Seminar on Commutative Algebra and Algebraic Geometry

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9/2

UNUSUAL Time/Place: 5pm in Evans 1015: JM Landsberg (Texas A&M; Simons Institute)

Tensors of minimal border rank

I will describe a problem motivated by computer science that quickly leads to questions about smoothability of zero dimensional schemes and new and interesting algebraic structures. In geometric language, the \(m\)-th secant variety to the Segre variety \(Sec_m(Seg(\mathbb{P}^{m-1} \times \mathbb{P}^{m-1}\times \mathbb{P}^{m-1}))\) is an orbit closure and one would like to know what points are on the boundary, or more coarsely, what are the components of the boundary? This is joint work with F. Gesmundo, J. Jelisiejew, T. Mandziuk, and A. Pal.

9/9

David Eisenbud

Summands of syzygies: a family of examples

A large class of interesting rings called Golod rings includes things like \(R = S/I^d\) for \(d >1\), where \(S\) is a regular local (or graded polynomial) ring containing a field \(k\) of characteristic 0 and \(I\) is a (homogeneous) ideal. Hai Long Dao and I were surprised to discover (first experimentally) that over a Golod ring, some sequence of syzygy modules of \(k\) contain \(k\) as a direct summand, and we gave a complete family of examples of the possible sequences. To do this we needed to understand a natural minimal \(S\)-free resolution of \(I^2\) when \(I\) is generated by a regular sequence. In this talk I'll explain this problem and its solution using a simple part of the representation theory of the general linear group.

9/16

Hannah Larson

Brill--Noether theory of special curves

Brill--Noether theory studies the maps of curves C to projective spaces. The classical Brill--Noether theorem (established by work of Eisenbud, Fulton, Geiseker, Griffiths, Harris, Lazarsfeld) describes the geometry of this space of maps when C is a general curve. However, the theorem fails for special curves, notably curves that are already equipped with some unexpected map to a projective space. The first case of this is when C is a low-degree cover of the projective line. For general such covers, the Hurwitz--Brill--Noether theorem (joint with E. Larson and I. Vogt) provides a suitable analogue. I'll also present results (joint with S. Vemulapalli) regarding the next natural case: when C is equipped with an embedding in the projective plane.

9/23

Frank-Olaf Schreyer

Smooth surfaces in \(\mathbb{P}^4\)

Hartshorne conjectured that the degree of a smooth rational surface in \(P4\) is bounded. This conjectured was answered positively by Ellingsrud and Peskine (1989): There only finitely many components of the Hilbert scheme of surfaces in \(P4\), whose general element correspond to a smooth surface, which is not of general type. Around that time there was a flourish of activities to construct and classify such surfaces, e.g. by Okonek, Ranestad, Decker-Ein-Schreyer, Popescu, Abo. The existence proofs in [DES] used Macaulay (classic) in an essential way. Now more then 25 years later, Macaulay2 is much faster and we should be able to go further. In the talk I will explain our three basic construction methods: Hilbert-Burch morphism with vector bundles random searches over finite fields Tate resolutions and will illustrate each of these techniques in striking examples.

9/30

Hannah Friedman

Counting Homogeneous Einstein Metrics

The problem of finding Einstein metrics on a compact homogeneous space reduces to solving a system of Laurent polynomial equations. We prove that the number of isolated solutions of this system is bounded above by the central Delannoy numbers and we describe the discriminant locus where the number of isolated solutions drops in terms of the principal \(A\)-determinant.

10/7

Noah Olander

A derived category analogue of the Nakai—Moishezon criterion

We will prove a necessary and sufficient condition for the set of tensor powers of a given line bundle on a variety to generate the derived category. The condition is strongly analogous to the Nakai—Moishezon criterion for ampleness, and provides many new examples of such line bundles. This is joint work with Daigo Ito.

Daigo Ito

A geometric study of tensor ampleness

We study a class of line bundles satisfying an analogue of the Nakai—Moishezon criterion, which we call tensor-ample line bundles. We will first observe the corresponding divisors form a cone in the Néron—Severi space and see how classical techniques provide more examples of tensor ample line bundles. In particular, we give a complete description of this cone for smooth projective surfaces and discuss applications to reconstruction problems in derived categories. This is joint work with Noah Olander.

10/14

Marcus Bläser

Generic identification in tree-shaped structural equation models

A structural equation model (SEM) is a multivariate statistical model that is determined by a so-called mixed graph. Nodes in the mixed graph model random variables. A mixed graph contains two types of edges, directed ones, which model direct dependencies between the random variables, and bidirected ones, which model hidden correlations. When the random variables are normally distributed and the dependencies are linear (a so-called linear SEM), then the covariances between the random variables are rational functions of the parameters that are associated to the edges of the mixed graph. In the first part of the talk with give a gentle introduction to the topic and explain how linear SEMs give rise to interesting questions in polynomial equation solving.
To distinguish correlation from causation, we want to determine whether the observed covariances uniquely determine the hidden parameters. A linear SEM is called generically identifiable, if the rational map that maps the parameters of the SEM to the observed covariances between the random variables is generically injective. The complexity of deciding whether a given SEM is generically identifiable is open. In this talk, we present a randomized polynomial time algorithm for the case when the directed edges of the SEM form a tree.
Joint work with Yasmine Briefs and Aaryan Gupta.

Tyler Chamberlain

Counting In-and-Circumscribed Triangles

10/21

Feiyang Lin

When are splitting loci Gorenstein?

Splitting loci are certain natural closed substacks of the stack of vector bundles on \(\mathbb{P}^1\), which have found interesting applications in the Brill-Noether theory of \(k\)-gonal curves. In this talk, I will explain which splitting loci, as algebraic stacks, are Gorenstein or \(\mathbb{Q}\)-Gorenstein. There are two main ingredients. One is an intersection-theoretic computation of class groups of splitting loci in certain affine extension spaces. The other is an explicit understanding of the normal bundle of an open dense substack of splitting loci, yielding a formula for the class of their canonical module.

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10/28

Sam Grushevsky

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11/4

David Zhang

Perfectoid Techniques in Commutative Algebra

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11/11

Shang Xu

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Cameron Chang

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11/18

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11/25

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12/2

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12/9

Yuhan Liu

On the Chow Rings of the Moduli Spaces \( \mathcal{M}_{5,8} \) and \( \mathcal{M}_{5,9} \)

We will introduce the Chow ring of moduli space \( \mathcal{M}_{g,n}\) and discuss the recent work which proves the rational Chow rings of \( \mathcal{M}_{5,8}\) and \( \mathcal{M}_{5,9}\) are tautological.

Robin Hartshorne

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