them to combinatorial and geometric properties. In particular, our methods of studying the dimension of the tropical complex leads to new formulas for homological invariants of toric edge ideals of bipartite graphs, which have been extensively studied in the commutative algebra community.

Root polytopes, tropical types, and toric edge ideals

The ACM property for unions of lines in P1×P2

We explore explicit virtual resolutions, as introduced by Berkesch, Erman, and Smith, for ideals of ﬁnite sets of points in P 1 × P 1 . Speciﬁcally, we describe a virtual resolution for a suﬃciently general set of points X in P 1 × P 1 that only depends on | X | . We also improve an existence result of Berkesch, Erman, and Smith in the special case of points in P 1 × P 1 ; more precisely, we give an eﬀective bound for their construction that gives a virtual resolution of length two for any set of points in P 1 × P 1 . © 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Virtual resolutions of points in P 1 × P 1

We give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal $(x_1,x_2,\cdots,x_m)^n$ of a polynomial ring in $m$ variables. We also give a combinatorial description of the Alexander duals of such polarizations. In the three variable case $m=3$ and also in the power two case $n=2$ the descriptions are easily visualized and we show that every polarization defines a (shellable) simplicial ball. We give conjectures relating to topological properties and to algebraic geometry, in particular that any polarization of an Artinian monomial ideal defines a simplicial ball.

Polarizations of powers of graded maximal ideals

We introduce the VirtualResolution package for the computer algebra system Macaulay2. This package has tools to construct, display, and study virtual resolutions for products of projective spaces. The package also has tools for generating curves in $\mathbb{P}^1\times\mathbb{P}^2$, providing sources for interesting virtual resolutions.

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The virtual resolutions package for Macaulay2

A determinantal facet ideal (DFI) is generated by a subset of the maximal minors of a generic $n\times m$ matrix indexed by the facets of a simplicial complex. We consider the more general notion of an $r$-DFI, which is generated by a subset of $r$-minors of a generic matrix indexed by the facets of a simplicial complex for some $1\leq r\leq n$. We define and study so-called lcm-closed and interval DFIs, and show that the minors parametrized by the facets of $\Delta$ form a reduced Grobner basis with respect to any diagonal term order in both of these cases. We also see that being lcm-closed generalizes conditions previously introduced in the literature, and conjecture that in the case $r=n$, lcm-closedness is necessary for being a Grobner basis. We also give conditions on the maximal cliques of $\Delta$ ensuring that lcm-closed and interval DFIs are Cohen-Macaulay. Finally, we conclude with a variant of a conjecture of Ene, Herzog, and Hibi on the Betti numbers of certain types of $r$-DFIs, and provide a proof of this conjecture for Cohen-Macaulay interval DFIs.

Ayah Almousa

Papers

Polarizations of powers of graded maximal ideals

The virtual resolutions package for Macaulay2

Rees Algebras of Closed Determinantal Facet Ideals

Root polytopes, tropical types, and toric edge ideals

Determinantal Facet Ideals for Smaller Minors