Algebra, Geometry, and Number Theory Seminar

Abstracts

Asilata Bapat - The Calogero--Moser space is a symplectic algebraic variety that deforms the Hilbert scheme of points on a plane. It can be interpreted in many ways, such as the parameter space of irreducible representations of a Cherednik algebra, or as a Nakajima quiver variety. In this talk, I will describe work in progress towards constructing quiver GIT compactifications of the Calogero-Moser space and the Hilbert scheme of points. I will also discuss some related combinatorics.

Anand Deopakur - In algebraic geometry, a 'ribbon' is a particular kind of non-reduced curve; locally, it looks like a the product of a smooth curve and the spectrum of the dual numbers. Strangely enough, understanding these peculiar objects has far-reaching implications towards understanding the algebra and the geometry of smooth curves and their moduli space. I will explain this connection, describe recent progress, and outline some future prospects.

Daniel Erman - We consider Noether normalization over a finite field from a probabilistic perspective, and use this prove several applications including: an effective Noether normalization result over a finite field, and a Noether normalization result over the integers. Our method involves a higher-dimensional variant of Poonen’s closed point sieve.

Reed Gordon-Sarney - Let k be a field, let G/k be a smooth connected linear algebraic group, and let X be a G-torsor over k. Generalizing a question of Serre, Totaro asked if the existence of a zero-cycle on X of positive degree d implies the existence of closed etale point on X of degree dividing d. We settle Totaro's question affirmatively for algebraic tori of rank less than or equal to 2 and present fundamental open questions that arise from this work.

Andrew Kustin - Let R be a non-negatively graded Cohen-Macaulay ring with R_0 a Cohen-Macaulay factor ring of a local Gorenstein ring. Let d be the Krull dimension of R, m be the maximal homogeneous ideal of R, and M be a finitely generated graded R-module. It has long been known how to read information about the socle degrees of the local cohomology module H_m^0(M) from the twists in position d in a resolution of M by free R-modules. It has also long been known how to use local cohomology to read valuable information from complexes which approximate resolutions in the sense that they have positive homology of small Krull dimension. In this talk, we read information about the maximal generator degree (rather than the socle degree) of H_m^0(M) from the twists in position d-1 (rather than position d) in an approximate resolution of M. This is joint work with Claudia Polini and Bernd Ulrich.

Tyler Lewis - Maurice Auslander and Idun Reiten in 1989 proved the classification for scrolls of finite Cohen-Macaulay representation type. We will show a family of rings with a simple codimension two Cohen-Macaulay singularity, which were classified by Anne Frübhis-Krüuger and Alexander Neumer in 2010, has infinite Cohen-Macaulay representation type. The proof is divided into two parts: existence, which uses a similar construction as Auslander and Reiten, and uniqueness, which relies on the computation of Fitting ideals.

Patrick McFaddin - The theory of algebraic cycles on homogenous varieties has seen many useful applications to the study of central simple algebras, quadratic forms, and Galois cohomology. Significant results include the Merkurjev-Suslin Theorem and Suslin's Conjecture, recently given a conditional proof by Merkurjev. Despite these successes, a general description of Chow groups (with coefficients) remains elusive, and computations of these groups are done in various cases. In this talk, I will give some background on K-cohomology groups of Severi-Brauer varieties and discuss some recent work on computing these groups for algebras of index 4.
Pre-talk (Central Simple Algebras and Severi-Brauer varieties): A central simple algebra over a field F is a twisted matrix algebra. The collection of all such algebras over F forms a group Br(F) which yields arithmetic information about F and appears naturally in class field theory. Similarly, a Severi-Brauer variety is a twisted form of projective space and has long history of study, dating back to Severi's work in the 1930's. In this talk, I will introduce these objects as well as the process of associating to any algebra A a Severi-Brauer variety SB(A) which encodes much of the structural information of A through its collection of left ideals. This process is analogous to forming the prime spectrum of a commutative ring.

Jackson Morrow - In Mazur’s celebrated 1978 Inventiones paper, he classified the torsion subgroups which can occur in the Mordell-Weil group of an elliptic curve over \mathbf{Q}. His result was extended to elliptic curves over quadratic number fields by Kamienny, Kenku, and Momose, with the full classification being completed in 1992. What both of these cases have in common is that each subgroup in the classification occurs for infinitely many elliptic curves; however, this no longer holds for cubic number fields. In 2012, Najman showed that there exists a unique (up to \overline{\mathbf{Q}}-isomorphism) elliptic curve whose torsion subgroup over a particular cubic field is $\mathbf{Z}/21 \mathbf{Z}$. This curve yielded the first sporadic example of a torsion subgroup.

In this talk, we will recall previous literature concerning torsion subgroups of elliptic curves over number fields, introduce new results about sporadic points on the modular curves $X_1(N)$ and $X_1(M,N)$, and discuss the tools used in the analysis of cubic points on these modular curves.

Pre-talk (Torsion subgroups of elliptic curves and explicit methods in arithmetic geometry): In this talk, we will discuss constructions of modular curves, torsion subgroups of elliptic curves over number fields, and some explicit methods in Diophantine finiteness.

Andrew Niles - It is a well-known result of Mumford that over an algebraically closed field of characteristic coprime to 6, the Picard group of the stack M_{1,1} of elliptic curves is cyclic of order 12. But only recently was this result extended to encompass more general base schemes; a 2010 result of Fulton and Olsson computes this Picard group whenever 2 is invertible on the base scheme, and for arbitrary reduced base schemes.

In this talk, we discuss how to partially extend the results of Mumford and Fulton-Olsson to certain stacks of elliptic curves equipped with level structure. We compute the Picard groups of the stacks Y_0(2) and Y_0(3), over any base scheme on which 6 is invertible.

Padmavathi Srinivasan - Conductors and minimal discriminants are two measures of degeneracy of the singular fiber in a family of hyperelliptic curves. In the case of elliptic curves, the Ogg-Saito formula shows that (the negative of) the Artin conductor equals the minimal discriminant. In the case of genus two curves, equality no longer holds in general, but the two invariants are related by an inequality. We investigate the relation between these two invariants for hyperelliptic curves of arbitrary genus.

Arul Shankar - A conjecture of Bhargava describes the density of discriminants of degree-n S_n number fields. Work of Kedlaya extends the reach of this conjecture to many other families of number fields. In this talk, we will consider the family of D_4 fields, the simplest family of nonabelian and non S_n fields. I will explain how traditional strategies fall short in analyzing this family, and how these shortcomings may be overcome by utilizing additional algebraic structure arising from the outer automorphism of D4. This is joint work with Ali Altug, Ila Varma, and Kevin Wilson.

Last semester's seminar.