Commutative Algebra & Algebraic Geometry
CAAG

Note: seminars at CUNY are now all virtual

The seminar will meet via ZOOM SESSIONS on selected Fridays 4:00-5:00 PM

To obtain the zoom link for our seminar for the Spring 2021 Semester (do not use the one from last semester!), please register here and you will be provided with a personal link and password to join the seminar on all the scheduled dates

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

Fei, Ye, Queensborough Community College (CUNY), Feye@qcc.cuny.edu

Spring 2021

Friday, Feb 26, 4:00PMSpeaker: Saeed Nasseh (Georgia Southern U)Title: Lifting theory for differential graded modules

Abstract: Lifting property of modules is a fundamental question in commutative ring theory, which is tightly connected to deformation theory of modules and has important applications in the theory of maximal Cohen-Macaulay approximations. This notion was studied by Auslander, Ding, and Solberg for modules and by Yoshino for complexes. In this talk, I will survey recent developments on the lifting theory for differential graded (DG) modules over strongly commutative and non-negatively graded DG algebras and describe the relation between this notion and the long-standing Auslander-Reiten conjecture. This investigation requires some new ideas using homotopy limits and diagonal ideals.

Friday, Mar 26, 4:00PMSpeaker: Takumi Murayama (Princeton University)Title: Boutot's theorem holds for quasi-excellent Q-algebras (click here for the recording of the talk)

Abstract: Let S be a regular ring. By work of Hochster-Roberts, Boutot, Smith, Hochster-Huneke, Schoutens, and Heitmann-Ma, we know that every pure subring of S is pseudo-rational, hence Cohen-Macaulay. This applies in particular to rings of invariants of linearly reductive groups. For pure maps R -> S of rings essentially of finite type over fields of characteristic zero, Boutot's result is even stronger: if S is pseudo-rational, then R is pseudo-rational. We show that Boutot's theorem holds more generally if R and S are quasi-excellent Q-algebras, which we use to give new proofs of the theorems of Hochster-Huneke and Schoutens. This solves a conjecture of Boutot and answers a question of Schoutens in the quasi-excellent case. To prove these results, we show that the Grauert-Riemenschneider vanishing theorem holds for Q-schemes, solving a conjecture of Boutot.

Friday, Apr 9, 4:00PMSpeaker: Linquan MaTitle: Globally +-regular varieties

Abstract: We generalize the theory of globally F-regular pairs to mixed characteristic, which we call +-regularity, and introduce certain stable sections of adjoint line bundles. This is inspired by recent work of Bhatt on the Cohen-Macaulayness of the absolute integral closure. We will discuss some applications of these results to birational geometry in mixed characteristic. Joint work with Bhargav Bhatt, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron, and Jakub Witaszek.

Speaker: James Myer (GC)Title: The alterations paradigm shift for the problem of resolution of singularities part III: the three point lemma (video recording of lecture).

Abstract: As we’ve seen in two previous talks, the construction given in Smoothness, Semistability, and Alterations by Aise Johan de Jong (1996) of an alteration of any variety by a regular variety begins by reducing immediately to the case of a family of curves whose generic fiber is smooth of genus g with at least 3 marked points on each irreducible component of each fiber. Consequently, there is a morphism from an open subset of the base of the family to the moduli stack of stable curves of genus g with n marked points. Morally, projection onto the domain factor from the closure of the graph of this morphism yields an alteration of the base over which our old family extends to a family of stable curves. Now, the old family whose fibers could have terrible singularities has been improved to a new family whose fibers have at-worst nodal singularities. There is a little issue, however: there is not necessarily a morphism from the new family onto the old family, only a not-everywhere-defined birational map!

To show that this birational map from the new to the old family is indeed a morphism, we study the projection onto the domain factor from the closure of the graph of the map. We may arrange that the new family is normal, and so the projection is a birational morphism onto a normal variety. If we can show that this morphism is finite, i.e. has finite fibers, Zariski’s Main Theorem will guarantee that the morphism is an isomorphism, and so we will have a morphism from the new family to the old family after all. While trying to show that the fibers of this morphism are finite — essentially, that there are no “contractions” — the 3 points we’ve marked on each irreducible component of each fiber will come to the rescue!

