Tag - Algebraic geometry

Yves Andrรฉ: On the canonical, fpqc and finite topologies: classical questions, new answers (and conversely)

29th April, 2021

Up to a finite covering, a sequence of nested subvarieties of an affine algebraic variety just looks like a flag of vector spaces (Noether); understanding this "up to" is a primary motivation for a fine study of finite coverings.

The aim of this talk is to give a bird's-eye view of some fundamental questions about them, which took root in Algebraic Geometry (descent problems etc.), then motivated major trends in Commutative Algebra (F-singularities etc.), and recently found complete solutions using p-adic methods (perfectoids). Rather than going into detail of the latter, the emphasis will be on synthesizing, from the geometric viewpoint, a rather scattered theme.

This is based on joint work with Luisa Fiorot.

Will Sawin: The Shafarevich Conjecture for Hypersurfaces in Abelian Varieties

18th March, 2021

Faltings proved the statement, previously conjectured by Shafarevich, that there are finitely many abelian varieties of dimension n, defined over a fixed number field, with good reduction outside a fixed finite set of primes, up to isomorphism. In joint work with Brian Lawrence, we prove an analogous finiteness statement for hypersurfaces in a fixed abelian variety with good reduction outside a finite set of primes. I will give a broad introduction to some of the ideas in the proof, which builds on p-adic Hodge theory techniques from work of Lawrence and Venkatesh as well as sheaf convolution in algebraic geometry.

Kirill Zainoulline: Algebraic Geometry

11th January, 2021

This is a 23-lecture course, with each lecture being around 80 minutes, given online by Kirill Zainoulline. It gives an introduction to algebraic geometry.

A brief overview of commutative algebra: rings and ideals, Nakayama's Lemma, localization, Krull-dimension, direct-limits, integral dependence.ย Toward algebraic varieties: Regular functions, algebraic sets, Hilbert's Nullstellensatz, Zariski topology, ringed spaces, affine and projective varieties.ย Toward sheaves and group schemes: functors of points, Grothendieck topologies, representable functors, group schemes, tori, Grassmannians, torsors and twisted forms, quadrics and Severi-Brauer varieties.

Asilata Bapat: A Thurston compactification of Bridgeland stability space

30th November, 2020

The space of Bridgeland stability conditions on a triangulated category is a complex manifold. We propose a compactification of the stability space via a continuous map to an infinite projective space. Under suitable conditions, we conjecture that the compactification is a real manifold with boundary, on which the action of the autoequivalence group of the category extends continuously. We focus on 2-Calabi-Yau categories associated to quivers, and prove our conjectures in the A2 and affine A1 cases.

Uriel Sinichkin: Enumeration of algebraic and tropical singular hypersurfaces

29th November, 2020

It is classically known that there exist (n+1)(d-1)n singular hypersurfaces of degree d in complex projective n-space passing through a prescribed set of points (of the correct size). In this talk we will deal with the analogous problem over the real numbers and construct, using tropical geometry, Ω(dn) real singular hypersurfaces through a collection of points in ℝℙn. We will also consider the enumeration of hypersurfaces with more than one singular point. No prior knowledge of tropical geometry will be assumed.

Alexander Carney: Heights and dynamics over arbitrary fields

15th October, 2020

Classically, heights are defined over number fields or transcendence degree one function fields. This is so that the Northcott property, which says that sets of points with bounded height are finite, holds. Here, expanding on work of Moriwaki and Yuan-Zhang, we show how to define arithmetic intersections and heights relative to any finitely generated field extension ๐พ/๐‘˜, and construct canonical heights for polarizable arithmetic dynamical systems ๐‘“:๐‘‹โ†’๐‘‹. These heights have a corresponding Northcott property when ๐‘˜ is โ„š or ๐”ฝ๐‘ž. When ๐‘˜ is larger, we show that Northcott for canonical heights is conditional on the non-isotriviality of ๐‘“:๐‘‹โ†’๐‘‹, generalizing work of Lang-Neron, Baker, and Chatzidakis-Hrushovski. Additionally, we prove the Hodge Index Theorem for arithmetic intersections relative to ๐พ/๐‘˜. Since, when Northcott holds, pre-periodic points are the same as height zero points, this has applications to dynamical systems. By the Lefschetz principle, these results can be applied over any field.

Francesca Carocci: The geometry of moduli spaces of stable maps to projective space and modular desingularisation via log blowups

7th September, 2020

An LMS online lecture course in moduli spaces.

Moduli spaces of stable maps have been of central interest in algebraic geometry for the last 30 years. In spite of that, the geometry of these spaces in genus bigger than zero is poorly understood, as the Kontsevich compactifications include many components of different dimensions meeting each other in complicated ways, and the closure of the smooth locus is difficult to describe.

In recent years a new perspective on the problem of finding better behaved compactifications, ideally smooth ones, has come from log geometry. This approach has proved successful in a series of examples and log geometry is now becoming a natural setting to study modular resolutions of moduli spaces.

The aim of this series of talks will be to see how log geometry techniques apply to give modular smooth compactifications of moduli spaces of stable maps to projective spaces in genus one and two; we will also explain why the latter are interesting from an enumerative point of view.

In more detail: we begin by studying the deformation theory and the global geometry of moduli spaces of genus one and two stable maps; we then give a brief introduction to log schemes, line bundles on log schemes and log blowups and conclude by exhibiting the log modification resolving the moduli spaces of maps in genus one and two and explaining their modular meaning.