Talks
Rational points on curves over finite fields: bounds, geometry, and maximality
Yves Aubry, University of French Polynesia
In this talk, we will focus on the number of rational points on projective
algebraic curves over a finite field. After recalling the Weil bound arising
from the Riemann Hypothesis, as well as Ihara's bound, which improves it when
the genus grows, we will present an approach that interprets these bounds in
the Neron-Severi group of the surface defined by the product of the curve with
itself. We will then study a new bound involving both the number of rational
points over the base field and over a quadratic extension, and in particular
the curves that attain this latter bound. In particular, we characterize their
zeta function and the isogeny class of their Jacobian, as well as the
connection with Ihara-maximal curves.
On moduli spaces of some threefolds of general type
Jungkai Chen, National Taiwan University
There are some recent advances in the study of threefolds of general type. With
the explicit description of minimal models and canonical models for threefolds
near the Noether line, it is possible to explore some aspects of moduli spaces
of threefolds of general type with small invariants. This talk is based on
joint work in progress with Yongnam Lee and Phin-Sing Soo.
Amitsur Subgroups in Arithmetic and Equivariant Geometry
Alexander Duncan, University of South Carolina
For a variety defined over a field, a line bundle over the algebraic closure
may fail to descend to a line bundle on the original variety. Similarly, if
there is a group action, that action may fail to lift to certain line bundles.
The Amitsur subgroup precisely measures this failure, taking values in the
Brauer group of the field (or in an equivariant analog). A key property of the
Amitsur subgroup is that it is an (equivariant) birational invariant. When all
overfields and subgroups are considered, it is fine enough to completely
distinguish between varieties in several interesting classes. I will discuss
how the language of Mackey functors can be used to describe and compute the
Amitsur subgroup, especially in the cases of Fano varieties, toric varieties,
and torsors of tori.
The Jacobi bound conjecture and the dimension conjecture
Taylor Dupuy, University of Vermont
The Jacobi Bound Conjecture and the Dimension Conjecture are fundamental open
problems in Differential Algebra. I will talk about recent progress on these
problems in joint work with David Zureick-Brown. The statements of these
conjectures follow. Let K be a differential field of characteristic zero. Let
R=K{x₁,...,xₙ} be the ring of differential polynomials in n differential
indeterminates. The Jacobi Bound Conjecture is a differential algebraic analog
of Bezout’s inequality for intersections of hypersurfaces in affine space.
Roughly it tells us how many constants of integration are required to solve a
system of n equations in n variables. The Jacobi Bound Conjecture states that
if I is the differential ideal in R generated by n differential polynomials in
R and P is a minimal prime differential ideal over I such that the Krull
dimension of R/P is finite then the Krull dimension of R/P is bounded by the
tropical determinant of the order matrix of the system. The Dimension
Conjecture states that if u₁,...,uᵢ are differential polynomials in R with i
Explicit birational geometry over imperfect fields and the Cremona group
Andrea Fanelli, University of Bordeaux
In this talk I will present joint projects with Fabio Bernasconi, Julia
Schneider, Stefan Schröer and Susanna Zimmermann aimed to study some aspects of
the birational geometry of regular algebraic surfaces over imperfect fields.
These objects naturally appear when one works with fibrations over
algebraically closed fields in positive characteristics. I will also discuss
applications to the Cremona group.
Smoothing toric Gorenstein Fano varieties
Matej Filip, University of Ljubljana
Given a Laurent polynomial with reflexive Newton polytope, we give a criterion
ensuring that it gives rise to a deformation family with smooth general fibre.
One-parameter deformations in this family correspond to mutations of the
Laurent polynomial. In dimension three, the smooth fibres arising from this
construction include all 98 deformation families of very ample smooth Fano
threefolds. This is an upcoming result that generalises already published work
"Laurent Polynomials and Deformations of Non-Isolated Gorenstein Toric
Singularities". Building on that framework, I will also outline evidence
towards a conjecture of Corti–Filip–Petracci predicting a one-to-one
correspondence between 0-mutable Laurent polynomials and the smoothing
deformation components of three-dimensional affine Gorenstein toric varieties.