Fall 2020

Friday, Oct 2Speaker: James Myer (GC)Title: The alterations paradigm shift for the problem of resolution of singularities

Abstract: Heisuke Hironaka solved the problem of resolution of singularities for a variety over a field of characteristic zero in 1964, although the story goes that his proof was so complicated as to stump Alexander Grothendieck! Efforts have been made since then to simplify the proof and see if it can be made to work for positive characteristic, but to no avail.

Regardless of characteristic:
The singularities of a curve are resolved in one fell swoop by the normalization, a powerful tool hailing from algebra made to work for us in geometry, by Jean-Pierre Serre’s Criterion for Normality, and we owe much to Oscar Zariski for teaching us about the normalization.  Joseph Lipman dealt with surfaces in as much generality as one could hope for.  Threefolds are rumored to have been handled by Vincent Cossart and Olivier Piltant, although their proof is quite long. Fourfolds and up are uncharted territory for the most part.

In his 1996 paper Smoothness, Semi-Stability, and Alterations, Johan de Jong introduces a paradigm shift for solving the problem of resolution of singularities by relaxing a resolution of singularities to what he calls an alteration. The distinction is that a resolution is a birational morphism (generically one-to-one), whereas an alteration is a generically finite morphism (generically finite-to-one). Every variety can be altered to a nonsingular variety regardless of the characteristic. In fact, de Jong’s technique paired up with Dan Abramovich’s geometric insight yields a proof that every variety over a field of characteristic zero admits a resolution of its singularities in a paper consisting of only twelve pages, see Smoothness, Semi-Stability, and Toroidal Geometry. de Jong’s technique relies on (at least) two ingenious ideas:
The first is the statement that there is a simple blowup of any variety that admits a morphism to a projective space of one less dimension whose fibers are curves ~ intuitively, any variety can be modified to be a family of curves. Taking the intermediate variety in what is known as the Stein factorization of the morphism instead of the target projective space ensures the curves in the family are connected.
The second is that the curves in the family of curves that results can be marked so as to become stable (in the sense of Pierre Deligne and David Mumford) so that it gives rise to morphism into the moduli space of stable curves, where we may take advantage of established facts about the moduli space of stable curves, including the fact that its compactification has curves with at worst nodal singularities.
My talk has the goal of introducing Johan de Jong’s paradigm shift, and indicating, to the extent that I can, the idea of the proof that every variety may be altered to a nonsingular variety. An emphasis will be placed on the picture that demonstrates that there is a simple blowup of any variety that admits a morphism to a projective space of one less dimension whose fibers are curves.

Friday, Oct 16

Speaker: James Myer (GC)Title: The alterations paradigm shift for the problem of resolution of singularities (part 2) (click here for a recording of the talk)

Abstract:See above.

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Friday, Oct 30
Speaker: Miodrag Iovanov (Univ. of Iowa)
Title: Diamond Rings - a class of rings close to Noetherian (click here for a recording of the talk)

Abstract. A ring is said to have property D(=diamond) if the injective hull E of any simple module is locally Artinian, that is, a finitely generated submodule of such an E is Artinian. Commutative Noetherian rings are known to have property D. This property is of interest in noncommutative rings, as Noetherian rings don't always have it, and it is closely related to outstanding questions in algebra such as Jacobson's conjecture (which states that the intersections of the powers of the Jacobson radical of a noncommutative Noetherian ring is zero).

Perhaps somewhat surprisingly, this has not been studied much for the commutative case. We present several motivating examples and questions from the non-commutative case, and then describe such recent results for the commutative case. It turns out rings with this property D are exactly rings which have "enough" modules of finite length. This simple observation leads to generalizations of several classical results of commutative Noetherian algebra - such that the Krull intersection theorem - to the generality of rings with property D, and which recover also results of Jans, Matlis, Vamos. On the other hand, we show that for complete rings (with a suitable natural topology consisting of ideals of co-finite length), property D is equivalent to the ring being Noetherian, which further points to property D being quite "close" to Noetherian.

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Friday, Nov 13
Speaker: Xiaolei Zhao (University of California, Santa Barbara)
Title: Title: Elliptic quintics on cubic fourfolds, O'Grady 10, and Lagrangian fiberations (click here for a recording of the talk).

