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)

Spring 2019

Friday, Feb 15
Speaker: Giovan Battista Pignatti (GC)
Title: Reinterpreting birational geometry through the theory of valuation rings.

Abstract: The Zariski-Riemann space ZR(K|k) of a field extension K|k is the set of all valuation rings between the two fields. It is naturally endowed with a topology. To every collection of valuation rings it is possible to associate a normal k-variety. Using a special class of open coverings of ZR(K|k), we will build a category which turns out to be equivalent to the category of all proper, normal k-varieties with birational morphisms. This allows to revisit many geometric notions only using the purely algebraic language of valuation rings. (The talk is based on a set of notes by Hans Schoutens which is still work in progress. Participation of the audience is welcomed and encouraged).

 

Friday, Mar 1
Speaker: Ben Blum-Smith (Eugene Lang College / The New School)
Title: Bounding the degree of generation of invariant fields

Abstract: It has long been of concern to invariant theorists to know something about the degrees in which invariant rings are generated. For example, a classical result of E. Noether states that if G is a finite group acting linearly on a polynomial ring over a field k, then the invariant ring is generated by polynomials of degree at most the order of G (subject to a restriction on the characteristic of k). In this talk, we consider the analogous question for invariant fields. This question is relevant to applications, for example in biological imaging. We show that various results from linear algebra on tensor decomposition have interpretations in terms of the degree of generation of invariant fields, and provide some notes toward a general theory in the case that G is abelian and k has characteristic 0.

 

Friday, Mar 15
Speaker: Fei Ye, QCC (CUNY)
Title: Introduction to D-modules

 

Friday, Mar 29, 4:00-5:00PM
Speaker: Mingyi Zhang (Northwestern University)
Title: Hodge ideals and Hodge filtration of a left D-module associated to Q-divisor with certain isolated singularities

Abstract: Given a singular subvariety Z on a smooth complex variety X, Saito studied a special D-module O_X(*Z) which underlines a mixed Hodge module. It carries a canonical Hodge filtration. In general, it is hard to describe this filtration. However, when Z has an isolated singularity given by a weighted homogeneous polynomial, Saito gave an explicit formula of the Hodge filtration. On the other hand, this D-module corresponds to a series of ideal sheafs, called Hodge ideals, associated wit the integral divisor Z. I will give a crash introduction of Hodge ideals and Hodge filtration of a D-module associated with Q-divisor, and how to generalize Saito's formula to the Q-divisor case.

 

Friday, April 12, 4:00-5:00PM
Speaker: Uma Iyer (BMCC and the Graduate Center)
Title: A question on path algebras

Abstract: Lunts and Rosenberg defined the algebra of differential operators on noncommutative algebras. One major source of noncommutative algebras are the path algebras of quivers. We introduce a study of the algebra of differential operators on the path algebra of an arbitrary finite quiver.

 

Tuesday, April 30, 5:00-6:00PM in room 5212 (note special time, place, and day)
Speaker: Nero Budur (University of Leuven)
Title: Jets, quivers, and SL_n(Z)

Abstract: Using jet schemes and quivers, we show that the space of representations of the fundamental group of a compact Riemann surface of genus at least two has rational singularities. We apply this to show that the number of irreducible complex representations of SL_n(Z) of dimension at most m grows at most as the square of m, for a fixed n>2.


 

Fall 2018

Friday, Sep 28 (oral defense)
Speaker: Giovan-Battista Pignatti (GC)
Title: Infinite directed unions of local quadratic transforms (abstract).

 

Friday, October 5
Speaker: Jai Laxmi  (Indian Institute of Technology, Bombay)
Title: Tate resolutions and deviations of graded algebras (abstract).

 

Friday, October 19
Speaker: Ethan Cotterill (Boston College)
Title: Real inflection points of real linear series on real hyperelliptic curves (joint with I. Biswas and C. Garay López)

Abstract: According to Plucker's formula, the total inflection of a linear series (L,V) on a complex algebraic curve C is fixed by numerical data, namely the degree of L and the dimension of V. Equipping C and (L,V) with compatible real structures, it is more interesting to ask about the total real inflection of (L,V). The topology of the real inflectionary locus depends in a nontrivial way on the topology of the real locus of C. We study this dependency when C is hyperelliptic and (L,V) is a complete series. We first use a nonarchimedean degeneration to relate the (real) inflection of complete series to the (real) inflection of incomplete series on elliptic curves; we then analyze the real loci of Wronskians along an elliptic curve, and formulate some conjectural quantitative estimates.

 

Friday, November 2
Speaker: Ben Blum-Smith (Eugene Lang College / The New School)
Title: Cohen-Macaulayness of invariant rings is determined by inertia groups

Abstract: If a finite group G acts on a Cohen-Macaulay ring A, and the order of G is a unit in A, then the invariant ring A^G is Cohen-Macaulay as well, by the Hochster-Eagon theorem. On the other hand, if the order of G is not a unit in A, then the Cohen-Macaulayness of A^G is a delicate question that has attracted research attention over the last several decades, with answers in several special cases but little general theory. In this talk we show that the statement that A^G is Cohen-Macaulay is equivalent to a statement quantified over the inertia groups for the action of G on A, acting on strict henselizations of appropriate localizations of A. In a case of long-standing interest — a permutation group acting on a polynomial ring — we show how this can be applied to find an obstruction to Cohen-Macaulayness that allows us to completely characterize the permutation groups whose invariant ring is Cohen-Macaulay regardless of the ground field. This is joint work with Sophie Marques.

