Commutative Algebra & Algebraic Geometry

CAAG






The seminar will meet on selected Fridays 4-5 PM at the CUNY Graduate Center in room 6417.
The CUNY Graduate Center is located in 365 Fifth Avenue, New York, NY 10016.

Organizers:
Laura Ghezzi, New York City College of Technology (CUNY), lghezzi@citytech.cuny.edu
Hans Schoutens, New York City College of Technology and the Graduate Center (CUNY), hschoutens@citytech.cuny.edu

(For previous semesters click here)


Fall 2017

Friday, October 6
Speaker: Malgorzata Marciniak, La Guardia Community College, CUNY
Title: Hessian Dynamics on the Hesse pencil
 
Abstract: Our ultimate goal is to answer the question of whether the group of rational points on an elliptic curve $C$ is related to the group of rational points on its Hessian $H(C)$. Prior to attacking this problem, we considered the Hessian dynamics on the pencil of cubic curves (called the Hesse pencil). Here we describe and analyze the Hessian dynamics in terms of projective coordinates on the space of parameters on the Hesse pencil. The presentation will be based on research activities conducted for the Beginners Explorations in Algebraic Geometry (BEAG) during the academic year 2016/17.


Friday, October 20
Speaker: Fei Ye
Title: Multiplier Ideals and Some Applications to Adjoint Line Bundles

Abstract: Multiplier ideals are associated with ideals on algebraic variety and satisfy certain vanishing theorems. In this talk, I will explain how we can associate a maximal ideal with a multiplier ideal. As applications, I will present two results on adjoint line bundles on smooth projective varieties of dimension 3.



Friday, November 3
Speaker: Hans Schoutens (NYCCT and GC)
Title: From families of Weyl algebras to Bernstein algebras

Abstract:

Example 1: let $x$ and $y$ be non-commuting variables over the affine line $\text{Spec}(k[t])$ and consider the `variety' with equation $xy-yx=t$ and coordinate ring $B_1$; its fibers are all Weyl algebras, except above the origin, which is a polynomial ring;

Example 2: the forgetfull functor from the category of (non-commutative) algebras to the category of modules with an alternating form (given by the commutator bracket) admits an adjoint, associating to a module with an alternating form its `universal alternating algebra' $B_2$;

Example 2bis: the forgetfull functor from the category of (commutative) algebras with a Poisson bracket to the category of modules with an alternating form admits an adjoint, associating to a module with an alternating form its `universal Poisson algebra' $B_3$.

I want to describe a framework that (i) will encompass the algebras $B_i$ described in these examples; and (ii) allows for a general structure theory as in the first example. This is what I call a Bernstein algebra: a filtered (non-commutative) algebra whose degree zero elements are central, generated by finitely many linear elements, whose commutators have degree zero. The algebras $B_1$ and $B_2$ are both Bernstein, whereas $B_3$ is realized as the associated graded ring of a Bernstein algebra. To obtain a good structure theory, we require that the Bernstein algebra be smooth, which will in particular imply that its associated graded ring is a polynomial ring (the above examples are of this type).




Friday, November 17
Speaker: Bart van Steirteghem
Title: Smooth affine spherical varieties: classification and application

Abstract: A smooth affine complex variety, equipped with an action of a reductive group G, is called spherical if its coordinate ring is multiplicity free as a representation of G. I will give an overview of the classification(s) of these varieties, including joint work with G. Pezzini and F. Knop. I will also show how, thanks to work of Knop's, the combinatorics of such varieties can be used to classify the (real) multiplicity free Hamiltonian manifolds of symplectic geometry. As an illustration, I will explain (following joint work with Pezzini and K. Paulus) how one can recover C. Woodward's result that every reflective Delzant polytope is the momentum polytope of such a manifold.



Friday, December 1
Speaker: Jai Laxmi  (Indian Institute of Technology, Bombay)
Title: Embeddings of Canonical Modules

Abstract: It is well-known that, for a Cohen-Macaulay local ring $S$ with a canonical module $\omega_S$, if $S$ is generically Gorenstein, then $\omega_S$ can be identified with an ideal of $S$, that is, $\omega_S$ embeds into $S$.
In this talk, we are concerned with a specific embedding of a canonical module of $R/I_{m,n}$ to itself, where  $I_{m,n}$ is an ideal generated by all square-free monomials of degree $m$ in a polynomial ring $R$ with $n$ variables. We discuss how to construct such an embedding using a minimal generating set of $\text{Hom}_R(R/I_{m,n}, R/I_{m,n})$.
 
This talk is based on a recent joint work with Ela Celikbas and Jerzy Weyman.