The cone theorem for Kahler varieties
Christopher Hacon, University of Utah
There has been substantial recent progress towards the minimal model program
for Kahler varieties. In this talk I will discuss a recent proof of the Cone
Theorem for Kahler varieties of arbitrary dimension (improving on previous work
of Cao-Horing) and related results such as the canonical bundle formula,
subadjunction and Wenhao Ou's recent breakthrough result on the
characterization of uniruled compact Kahler manifolds.
Automorphism groups of toroidal horospherical varieties
DongSeon Hwang, Ajou University
We present our recent work on the structure of the identity component of the
automorphism group of a smooth, complete, toroidal horospherical variety by
generalizing the notion of Demazure roots using the toric bundle structure. In
particular, we provide a criterion for the reductivity of Aut⁰(X) in terms of
an analogous notion of Demazure roots for such toric bundles, i.e., projective
toric bundles over rational homogeneous spaces. As an application, we prove the
K-unstability of certain ℙ¹-bundles over rational homogeneous spaces. This is
joint work with Lorenzo Barban and Minseong Kwon.
Pluripotential theory on curves
Sione Ma'u, University of Auckland
The extremal (green) function, Chebyshev constant, and Robin constant
associated to a compact nonpolar subset of the complex plane are well-known
potential-theoretic quantities. Extensions of the latter quantities
(directional Chebyshev and Robin constants) to certain affine algebraic curves
have been studied using pluripotential theory. In this talk, I will present a
couple of recent results connecting them to the extremal function. In
particular, directional Chebyshev polynomials asymptotically give the extremal
function. This is joint work with Norm Levenberg.
Cluster type varieties
Joaquin Moraga, University of California Los Angeles
In this talk, I will introduce a new class of algebraic varieties that has been
of recent interest; cluster type varieties. These are compactifications of
algebraic tori in such a way that the volume form has no zeros on the
compactifications. The second aim of this talk is to show some main properties
and characteristics of cluster type varieties. Finally, we will show some new
results towards the understanding of cluster type varieties from the
perspective of Cox rings.
Fun with Mac Lane Valuations
Andrew Obus, University of New York
Mac Lane's technique of "inductive valuations" on rational function fields is
over 85 years old, but has only recently been used to attack problems about
arithmetic surfaces. This first half of the talk will be an explicit, hands-on
introduction to the theory, requiring no background beyond the definition of a
discrete valuation. We will then discuss how the theory helps us work with
models of the projective line over a discretely valued field, and how the
insights gained can be used to explicitly resolve certain types of
singularities of arithmetic surfaces.
A few observations on algebraic dynamics of Calabi-Yau manifolds
Keiji Oguiso, University of Tokyo
I would like to discuss how one can use MMP to study algebraic dynamical
aspects of Calabi-Yau manifolds, namely, primitivity and the existence of
Zariski-dense orbit of birational automorphisms and biregular automorphisms of
some Calabi-Yau manifolds (Calabi-Yau manifolds of Wehler type and Calabi-Yau
threefolds with c₂-contractions) with relevant open problems.
Singular supports and characteristic cycles of etale sheaves in positive characteristic
Takeshi Saito, University of Tokyo
As an analogue of microlocal analysis for D-modules, the singular supports and
characteristic cycles of etale sheaves on a smooth scheme over a perfect field
are defined on the cotangent bundle. I will recall a summary of the theory and
discuss open questions.
Homological stability and weak approximation
Sho Tanimoto, Nagoya University
Motivated by a topological proof of Manin's conjecture over global function
fields, Ellenberg and Venkatesh envisioned homological stability for the space
of sections of Fano fibrations. In this talk we discuss this property in the
context of weak approximation and establish such a stability for certain Fano
fibrations. This is joint work with Yuri Tschinkel.
Characterization of products of projective spaces via the nef complexity
Kiwamu Watanabe, Chuo University
The nef complexity, introduced by Yoshinori Gongyo, is a numerical invariant
that measures how far a Fano variety is from being a product of projective
spaces. In this talk, I will explain a divisorial characterization stating that
the nef complexity is always nonnegative, and it vanishes precisely for
products of projective spaces. I will also discuss explicit computations,
several applications, and further related developments. This is joint work with
Joshua Enwright, Stefano Filipazzi, Yoshinori Gongyo, Joaquin Moraga, Roberto
Svaldi, and Chengxi Wang.