Abstract: We will show how to construct hyperkähler manifolds of O'Grady 10 type out of smooth cubic fourfolds. Applications to classical algebraic geometry questions on intermediate jacobian fibrations of families of cubic threefolds, and quintic elliptic curves on cubic fourfolds will be explained. This is based on joint work with Chunyi Li and Laura Pertusi.

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Friday, Dec 4
Speaker: Federico Caucci (University of Florence)
Title: The basepoint-freeness threshold of a polarized abelian variety (click here for a recording of the talk).

Abstract: We will introduce the basepoint-freeness threshold of a polarized abelian variety, as defined by Jiang and Pareschi, and explain its relation with the syzygies of the section algebra of the polarization. After an overview on equations defining abelian varieties and higher syzygies, we will mainly focus on a conjectural upper bound for this threshold which has been very recently studied by Ito and Jiang. If true, it would have a direct application to syzygies of (primitive) polarized abelian varieties, that can be seen as a revised and extended version of the generalized Fujita's freeness conjecture, in the case of abelian varieties.

Spring 2020

Friday, Feb 14
Speaker: Daniele Bartoli (Università Degli Studi di Perugia)
Title: Applications of curves over finite fields to polynomial problems

Abstract: Algebraic curves over finite fields are not only interesting objects from a theoretical point of view, but they also have deep connections with different areas of mathematics and combinatorics. In fact, they are important tools when dealing with, for instance, permutation polynomials, APN functions, planar functions, exceptional polynomials, scattered polynomials, Moore-like matrices. In this talk I will present some applications of algebraic curves to the above mentioned objects.

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Friday, Feb 28
Speaker: Jai Laxmi (University of Connecticut)
Title: Spinor structures on free resolutions of codimension four Gorenstein ideals

Abstract: The problem of classifying Gorenstein ideals of codimension four was around ever since Buchsbaum-Eisenbud classified Gorenstein ideals of codimension three. The first result related to the structure of codimension four case were obtained by Kustin and Miller. We analyze the structure of spinor coordinates on resolutions of Gorenstein ideals of codimension four. As an application we produce a family of such ideals with seven generators which are not specializations of Kustin-Miller model.

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Friday, Mar 13(CANCELED)

Speaker: Malgorzata Marciniak (Laguardia)

Title: Analyzing the Game of Cycles

Abstract: Playing the game of cycles (as introduced in Su's 2020 book Mathematics for Human Flourishing) can be interpreted as converting a planar non-directed graph into a planar directed graph. Thus, every stage of the game can be represented as a graph of mixed type with a suitable adjacency matrix. The rules of the game and the winning moves have interpretations in terms of that matrix, which suggests another way to analyze the game and its strategies.

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Friday, Mar 27(CANCELED)
Speaker: Wim Veys (University of Leuven)
Title: Zeta functions and the monodromy conjecture

Abstract: The monodromy conjecture is a mysterious open problem in singularity theory. It relates arithmetic and topological/geometric properties of a multivariate polynomial over the integers. The case of interest is when the zero set of the polynomial has singular points. We will present some history, motivation, and related results.

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Friday, Apr 24
Speaker: James Meyer (GC)
Title: TBA

Abstract:

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Fall 2019

Friday, Sep 6
Speaker: Bart Van Steirteghem (FAU Erlangen-Nürnberg and CUNY)
Title: Momentum polytopes of projective spherical varieties

Abstract: Spherical varieties are a natural generalization of toric varieties, where the acting torus is replaced by a general connected reductive group. They also include flag varieties and symmetric varieties. In this talk, I will explain how one can use their rich combinatorics to characterize the momentum polytopes of projective spherical varieties. I will also discuss an application to the Kählerizability problem for compact multiplicity free Hamiltonian manifolds. This is joint work with S. Cupit-Foutou and G. Pezzini.

