General information
This the webpage of the seminar on deformation theory to be held during winter semester 2020/2021 at University of Amsterdam.
Intuitively, we will be interested in infinitesimal changes of algebraic varieties and various structures on them such as subschemes, line bundles and sheaves. This might considered as local analysis of appropriate moduli spaces, e.g., Hilbert scheme, Picard schemes, moduli of varieties.
We will meet according to the following schedule:
Date: Every Wednesday at 11:00($+\varepsilon$)
Location: Kortewegde Vries Institute for Mathematics, room F3.20
Online: On Zoom [Meeting ID: 978 3804 1835 Password: 396626]
Programme of the seminar:
Here comes a more specific programme of the seminar. Click for more details on the talk contents and some supplementary references.

[Wed, 16.09.2020] Introduction to deformation theory
(Maciek Zdanowicz)
Short explanation of what deformation theory is. Some simple examples.

[Wed, 23.09.2020] Structures over dual numbers
(Wouter Rienks) [Recording and Wouter's notes]
This talk will be completely based on Hartshorne's Deformation Theory Chapter 2. We will analyse the guts of $k[\varepsilon]/\varepsilon^2$deformations of subschemes, line bundles and vector bundles. This might be thought as the computation of tangent spaces of Hilbert schemes, Picard schemes and moduli spaces of vector bundles.

[Wed, 30.09.2020] Infinitesimal lifting property and deformations of smooth varieties
(Dirk van Bree) [Recording and Dirk's notes]
We will prove that smooth affine schemes admit no nontrivial $k[\varepsilon]/\varepsilon^2$deformations. Using that and a Cech cohomology argument, we will show that for a projective variety $X$ such deformations are parametrized by $H^1(X,T_X)$. We will then contrast this with potential nonexistence of nonequicharacteristic deformations from $\mathbb{Z}/p$ to $\mathbb{Z}/p^2$. A potential set of references for the talk is a large chunk of Hartshorne's Chapter 4 and Sernesi's Chapter II "Infinitesimal deformations".

[Wed, 07.10.2020] Functors of Artin rings and formal moduli
(Renjie Lyu) [Renjie's notes]
We will introduce deformations of different structures on algebraic varieties over more general Artinian rings. We will then combine them into gadgets called deformation functors and analyse their abstract properties such as existence of a hull and prorepresentability. This is wellexplained in FGA Explained (Chapter 6) or Hartshorne's Chapter 15.

[Thu, 15.10.2020, 10:00] Schlessinger prorepresentability and hull existence criteria
(Mike Daas)
[Recording of Mike's talk]
We will prove Schlessinger's criterion for prorepresentability and existence of a hull. Then we will provide a counterexample for prorepresentability of all deformation functors of projective varieties. This might be based on basically any reference, e.g., Chapter 15 and 16 of Hartshorne's book or the original paper ``Functors of Artin rings'' by Schlessinger.

[Wed, 21.10.2020] Tangent and obstruction theories
(Kees Kok) [Recording and Kees' notes]
In this talk, we want to present some special conditions imposed on deformation functors which are sufficient for existence of a hull. We will then use them to prove existence of a hull for multiple types of deformation functors (e.g., deformations of smooth projective varieties). A nice set of references is FGA Explained Chapter 6 and some parts of Hartshorne's Deformation Theory. It is also worth looking into Schlessinger's paper ``Functors of Artin rings''.

[Wed, 28.10.2020] Exal category and naive cotangent complex
(Emelie Arvidsson) [Emmi's notes]
In this talk we will consider deformations of general (not necessarily smooth)rings and algebras. We will define the category ${\rm Exal}_A(B,M)$ of extension of an $A$algebra $B$ be a $B$module $M$ and then classify them by the group ${\rm Ext}^1({\rm NL}_{B/A},M)$ for a certain natural complex ${\rm NL}_{B/A}$ (isomorphic to $\Omega^1_{B/A}$ for $B/A$ smooth) called the naive cotangent complex. We will then relate this to our previous discussions. Stacks Project includes a pretty clear account on the topic.

[Wed, 4.11.2020] Cotangent complex and its deformation theoretic applications
(Maciej Zdanowicz) [Recording and Maciek's notes]
We will first emphasize what was lacking for existence of a tangent and obstruction theory for general deformations as in the previous talk. We will overcome the previous obstacle by introducing the cotangent complex axiomatically and indicating how this can lead to a wellbehaved deformation theory.

[Wed, 11.11.2020] Construction of the cotangent complex
(Lenny Taelman) [Recording and Lenny's notes]
We will construct the cotangent complex using homotopical algebra.

[Wed, 18.11.2020] SchlessingerLichtenbaum complex and deformations of singularities
(Olivier de Gaay Fortman) [Recording and Olivier's notes]
In the talk, we will explicate the second truncation $\tau_{\geq 2}L_{B/A}$ of the cotangent complex which is sufficient for most deformation theoretic considerations. We will show how this can be applied for some computations concerning deformations of singularities. This is wellexplained in Hartshorne's Deformation Theory Chapter 3 and Stacks Project.

[Wed, 25.11.2020] Murphy laws on moduli spaces
(Simon Pepin Lehalleur) [Recording of Simon's talk]
Following Vakil's paper we will prove that arbitrarily complicated singularities show up on very reasonable moduli spaces such as moduli of general type varieties.

[Wed, 2.12.2020] BogomolovTianTodorov theorem
(Emma Brakkee) [Recording and Emma's notes]
We will prove that in characteristic zero the deformations of varieties with trivial canonical class are unobstructed (the formal moduli space of such varieties is smooth). In the course of the proof we will introduce the $T_1$lifting property sufficient for smoothness, which we will then verify in our context. The talk might be based on the releveant part of Daniel Litt's notes on deformation theory.

[Wed, 9.12.2020] Overview of singularities of Hilbert schemes of points
(Sergej Monavari) [Recording and Sergej's notes]

[Wed, 16.12.2020] Lifting from characteristic $p$ to characteristic zero
(Jack Davies) [Recording and Jack's notes]
We will discuss the problem of nonequicharacteristic deformations (liftings) of characteristic $p>0$ varieties. In particular, based on Illusie's part of FGA Explained we will present an example of a nonliftable variety.

[Wed, 06.01.2021] On the birational nature of lifting
(Przemysław Grabowski) [Recording and Przemek's notes]

[Wed, 13.01.2021] Bend and break & Mori's resolution of Hartshorne's conjecture
(Diletta Martinelli)
[Meeting ID: 978 3804 1835 Password: 396626]
References
In the talks we will likely follow some of the following references.
We will also discuss some of the following papers:
 C. Liedtke, M. Satriano  On the birational nature of lifting
 R. Vakil  Murphy’s law in algebraic geometry: Badlybehaved deformation spaces
 J. Jelisiejew  Pathologies on the Hilbert scheme of points
Organizers:
The seminar is run by Lenny Taelman and Maciek Zdanowicz
All inquiries should be directed to m.e.zdanowicz [at] uva [dot] nl
COVID19 regulations
 There can be at most 10 people in the seminar room (1 speaker and 9 audience members). At all times keep 1.5m distance.
 If you come to the seminar, you must also book a desk at the institute.
 Every talk can be followed from home using zoom (and you can ask questions from home). Also the speaker can choose to give the talk from home
 If you have any of the symptoms (cold, fever, sore throat, loss of smell or taste, ...) then stay home (and get tested)
The ventilation of the seminar room has been tested, and based on that we are allowed to be 10 people in the room. However: because of the ventilation it may get chilly in the room, so bring a sweater. If more than 10 people want to come to the seminar, we will set up a rotation schedule.