IPM Algebraic Geometry Seminar
IPM is holding a biweekly zoom Algebraic Geometry seminar (Fall/Winter 2021-22).
Where: Zoom
Meeting ID: 908 611 6889
Passcode: 362880
Seminar mailing list: google group
Poster: here
Talks
Speaker: Hassan Haghighi (K. N. Toosi University of Technology)
Title : Unexpected hypersurfaces: some examples, a few constructions
Abstract : In recent years, a novel attitude to the classical
problem of identifying and classifying special linear systems in projective n
space, has been emerged. For a subvariety Z of the projective n space with
defining ideal I, let P_1,…,P_s be general distinct points in this space and
let m_1,…,m_s be positive integers which at least one of them is greater than
one. On the subspace of those elements of degree d part of the homogeneous
ideal I which vanish at P_i with multiplicity at least m_i, each fat point
m_iP_i defines a specific number of linear relations on this subspace. For a
given set of points P_i with multiplicity m_i, it is expected that these linear
equations to be linearly independent. If it is not the case, then one says that
the variety Z admits an unexpected hypersurface with respect to the fat point
subscheme defined by these fat points, and this linear subspace is called a
special linear system on the variety Z. Each element of this subspace defines a
hypersurface, known as an unexpected hypersurface. In this talk, we review some
interesting examples which brought into the scene with this new approach and
explain some existing methods to construct unexpected hypersurfaces.
Date: March 8, 2022
Video link
Passcode: AN*b80EX
Speaker: Ali Bajravani (Azarbaijan Shahid Madani University)
Title : Stable vector bundles on curves and their Brill-Noether theory
Abstract : We discuss some stricking properties of stable vector bundles over curves,
which are frequently used in moduli and Brill-Noether arguments of these bundles. Then, after a quick
historical surf in the topic, we give an upper bound for dimensions of Brill-Noether schemes of rank 2 stable vector bundles.
Date: February 22, 2022
Video link
Passcode: T?a6.X^2
Speaker: Amir Nasr (IPM)
Title : Toric quiver varieties
Abstract : We discuss the smoothness of toric quiver varieties.
When a quiver Q is defined with the identity dimension vector, the corresponding quiver variety is also a toric variety.
So it has a fan representation and a quiver representation. I consider only quivers with canonical weight and we
classify smooth such toric quiver varieties. I show that a variety corresponding to a quiver with the identity
dimension vector and the canonical weight is smooth if and only if it is a product of projective spaces or their blowups.
Date: February 8, 2022
Video link
Passcode: i#is9m6@
Speaker: Esmail Arasteh Rad (IPM)
Title : Rapoport-Zink spaces for local ℙ-shtukas
Abstract : Rapoport-Zink spaces for p-divisible groups are local counterparts for Shimura varieties.
According to the dictionary between function fields and number fields, they correspond to the RZ-spaces for local
P-shtukas. We review the construction of these moduli spaces and then discuss our approach for computing the semi-simple
trace of Frobenius on their (nearby-cycles) cohomology.
Date: January 25, 2022
Video link
Passcode: Uv1zkWV=
Speaker: Tahereh Aladpoosh (IPM)
Title: Postulation of generic lines and a multiple line in ℙ^n
Abstract : A well-known theorem by Hartshorne and Hirschowitz states that a generic
configuration of lines has good postulation. So what about non-reduced configurations? Can adding a multiple
line to the configuration still preserve it’s good postulation? This is the question we mainly deal with in this talk.
In the first part of the talk we introduce the postulation problem for projective schemes, then we discuss the problem
for the family of schemes supported on generic linear configurations, which are the ones of particular interest.
In the second part of the talk we focus on the postulation of generic lines and one multiple line in projective space.
We give our main theorem providing a complete description to the case of lines and a double line, then we propose
a conjecture to the general case, finally we discuss what is known about the conjecture and more recent results on it.
Date: December 28, 2021
Video link
Passcode: 5d36+Lu#
Speaker: Abbas Nasrollah Nejad (Institute for Advanced Studies in Basic Sciences)
Title: The relation type of singular space of hypersurfaces
Abstract : In this talk, we will introduce the notion of relation type of formal and analytic algebras
and show that it is well defined by using of Andre\'-Quillen homology. In particular, the relation type is an
invariant of an affine algebraic variety and a complex space germ. We will discuss and essay to explain the
relation type of singular subscheme of isolated hypersurface singularities. This talk is based on joint ongoing work with Maryam Akhavin.
Date: December 14, 2021
Video link
Passcode: uS6MSyh@
Speaker: Amin Gholampour (University of Maryland)
Title: 2-dimensional stable pairs on 4-folds
Abstract : I will discuss a 2-dimensional stable pair theory of nonsingular
complex 4-folds that is parallel to Pandharipande-Thomas' 1-dimensional stable pair
theory of 3-folds. The stable pairs of a 4-fold are related to its 2-dimensional subschemes
via wall-crossings in the space of polynomial stability conditions. In Calabi-Yau case, Oh-Thomas
theory is applied to define invariants counting these stable pairs under some restrains. This is a joint work with Yunfeng Jiang and Jason Lo.
Date: November 30, 2021
Video link
Passcode: Wa9HhC@F
Speaker: Amir Jafari (Sharif University of Technology)
Title: Grothendieck Galois theory and some of its applications in combinatorics
Abstract : This is going to be a report of my ongoing joint research project
with Mr. Moghaddamzadeh on finite projective geometries. However, a good portion of the talk will be
spent on explaining Grothendieck's generalizations of Galois theory.
Date: November 16, 2021
Video link
Passcode: !%28L$pp
Speaker: Kiumars Kaveh (University of Pittsburgh)
Title: Vector bundles on toric varieties
Abstract : In this talk we review construction of toric varieties and classification of (torus equivariant)
line bundles and vector bundles on them (after Klyachko). We interpret Klyachko's data of a vector bundle as
a "piecewise linear map" into the Tits building of the general linear group GL(r).
This "building" perspective helps to approach many questions about vector bundles on toric varieties in a new light.
As an application of this idea, we obtain a classification of (torus equivariant) vector
bundles on toric schemes in terms of "piecewise affine maps" to the Bruhat-Tits building of GL(r).
This is work in progress with Chris Manon and Boris Tsvelikhovsky. I try to cover most of the background material.
Date: November 2, 2021
Video link