 

Friday, November 16
Speaker: Hans Schoutens (CUNY)
Title: Total blow-up ring along a valuation.

Abstract: Given a smooth variety $X$, any valuation on its function field has a unique center $x\in X$. This then determines a unique local blowing up at $x$ along the valuation. Abhyankar showed that when $X$ has dimension two, iterating this construction recovers the valuation ring; later, his student Shannon gave counterexamples in dimension three. So, two natural questions emerge: when do we get the valuation ring, and if we don't, what do we get? I will concentrate on the latter question, continuing the discussion of Giovan-Battista from last month. (This is joint work with Loper, Heinzer, Olberding and Toeniskotter).

 

 

Friday, November 30
Speaker: Chuanhao Wei (Stonybrook University)
Title: Zeros of log-one-forms and families of log-varieties

Abstract: I will introduce the result about the relation between the zeros of holomorphic log-one-forms and the log-Kodaira dimension, which is a natural generalization of Popa and Schnell's result on zeros of one-forms. Some geometric corollaries will be stated, e.g. algebraic Brody hyperbolicity of log-smooth family of log-pairs of log-general type, which also serves as a motivation to the main theorem.

 

 


Spring 2018

Friday, February 2
Speaker: Jai Laxmi  (Indian Institute of Technology, Bombay)
Title: Decomposing Gorenstein Rings as Connected Sums

Abstract: In 2012, Ananthnarayan, Avramov and Moore gave a new construction of Goren-
stein rings from two Gorenstein local rings, called their connected sum. Given a Gorenstein
Artin ring, we investigate conditions on the ring which force it to be indecomposable as
a connected sum. We will see characterizations for Gorenstein Artin local rings to be de-
composable. Finally, we will show that the indecomposable components appearing in the
connected sum decomposition are unique up to isomorphism.

This is joint work with H. Ananthnarayan, E. Celikbas and Z. Yang.

 

Friday, February 16
Speaker: Andrew Stout  (BMCC)
Title: An Introduction to Algebraic Spaces

Abstract: Grothendieck's celebrated results on the existence of Hilbert spaces as schemes required projectivity assumptions -- a fact that one can detect proved frustrating as one reads through the details in FGA. Inspired by ideas from deformation theory, M. Artin found that we may remove projectivity if we expand the category of schemes to the new category of algebraic spaces. Roughly speaking, a quasi-separted algebraic space is a sheaf in the etale topology which contains a dense open scheme. The density assumption and the fact that the etale topology is invariant under infinitesimal thickenings makes it possible to extend geometric properties of schemes to these functors. Although these functors are not geometric spaces, they behave well enough to allow for an extension of most major results from scheme theory to algebra spaces (e.g., Stein factorization, Chow's lemma, Finiteness of Cohomology, GAGA). Furthermore, every algebraic space may also be viewed as a quotient of a scheme modulo an etale equivalence relation, which immediately leads to numerous examples. This material also serves as motivation to the larger theory of algebraic stacks. In this talk, I will offer an accesible overview of the material above as time permits.

 

Friday, March 2
Speaker
: Laura Ghezzi  (CityTech)
Title: The canonical degree and almost Gorenstein rings

Abstract: This talk is a sequel to a previous one on “Invariants of Cohen Macaulay rings associated to their canonical ideals”. The big picture is to refine our understanding of Cohen-Macaulay rings (which are not Gorenstein). Let R be a Cohen-Macaulay local ring of dimension one with a canonical ideal. First we review the definition of canonical degree and the necessary background. We then show that almost Gorenstein rings are exactly those rings that have minimal canonical degree. We also discuss generalizations of this result and open questions.



 
Friday, April 13
Speaker
: Shizhuo Zhang (Indiana University)
Title: Exceptional collection of line bundles on rational surfaces and its application to several conjectures.

Abstract: After recalling the definition of (strong) exceptional collection of line bundles on smooth projective surface X, I will introduce a technical tool called toric system coming from exceptional collection invented by L.Hille and M.Perling and various operations on it. Hille and Perling conjectured that all the full strong exceptional toric system is coming from augmentation(an analogue operations of blow up of a point) from the ones on P^2 and Hirzebruch surfaces. I will show that the conjecture holds for cyclic strong exceptional toric systems on any rational surface. As a result, I will show that the existence of cyclic strong exceptional toric systems will imply the rationality of smooth projective surfaces and the anticanonical divisor -K_X is big and nef. Then, I will give a counter example of the conjecture on a weak del Pezzo surface of degree 2. I will talk about the application of toric systems in several conjectures of D.Orlov on dimension of $D^b(coh X)$ for smooth variety, rationality of surfaces, classifications of 2-hereditary tilting bundles and Brill-Noether problems on rational surfaces, and moduli spaces of quiver representations.

 

Friday, April 27
Speaker
: Ruijie Yang (Stonybrook)
Title: On irrationality of hypersurfaces in projective spaces

Abstract: A classical question in algebraic geometry is to determine whether or not a variety is rational. In this talk, I will focus on its opposite side: how "irrational" a variety can be? In particular, I will discuss my work on certain measures of irrationality of hypersurfaces of large degree in projective spaces.

 



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.