Algebra, Geometry, and Number Theory Seminar

Abstracts

Mohammed Alabbood - The 27 lines on a smooth cubic surface in P³

In this seminar, we will discuss the problem of existence of a line on a non-singular cubic surface S in P³ by using the resultant and the polar to construct a polynomial whose roots will be the lines in S. We will also present a way to label these lines in terms of two fixed disjoint lines on a cubic surface S in such a way that we can determine all the remaining 25 lines on S.

Matthew Ballard - The derived geometry of birational maps

We will discuss how to assign an integral transform of derived categories to any D-flip. This involves using a little homotopical machinery and yields a functor with pleasant homological properties. Some examples will be discussed.

Pretalk: I will talk about flips and flops and expectations for how they change the derived category.

Dori Bejleri - Motivic Hilbert zeta functions of curves

The Grothendieck ring of varieties is the target of a rich invariant associated to any algebraic variety which witnesses the interplay between geometric, topological and arithmetic properties of the variety. The motivic Hilbert zeta function is the generating series for classes in this ring associated to a certain compactification of the unordered configuration space, the Hilbert scheme of points, of a variety. For curves with planar singularities, there is a large body of work detailing beautiful connections with enumerative geometry, representation theory and topology. In this talk I will discuss the behavior of the motivic Hilbert zeta function of a reduced curve with arbitrary singularities.

Candace Bethea - Naive homotopy classes of endomorphisms of P¹

The aim of this talk is to explain Cazanave's computation the naive homotopy classes of rational maps P¹ → P¹ using symmetric bilinear forms. The pretalk will introduce rational functions on P¹ and the naive direct sum operation. The talk will introduce the Bézout form, and I will discuss the isomorphism between naive homotopy classes of endomorphisms of P¹ and the monoid of stable isomorphism classes of non-degenerate symmetric bilinear forms under direct sum.

Michael Burr - Understanding Algorithmic Complexity Using Algebraic Geometry

Many algorithms in computer science take a polynomial system as input. The algorithmic complexity of these algorithms is often expressed in terms of the complexity of the polynomial system (e.g., the degree and bit-size of the coefficients). In this talk, I will present an alternate way to understand the algorithmic complexity using the geometry of (the embedding of) the underlying algebraic variety using a technique called continuous amortization. This is a uniform technique that has led to state-of-the-art algorithmic complexity bounds. I will not assume a computer science background, and I will try to provide some insight into how pure mathematicians can successfully translate their tools to applied problems.

Pretalk: This will be a "what is"-style talk on homotopy continuation. Homotopy continuation is the main tool in the (relatively new) field of numerical algebraic geometry. Homotopy continuation represents an application of tools from analysis and computation to approximate roots of polynomial systems. This tool has been successful in solving many applied problems including those when more algebraic methods (such as Groebner bases) fail. In this talk, I will provide an introduction to the theory and practice of homotopy continuation with minimal prerequisites.

Pete Clark - Densities of sets of integers represented by quadratic forms

Given an integral quadratic form, one seeks to determine the set of integers that it represents. Though classical -- sums of squares were handled by Fermat, Lagrange and Legendre-Gauss -- in general the problem remains open, notwithstanding the algebraic, arithmetic and analytic theories developed to address it.

In this talk we investigate the size of the set of integers represented. The main result is a "Near Hasse Principle" -- an integral quadratic form represents 100% of the integers that it locally represents. This leads to an asymptotic for the number of integers of absolute value at most X represented. When there are at least three variables, this amounts to determining the density of the set of represented integers. We then address -- but do not fully resolve -- the inverse problem of which densities arise.

This is joint work with Paul Pollack, Jeremy Rouse and Kate Thompson.

