GradMoCCA ☕ a Graduate Meeting on
Combinatorial Commutative Algebra


First Session

Michael Perlman: Introduction to local cohomology as a D-module

Abstract: Through a series of concrete examples, we will introduce the local cohomology modules associated to subvariety in affine space. Our main focus will be on structural properties of these modules over the Weyl algebra of differential operators D, and connections to the theory of singularities. We will discuss tools from representation theory that may be used to carry out calculations in the presence of a group action

David Carey: Intersection Complexes: a family of complexes with pure Betti diagrams

Abstract: Betti diagrams are an important algebraic invariant of graded modules over polynomial rings. The study of pure Betti diagrams is of particular significance, because when we consider the rational cone generated by all Betti diagrams, the extremal rays of this cone turn out to be the pure diagrams corresponding to Cohen-Macaulay modules.

In this talk I will present a family of simplicial complexes whose Stanley-Reisner ideals have pure Betti diagrams. I will start by reframing the problem of finding pure Betti diagrams in entirely combinatorial terms. Then I will go on to define the family of intersection complexes, with the use of some examples. And finally I will show why their corresponding Betti diagrams are pure.

Greg Taylor: Asymptotic syzygies of secant varieties of curves

Abstract: We discuss the asymptotic behavior of the minimal free resolution of the secant variety of a smooth curve. In particular, we discuss the asymptotic purity of the Boij-S ̈oderberg decomposition and its corollaries.

Second Session

Aleksandra Sobieska: Subcomplexes of Certain Free Resolutions

Abstract: What are the subcomplexes of a free resolution? This question is simple to state, but the naive approach leads to a computational quagmire that is infeasible even in small cases. In this talk, I invoke the Bernstein–Gelfand–Gelfand (BGG) correspondence to address this question for free resolutions given by two well-known complexes, the Koszul and the Eagon–Northcott. This novel approach provides a complete characterization of the ranks of free modules in a subcomplex in the Koszul case and imposes numerical restrictions in the Eagon–Northcott case. This is joint work with Maya Banks.

Kuang Yu Wu: Affine subspace concentration conditions

Abstract: We define a new notion of affine subspace concentration conditions for lattice polytopes, and prove that they hold for smooth and reflexive polytopes with barycenter at the origin. Our proof involves considering the slope stability of the canonical extension of the tangent bundle by the trivial line bundle and with the extension class $c_1(T_X)$ on Fano toric varieties

Lauren Cranton Heller: Characterizing multigraded regularity on products of projective spaces

Abstract: The Castelnuovo-Mumford regularity of a sheaf on projective space can be described in terms of the minimal free resolutions of the corresponding graded module and its truncations. This characterization does not hold for the multigraded generalization of regularity defined by Maclagan and Smith. I will present an analogous criterion for regularity on products of projective spaces and apply it to the case of complete intersections.

Byeongsu Yu: When are $\mathbb{Z}^{d}$-graded modules of affine semigroup rings Cohen-Macaulay?

Abstract: We give a new combinatorial criterion for $\mathbb{Z}^{d}$-graded modules of affine semigroup rings to be Cohen-Macaulay, by computing the homology of finitely many polyhedral complexes. This provides a common generalization of well-known criteria for affine semigroup rings and monomial ideals in polynomial rings. This is joint work with Laura Matusevich.

Third Session

Sergio Da Silva: Which toric ideals of graphs are geometrically vertex decomposable?

Abstract: A Gröbner degeneration is a useful method to reduce problems involving ideals to the monomial ideal setting. In the square-free case, we can associate a simplicial complex to a monomial ideal and ask whether this complex is vertex decomposable. Another tool in this direction, called a geometric vertex decomposition, is a generalization of this technique to non-monomial ideals. It was first introduced by Knutson-Miller-Yong to study diagonal degenerations of Schubert varieties. Later uses of the concept mostly occurred in the context of Schubert geometry, until very recent work of Klein-Rajchgot, which established a connection to liaison theory. The interplay between these two theories can be used to analyze degenerations, construct Gröbner bases, and find families of ideals which are glicci (in the Gorenstein liaison class of a complete intersection). In this talk, I will review the Stanley-Reisner correspondence and discuss vertex decomposability. I will then introduce the notion of a geometric vertex decomposition and highlight its applications to toric ideals of graphs

Karthik Ganapathy: The infinite-variable polynomial ring in positive characteristic

Abstract: GL-equivariant modules over infinite-variable polynomial rings have found applications in topology, algebraic statistics and combinatorics. The structure theory of such modules is well understood in characteristic zero by the work of Sam–Snowden and others. I will talk about ongoing work to extend their results to positive characteristic emphasizing the key differences.

Soohyun Park: Matroidal Cayley-Bacharach and ranks of covering flats

Abstract: A finite collection of points in P^n is said to satisfy the degree a Cayley-Bacharach property CB(a) if any degree a hypersurface passing through all but one of the points passes through all of them. In recent work, Levinson and Ullery show that a set satisfying CB(a) with small size (relative to a) can be covered by a collection of low-dimensional linear subspaces. Motivated by the combinatorial structure of their proof, they ask whether a matroidal analogue of their result holds. We show that it does not hold and explore combinatorial properties of matroids which satisfy the matroidal Cayley-Bacharach property MCB(a). This includes connections to the geometry of generalized permutohedra and covers of directed graphs by maximal paths.

Maya Banks: Sliding Into DMs: A brief intro to differential modules

Abstract: A differential module is a module equipped with a square-zero endomorphism and is a natural generalization of a chain complex. Arising in commutative algebra, algebraic geometry, and topology, differential modules are interesting objects of study in their own right while also providing a natural setting in which to generalize many classical results and constructions. In this talk, we’ll first introduce the basic structure and properties of differential modules, then explore how they behave as tools of generalization. In particular, we’ll investigate when results about familiar objects like complexes and resolutions can be generalized to differential modules, and what the ability or inability to generalize tells us about the results and structures involved.