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Friday, Sep 20, 4PM
Speaker: Hans Schoutens (CityTech and the CUNY Graudate Center)
Title: Generalized Hasse-Schmidt derivations

Abstract: Hasse-Schmidt derivations are certain types of higher order differential operators (introduced by Hasse and Schmidt and studied, among others, by Matsumura). In characteristic zero, they are ubiquitous, but their presence in positive characteristic is more subtle: they are no longer classical differential operators. Vojta has used them to construct arc schemes. In this talk, I will discuss a generalization of Hasse-Schmidt derivations and show the corresponding connection with generalized arc schemes (which I introduced a decade ago, and are used, among others, by Mustata and Stout). The recipe is quite simple: replace in the definitions $k[t]/t^nk[t]$ by an arbitrary local Artinian k-algebra.

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Friday, Oct 4, 4PM
Speaker: Ray Hoobler (CUNY)
Title: Bases of Azumaya algebras

Abstract: It is well known that any central separable algebra over a field is a matrix ring over a division algebra. If X is a regular scheme I will show that there is an essentially unique reflexive (semi-stable if X is projective), maximal order in the division algebra component of the generic point of any Azumaya algebra on X. As an application I shall show that for a projective smooth scheme X over a field of characteristic 0, an Azumaya algebra admits a flat connection if and only if its second Chern class vanishes.

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Friday, Oct 18, 4:15PM
Speaker: Hans Schoutens (CityTech and the CUNY Graudate Center)
Title: Using Hasse-Schmidt derivations to construct small MCMs.

Abstract: This talk complemens my talk from last month, where I introduced (general) Hasse-Schmidt derivations. Today I will sketch the argument how the existence of some nice' Hasse-Schmidt derivation over a complete three-dimensional local ring of positive characteristic yields the existence of a small MCM (i.e., a finitely generated module of depth three), resolving Hochster's conjecture for this special case.

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Friday, Nov 1, 4:15PM
Speaker: Ben Blum-Smith (Fordham and New School)
Title: Square-free Gröbner degenerations and cohomologically full rings

Abstract: Let I be a homogeneous ideal in a polynomial ring S := K[x_1,…,x_n]. A choice of monomial order on S specifies an initial ideal in(I) for I. The ring S/in(I) is the special fiber in a flat family whose generic fiber is S/I, and it has long been understood that S/I is “at least as good" as S/in(I) according to various notions of “good behavior" for rings. For example, the depth of S/I is at least the depth of S/in(I); thus S/I is Cohen-Macaulay if S/in(I) is Cohen-Macaulay. Because of this principle, S/in(I) is called a “Gröbner degeneration” of S/I. In full generality, the converse of this principle is false, as implied by the term “degeneration” — good behavior of S/I need not pass to S/in(I). However, in 1987, Bayer and Stillman showed that in an important special situation, namely when I is in “generic coordinates,” the relationship between S/I and S/in(I) is quite tight. In 2018, Aldo Conca and Matteo Varbaro announced a theorem that if the monomial ideal in(I) happens to be generated by squarefree monomials, then the relationship is tighter still. The result established and sharpened a conjecture of Herzog, and allowed the authors, as a corollary, to resolve a long-standing mystery in the theory of algebras with straightening law (ASLs). The proof is a first major application of the notion of cohomologically full rings, introduced very recently by Hailong Dao, Alessandro De Stefani, and Linquan Ma. In this talk, we will discuss the results of Conca and Varbaro, the proof methods, and speculate about possible applications.

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Friday, Nov 8, 4:15PM
Speaker: Roger Wiegand (University of Nebraska)
Title: Vanishing of Tor over fiber products

Abstract: Let  (S,m,k)  and  (T,n,k)  be local rings, and let  R   denote their fiber product over their common residue field  k.  (Thus R = {(s,t) | p(s) = q(t), where p:S —>k and q:T—>k are the reductions modulo the maximal ideals.) Inspired by work of Nasseh and Sather-Wagstaff, we explore consequences of vanishing of  Tor^R_m(M,N)  for various values of m, where  M  and  N  are finitely generated R-modules.  For instance, assume that neither  S  nor  T  is a discrete valuation domain and that  Tor^R_m(M,N) = 0 for some  m > 5.  Then at least one of  M, N  has projective dimension at most one.   This is joint work with Thiago Freitas, Victor Hugo Jorge Pérez, and Sylvia Wiegand.

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Friday, Nov 15, 4:15PM
Speaker: Ben Blum-Smith (Fordham and New School)
Title: Square-free Gröbner degenerations and cohomologically full rings (Part II)

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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.