Dustin Cartwright - Dual complexes of uniruled varieties

The dual complex of a semistable degeneration records the combinatorics of the intersections in the special fiber. The dual complex is homotopy equivalent to the Berkovich analytification, which is a non-Archimedean analogue of the analytic topology on a complex algebraic variety. I will talk about the following relation between the geometry of the general fiber and the topology of the dual complex, and thus the analytification: in arbitrary characteristic, the dual complex of an n-dimensional uniruled variety has the homotopy type of an (n-1)-dimensional simplicial complex.

Pretalk - Various properties of varieties containing the word "rational": A rational variety is on that contains a dense open set isomorphic to an open subset of projective space. Although this definition is easy to state, it turns out to be difficult to verify in examples. Instead, algebraic geometers have noticed that it's easier to classify varieties not in terms of their rationality, but in terms of the existence of rational curves in the varieties. This leads to the notion of rationally connected varieties, ruled varieties, and other variations on these concepts.

Alexander Duncan - Cubic surfaces in positive characteristic

I describe the possible automorphism groups of a cubic surface over an algebraically closed field of arbitrary characteristic. We will see that, while the actual groups that appear in different characteristics can vary considerably, the differences can all be explained by a handful of simple geometric observations.

Pretalk: Given two curves in the complex projective plane, Bezout's theorem provides a simple formula for the number of intersection points (counting multiplicities). The goal of intersection theory is to generalize this idea to intersections of subvarieties of more general varieties. I will discuss the intersection theory of surfaces with an eye towards describing the configuration of 27 lines on a cubic surface.

Blake Farman - Noncommutative projective schemes and DG categories

In their 1994 paper, Noncommutative Projective Schemes, Michael Artin and J.J. Zhang introduce a generalization of usual projective schemes to the setting of not necessarily commutative algebras over a commutative ring. Gonçalo Tabuada in 2005 endows the category of differential graded categories with the structure of a model category and in 2007 Toën shows that its homotopy category is symmetric monoidal closed. In this talk, we’ll give a brief overview of these results, adapting Artin and Zhang’s noncommutative projective schemes for the language of DG categories, and discuss a “geometric” description of this internal Hom for two noncommutative projective schemes.

Eloisa Grifo - Symbolic powers and the Containment Problem

Symbolic powers consist of the functions that vanish up to a certain order at each point in a given variety. The Containment Problem for symbolic and ordinary powers of ideals asks when the containment I^(a) ⊆ I^b holds. If I is a radical ideal in a regular ring, a famous result by Ein-Lazersfeld-Smith, Hochster-Huneke and Ma-Schwede partially answers this question. Harbourne proposed an improvement on this result, which unfortunately does not hold in full generality. However, in this talk we will discuss versions of Harbourne’s Conjecture that do hold. In particular, we will discuss some joint work with Craig Huneke on the case when R/I is an F-pure ring, and give evidence that Harbourne’s conjecture may always hold eventually.

Bastian Haase - Gerbe patching over arithmetic curves with a view towards homogeneous spaces

We will discuss patching techniques and local-global principles for gerbes over arithmetic curves. The patching setup is the one introduced by Harbater, Hartmann and Krashen. The results obtained for gerbes can be viewed as a 2-categorical analogue on their results for torsors. Along the way, we also discuss bitorsor patching and local global principles for bitorsors. As an application of these results, we will study local-global principles for homogeneous spaces through their quotient stacks.

Pretalk: Field Patching

In this talk, we will give an introduction to the patching technique introduced by Harbater, Hartmann and Krashen. This technique allows to study algebraic objects such as quadratic forms or central simple algebras over a fixed field via studying the same objects over a finite system of overfields. After discussing the general framework, we will focus on the case of arithmetic curves and discuss a couple of recent results obtained via this technique.

Jesse Kass - How to count lines on the cubic surface arithmetically?

A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. Over the real numbers, the number of lines on a smooth cubic surface depends on the surface, but Segre showed that a certain signed count of lines is the same for such surfaces. In my talk, I will explain Segre’s result and then extend that result to an arbitrary field. This is an application of A1-homotopy theory.

