Algebraic Geometry/Commutative Algebra Seminar, 2020–2021

To volunteer to give a talk, or for anything else regarding the seminar, contact Claudiu Raicu or Eric Riedl.

Spring Schedule

The seminar will meet on Thursdays, 3:00–4:00pm unless otherwise noted. Related events are also listed below.

Abstracts

Feb. 25, 2021

Speaker

Paolo Mantero (Arkansas)

Title

Interpolation problems

Abstract

Homogeneous polynomial interpolation problems are challenging problems in Algebraic Geometry. They ask for information regarding hypersurfaces of given degrees passing with given multiplicity through a given set X of points in a projective space. A theorem by Zariski and Nagata allows one to translate the problem into a question about the Hilbert function of symbolic powers of the ideal I_X defining the set of points, so the problem can be tackled with Commutative Algebra tools as well. In the present talk we will review the history of some interpolation problems, state classical and recent results and discuss open problems and conjectures.

Mar. 4, 2021

Speaker

Eric Riedl (Notre Dame)

Title

Linear sections of hypersurfaces and the Lang Conjectures

Abstract

The Lang Conjectures predict that on general hypersurfaces X in P^n of degree d > n+1, there are special subvarieties that contain all the rational curves, entire curves, elliptic curves, and all but finitely many of the rational points. For d > (3n+1)/2, we identify a candidate subvariety of X that might satisfy the Lang conjectures. We prove that it contains all rational and elliptic curves, and show that if we assume the Lang Conjecture, this subvariety will contain all entire curves as well. This subvariety is defined in terms of lines meeting the hypersurface set-theoretically in only two points.

Mar. 25, 2021

Speaker

Felix Janda (Notre Dame)

Title

Tautological classes from complete intersections

Abstract

The tautological ring is a certain subring of the Chow (or cohomology) ring of the moduli space of curves that contains most Chow cycles of interest. In 2004, Faber and Pandharipande proved that for any d, the cycles obtained by considering the locus of curves admitting a morphism of degree d to P^1 are tautological, and asked the question whether the same is true if we replace P^1 by a different variety. We will answer this question for a certain class of complete intersections in projective space. This is based on joint work with Q. Chen and Y. Ruan.

Apr. 1, 2021

Speaker

Gianluca Pacienza (Lorraine)

Title

Deformations of rational curves on primitive symplectic varieties and applications

Abstract

I will start by explaining why it is worth studying « singular » holomorphic symplectic varieties and rational curves on them. Then I will talk about a joint work with Ch. Lehn and G. Mongardi in which we study the deformation theory of rational curves on a (possibly singular) primitive symplectic variety and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge locus. As applications of our technique, I will present the extension of Markman's deformation invariance of prime exceptional divisors to this singular framework and provide existence results for uniruled ample divisors on primitive symplectic varieties which are locally trivial deformation of any moduli space of sheaves on a projective K3 surface or fibers of the Albanese map of those on an abelian surface.

Apr. 8, 2021

Speaker

Colleen Robichaux (UIUC)

Title

Castelnuovo-Mumford regularity and Kazhdan-Lusztig varieties

Abstract

We give an explicit formula for the degree of a vexillary Grothendieck polynomial. We apply our work to compute the Castelnuovo-Mumford regularity of certain matrix Schubert varieties. We also derive formulas for the regularity of Kazhdan-Lusztig varieties coming from open patches of Grassmannians as well as the regularity of mixed one-sided ladder determinantal ideals. This is joint work with Jenna Rajchgot and Anna Weigandt.

Apr. 15, 2021

Speaker

Alexandra Seceleanu (Nebraska)

Title

Canonical resolutions over Koszul algebras

Abstract

Koszul algebras are a class of (not necessarily commutative) algebras which show up naturally and abundantly in algebra and topology. An interesting feature of Koszul algebras is that they appear in pairs - every Koszul algebra has a dual algebra, which is also Koszul. One can use this duality to construct free resolutions. We focus on constructing explicit resolutions for the powers of the maximal ideal of a Koszul algebra. This generalizes a result of Buchsbaum and Eisenbud, which applies to the case where the Koszul algebra under consideration is a polynomial ring. The results are joint with Eleonore Faber, Martina Juhnke-Kubitzke, Haydee Lindo, Claudia Miller, and Rebecca R.G.

Apr. 22, 2021

Speaker

Vinh Nguyen (Purdue)

Title

Lech's Inequality for the Buchsbaum-Rim Multiplicity and Mixed Multiplicities

Abstract

The Hilbert-Samuel multiplicity of m-primary ideals is an important invariant of Noetherian local rings. In some sense it measures the singularities of these rings. Singularities are detected by seeing how much the multiplicity differs from the colength. For instance, a ring is Cohen-Macaulay if the colength of a parameter ideal equals its multiplicity. One is then interested in bounding the multiplicity of an ideal, possibly by it's colength. Lech proved such a bound. He originally proved this result to establish his conjecture in low dimensions. The inequality itself is of interest however, and it is far from sharp. A recent improvement to Lech's bound is due to Huneke, Smirnov, and Validashti. They improved it by giving an analogous bound for the multiplicity of a given ideal times the maximal ideal. In this talk I'll give a brief discussion of multiplicities, along with it's generalizations. I'll then talk about Lech type bounds for the different multiplicities. This is joint work with Kelsey Walters.

Apr. 27, 2021

Speaker

Eloísa Grifo (Riverside)

Title

A test module for complete intersections

Abstract

A local ring is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: k has finite projective dimension if and only if R is regular. This characterization can be extended to a characterization of complexes of R-modules, and phrased in the language of derived categories. Pollitz gave a similar homological characterization for complete intersections, although his characterization lives in the world of complexes. In this talk, we will discuss a return to the world of modules, and embark on a search for a finitely generated module that witnesses the (non-)complete intersection property of R. This is joint work with Ben Briggs and Josh Pollitz.