UMN Commutative Algebra and Algebraic Geometry Seminar, Spring 2022

A standard graded K-algebra R is Koszul if K has a linear free resolution over R.  While it is sufficient for the defining ideal to have a defining ideal generated by a quadratic Groebner basis (in some system of coordinates and with respect to some monomial order – such algebras are called G-quadratic), not all Koszul algebras arise this way.  In the absence of a quadratic Groebner basis, one can look for a G-quadratic algebra A such that the algebra in question is A modulo a regular sequence of linear forms – such algebras are called LG-quadratic.  These notions are well-studied on the commutative side but less so for quotients of exterior algebras.  We show that the implications G-quadratic => LG-quadratic => Koszul hold for quotients of exterior algebras and both implications are strict.  This is joint work with Zach Mere.

Vector bundles over a space are analogous to finitely generated modules over a ring.

In 1964, G. Horrocks proved that a vector bundle on a projective space P^n splits as a sum of line

bundles if and only if it has no intermediate cohomology. Then in 2015 Eisenbud-Erman-Schreyer,

used the BGG correspondence for products of projective spaces to prove a version of this criterion

with an additional hypothesis on the Betti numbers.

I will give a simpler, 5-minute proof of that result using an appropriate Fourier-Mukai transform and

show how this can be extended to products of weighted projective spaces, Hirzebruch surfaces,

and hopefully other toric varieties. This is based on ongoing work with Michael Brown and Lauren

Cranton Heller.

Given a self map $f$ of a projective variety $X$ over a number field, one typically expects the iterated preimages of an invariant subvariety to become more complicated. An arithmetic incarnation of this, asked by Matsuzawa, Meng, Shibata, and Zhang is if the rational points of these preimages stabilize. We answer their question for etale maps and relate their problem to the following cancellation statement: when does there exist $N$ such that for all $n\geq N$ and all rational points $x$ and $y$, if $f^n(x)=f^n(y)$, then $f^N(x)=f^N(y)$? We show such cancellation phenomena hold for smooth projective curves. This is joint work with Jason Bell and Yohsuke Matsuzawa.

Buchweitz-Greuel-Schreyer conjectured that the minimal rank of a matrix factorization is, roughly, 2^e where e is one half the codimension of the singular locus of f.  I will discuss a proof that this results holds for generic polynomials.  The proof introduces a notion of the secondary strength of a polynomial, and uses a variant of the ultraproduct technique of Erman, Sam, and Snowden.

The Eagon-Northcott complex of a map of finitely generated free modules has been an interest of study since 1962, as it generically resolves the ideal of maximal minors of the matrix that defines the map. In 1975, Buchsbaum and Eisenbud described a family of generalized Eagon-Northcott complexes associated to a map of free modules, which are also generically minimal free resolutions. As introduced by Berkesch, Erman, and Smith in 2020, when working over a smooth projective toric variety, virtual resolutions, rather than minimal free resolutions, are a better tool for understanding the geometry of a space. I will describe sufficient criteria for the family of generalized Eagon-Northcott complexes of a map to be virtual resolutions, thus adding to the known examples of virtual resolutions, particularly those not coming from minimal free resolutions.

We introduce the notion of strongly Lech-independent ideals and use this notion to derive inequalities on multiplicities of ideals. In particular, we prove that if we have a flat local extension of Noetherian local rings from (R,m) to (S,n) such that the dimensions of R and S are equal, the completion of (S,n) is the completion of a standard graded ring (S_g, n_g) over a field k, and the completion of I=mS is extended from a homogeneous ideal I_g, then Lech's conjecture holds.

⛱ 😎🌷 

I will present some recent progress in the study of Hilbert schemes of d points in affine space, and the related (local) punctual Hilbert schemes at fixed p in affine n-space.  Specifically, I will discuss some results on elementary components of Hilbert schemes of points and tie these to a question posed by Iarrobino in the 80's: does there exist an irreducible component of the punctual Hilbert scheme of dimension less than (n-1)(d-1)?  I will answer this question by describing a new infinite family of irreducible components satisfying this bound, when n=4.  A secondary family of elementary components also arises, providing further new examples of elementary components of Hilbert schemes of points, and improving our knowledge surrounding a folklore question on the existence of certain Gorenstein local Artinian rings.

This is joint work with Matt Satriano (U Waterloo).