All work is joint with Kirsten Wickelgren.

Andrew Kustin - Use a Macaulay inverse system to detect an embedded deformation

Let k be a field of characteristic not equal to 2, P be a standard-graded polynomial ring in four variables over k, and A=P/I be an Artinian Gorenstein k-algebra of embedding dimension four which is defined by six homogeneous forms and has socle degree three. We use the homogeneous Macaulay inverse system that corresponds to I to prove that A has an embedded deformation. In other words, we prove that there is an ideal J in P and an element f in P with f regular on P/J and I=(J,f). (It quickly follows that f is a quadratic form in P and that J is a 5-generated grade three Gorenstein ideal generated by the maximal order Pfaffians of a 5 by 5 alternating matrix of linear forms.)

The previous best results describing the structure of six-generated grade four Gorenstein ideals assume that P/I is a generic complete intersection. We treat the case when P/I is a graded Artinian algebra over a field at the expense of assuming that the socle degree of P/I is three.

The talk is about joint work with Sabine El Khoury from the American University of Beirut.

Pretalk: If one is working in characteristic zero, then the Macaulay Inverse System for an Artinian Gorenstein ring P/I, with socle degree 3, is a homogeneous form of degree three in the polynomial ring generated by the four partial derivative operators. So, in some sense, the talk is about cubic surfaces in projective 3-space. In the pre-talk I will explain how to use Divided Powers to make the notion of Macaulay Inverse System both coordinate free and characteristic free.

Patrick McFaddin - Exceptional collections on toric varieties

Exceptional collections of a variety X are effectively decompositions of the derived category of X, analogous to an (semi-)orthonormal basis of a vector space with inner-product. The existence of exceptional collections for smooth projective toric varieties defined over the complex numbers was settled affirmatively by Kawamata using the framework afforded by the toric Minimal Model Program. More recently, Ballard, Favero, and Katzarkov have given another proof using Variation of GIT. The study of toric varieties defined over arbitrary fields (so-called arithmetic toric varieties) has been taken up by a number of authors, although much less is known about their derived geometry. In this talk, we will discuss an effective Galois descent result for such collections and provide applications to arithmetic toric varieties of dimension two, three, and four. This is joint work with M. Ballard and A. Duncan.

Keller Vandebogert - Properties of Local Rings with Decomposable Maximal Ideals

This talk will start with two conditions due to a paper of Huneke and Jorgensen called the Auslander Condition and the Uniform Auslander Condition, and show that neither of these properties is necessarily preserved under localization. After some discussion of the constructions demonstrating this, we will move on to an examination of fiber products of finite Cohen-Macaulay type. We start with a relation between rings realized as nontrivial fiber products and induced isomorphisms between integral closures in the total fraction fields, and look more closely at how module structure over a nontrivial fiber product gives information about the structure over the individual pieces of that fiber product, and vice versa. Lastly, the main result for the talk will be stated and if time permits further discussion will ensue.

Pretalk: We will go over the necessary (nontrivial) terminology to digest the material of the main talk. This will include, but is not limited to: fiber products, Poincare/Bass series, hypersurfaces, depth/codepth/type/embedding codepth, Hilbert-Samuel multiplicity, semidualizing modules, and some examples for each of the aforementioned terms.

Xiaofei Yi - Algebraic Linkage

Two projective varieties in P³ are directly linked if their union is a complete intersection. Two projective varieties are linked if they can be connected by a sequence of direct links. Two ideals I and J are directly linked if there is a regular sequence α in the intersection such that I:(α)=J and J:(α)=I. And they are linked if they can be connected by a sequence of direct links. Some properties are invariant under linkage, but some properties require a "good" linkage class to be invariant. In the seminar, we will discuss some properties invariant under linkage class, and then discuss a technique to construct a good linkage class. If time allows, we will also discuss the linkage in the homogeneous case.

Last semester's seminar.