Over any infinite field, the generic determinantal rings are known to be fixed subrings of the action of the general linear group on a polynomial ring. Since the general linear group is linearly reductive in characteristic zero, these generic determinantal rings are direct summands of polynomial rings. In positive characteristic, these determinantal rings have many of the same properties—same generators and relations, Cohen Macaulayness, rational singularities—even though the general linear group is no longer linearly reductive in this case. In this talk we investigate if these determinantal rings continue to be direct summands of polynomial rings in characteristic p>0. We will also encounter some interesting varieties related to linear algebra along the way.

This is joint work with Mel Hochster, Vaibhav Pandey, and Anurag Singh. 

Let E be a finite set. An embedding of a \CC-vector space W into \CC^E induces a (homogeneous) configuration hypersurface in the ring \CC[x_e|e\in E]  by restricting the generic diagonal bilinear form \sum x_e \hat e\otimes\hat e to V as its discriminant locus. (The hat denotes the dual vectors). The hypersurface is defined by the configuration polynomial, the determinant of this restricted bilinear form, defined up to a factor corresponding to a choice of basis in W.

If G is a graph on the edge set E then the incidence matrix can be read as a map from W=\CC^V to \CC^E, and is a natural way to obtain a graph hypersurface. The configuration polynomial is then known as the graph polynomial of G, and has a very simple-looking description in terms of G. These polynomials appear naturally in the theory of Feynman amplitudes and their geometric properties are interesting to physicists. A conjecture of Aluffi adresses the question what the Euler characteristic of the projective complement of the graph hypersurface is.

In the talk, we explain configuration hypersurfaces, and why Aluffi made the conjecture, based on torus actions that arise in special situations. We show how certain matroid operations reflect in the topology of the graph polynomial.  Inspired by torus actions discovered by Mueller-Stach and Westrich, we then give a new way of making torus actions on these hypersurfaces, and discuss how this leads to a wealth of new graphs whose Euler characteristic of the graph polynomial is computable.

When trying to apply the machinery of derived categories in an arithmetic setting, a natural question is the following: for a smooth projective variety $X$ to what extent can $D^b(X)$ be used as an invariant to answer rationality questions? In particular, what properties of $D^b(X)$ are implied by $X$ being rational, stably rational, or having a rational point? On the other hand, is there a property of $D^b(X)$ that implies that $X$ is rational, stably rational, or has a rational point?

In this talk, we will examine a family of arithmetic toric varieties for which a member is rational if and only if its bounded derived category of coherent sheaves admits a full étale exceptional collection. Additionally, we will discuss the behavior of the derived category under twisting by a torsor, which is joint work with Matthew Ballard, Alexander Duncan, and Patrick McFaddin.

Direct summands (or more generally, pure subrings) of regular rings possess many desirable properties, and computations which are intractable for arbitrary rings are often more feasible for such rings. For example, associated primes of local cohomology or rings of differential operators, both of which are quite challenging to understand generally, are much better understood for such rings. While certain classes of rings (e.g., rings of invariants of linearly reductive groups) occur naturally as direct summands, in general it appears difficult to decide when a ring can be recognized as a direct summand of a regular ring. In this talk, we’ll first review some known examples of which rings are or aren’t of this form, and then present some new results indicating when a homogeneous ring cannot be recognized as a finite graded summand of a polynomial ring.

The Taylor resolution is a fundamental object in the study of free resolutions over the polynomial ring, due to its explicit formula, cellular/combinatorial structure, and applicability to any and all monomial ideals. In this talk, I will present a generalization of the Taylor resolution to hypersurface rings and describe how this construction can be iterated to produce a Taylor resolution over monomial complete intersection rings.

Support varieties over a complete intersection ring have found numerous, far-reaching applications in local algebra. In this talk I’ll discuss joint work with Ben Briggs and Daniel McCormick where we introduce a higher order support theory generalizing the classical support theory. This higher order support theory is applied to reveal symmetries in free resolutions over a complete intersection ring. Namely, for a finitely generated module over a complete intersection ring, its sequences of Bass and Betti numbers are eventually modeled by quasi-polynomials of period two; the leading terms of these two polynomials are independent of parity, respectively. I will show how this higher order support theory can be applied to establish the leading terms of these two quasi-polynomials actually coincide.