Reading seminar in algebraic geometry.
Spring 2023: Topics in algebraic geometry ("My favorite things")
The topic for the reading seminar in algebraic geometry this semester will be "My favorite things" (lederhosen optional). The idea is that speakers will present an example or theorem or technique etc in algebraic geometry that they find inspiring and try to explain it in an accessible way, with plenty of insider details that would not appear in a textbook.
The seminar will consist of talks by faculty followed by talks by graduate students. We will attempt to make the seminar accessible to everyone.
The seminar will meet on Fridays, 2:30PM--3:30PM, in LGRT 1322.
Tentative Schedule
2/17 Paul. The Morrison cone conjecture.
The Morrison cone conjecture asserts that the automorphism group of a Calabi--Yau 3-fold X acts on the closure of the Kahler cone with rational polyhedral fundamental domain. (Here we say a cone is rational polyhedral if it is generated by finitely many integral vectors.) In particular, if the closure of the Kahler cone is not rational polyhedral then the automorphism group is infinite. While the cone conjecture has been proved in some related settings, the original conjecture is wide open. The motivation comes from mirror symmetry, where one expects that the moduli space of the mirror Calabi--Yau Y has a limit point which can be described in terms of the action of Aut X on the Kahler cone using toric geometry. It is however of compelling interest in birational geometry and moduli theory independent of mirror symmetry. I'll give an overview of the conjecture and evidence in favor and against it with minimal prerequisites, in particular, I will define all the terms in this abstract.
2/24 Wendelin. Toric Geometric Invariant Theory.
Geometric Invariant Theory (GIT) is a way to take quotients X/G in algebraic geometry, which very roughly proceeds by removing some “unstable” points from X determined by a “stability condition” before taking the quotient by a group G. A familiar example is projective space P^{n−1}, which is the quotient of the vector space C^n minus the origin by the diagonal action of C^∗. More generally, quotients of C^n by an action of (C^*)^r give rise to toric varieties, and I will explain how in this context, GIT and stability can be understood very explicitly in terms of a gadget called the secondary fan. These toric varieties share many of the properties of projective space and can therefore often be studied very explicitly, providing a useful testing ground for whatever theorem you might want to prove.
3/3 Eyal. Semi-homogeneous vector bundles over abelian varieties
A vector bundle E over an abelian variety is semi-homogeneous if every translate of E is isomorphic to the tensor product of E with a line bundle. The vector bundle E is simple if End(E) is one-dimensional. Atiyah proved that every simple vector bundle over an elliptic curve is semi-homogeneous. This is no longer true over abelian varieties of dimension > 1. Nevertheless, semi-homogeneous vector bundles are particularly important. For example, if an equivalence F:D(A)→D(B) between the derived categories of two abelian varieties A and B maps points to objects of non-zero rank, then up to a shift the kernel of F is a simple semi-homogeneous vector bundle over the product A×B
(a result of Orlov). We will present the 1978 theorem of Mukai that a simple semi-homogeneous vector bundle E over an abelian variety A is determined, up to translation, by its slope c_1(E)/rank(E). Furthermore, every class in the rational Neron-Severi vector space NS(A)_Q is the slope of a simple semi-homogeneous vector bundle.
3/10 Kristin. Blow ups, minimal resolutions, and resolution of singularities.
Driven by examples, we'll explore what happens when blowing up singular points on varieties. We'll define resolution of singularities and see several examples of why the existence of a resolution algorithm is such a difficult problem. We'll discuss Hironaka's result and, if time permits, discuss weighted blow-ups and functorial resolutions using weighted blow-ups. There will be many pictures!
3/17 Spring break
3/24 Paul. Log Calabi--Yau manifolds and mirror symmetry
A Calabi--Yau manifold is a smooth complex projective variety such that the top exterior power of the holomorphic tangent bundle is trivial.
Mirror symmetry posits that Calabi--Yau manifolds come in mirror pairs X and Y
such that the complex geometry of X is equivalent to the symplectic geometry of Y and vice versa. The Strominger--Yau--Zaslow conjecture asserts that mirror pairs X,Y admit dual Lagrangian torus fibrations. The homological mirror symmetry conjecture of Kontsevich asserts that the derived category of holomorphic vector bundles on X is equivalent to the Fukaya category of Lagrangian submanifolds of Y.
Roughly speaking, a log Calabi--Yau manifold is a noncompact Calabi--Yau manifold with controlled behaviour at infinity, such that both the SYZ and HMS conjectures extend to the log Calabi--Yau setting (and are often more tractable than in the compact case). We will give an introduction to log Calabi--Yau manifolds and mirror symmetry for them, based on examples.
3/31 Arthur. The Hilbert scheme of n points in C^2 and link homology
4/7 Chunlin. The McKay correspondence as an equivalence of derived categories
We will discuss the McKay correspondence relating the representation theory of a finite subgroup G of SL(n,C) for n=2 or 3 to a minimal resolution of the quotient singularity C^n/G, interpreting it as an equivalence of derived categories following the work of Bridgeland-King-Reid
Bridgeland--King--Reid and Kapranov--Vasserot.
4/14 Pranav Ramakrishnan. The Arithmetic of Elliptic Pairs
The theory of Elliptic Pairs, as investigated in a paper by Castravet, Laface, Tevelev, and Ugaglia, provides useful conditions to determine polyhedrality of the pseudo-effective cone, which give rise to interesting arithmetic questions when reducing the variety modulo p. In this paper, we examine one such case, namely the blow-up X of 9 points in P^2, and study the density of primes p
for which the pseudo-effective cone of the reduction of X modulo p is polyhedral. This problem reduces to an analogue of Artin’s Conjecture on primitive roots like that investigated by Stephens and then Moree and Stevenhagen. As a result, we find that the density of such polyhedral primes hover around a higher analogue of the Stephens constant under the assumption of the Generalized Riemann Hypothesis.
4/21 Joe. Derived equivalences of hyperkaehler varieties.
I will describe some results on derived equivalences of hyperkaehler varieties, in particular a recent construction of a derived equivalence between generalized Kummer varieties.
4/28 AGNES @ Stony Brook
5/5 Yuxuan. Birational Hyperkahler Varieties.
I will talk about the Mukai flop for hyperkahler varieties. It is an important example in the birational geometry of hyperkahler varieties. I will also mention some examples and applications in research on moduli spaces of marked hyperkahler varieties and the generalization of the global Torelli theorem.
5/12 Ethan. Toric varieties and Stanley's theorem
One problem in combinatorics is to characterize the number of vertices, edges and higher dimensional faces of a convex simplicial polytope in Euclidean n-space. Strikingly, Stanley solved this problem by using toric varieties and deep facts on the topology of toric varieties such as Poincare duality and the hard Lefschetz theorem. In this talk, I will introduce toric varieties and explain Stanley's proof.
Links to the seminar from previous semesters.
Birational geometry and the minimal model program, Spring 2022
Fano varieties, Fall 2021
Mirror symmetry, Spring 2020
K3 surfaces, Fall 2019
The minimal model program, Spring 2019
Non-commutative K3 surfaces, Fall 2018
Topics in algebraic geometry, Spring 2018
Surface singularities, Fall 2017
Birational geometry, Spring 2017.
Algebraic surfaces, Fall 2016.
Fano varieties, Fall 2015
The Hitchin system, character varieties, and related topics, Spring 2015
Holomorphic symplectic varieties and Hyperkahler manifolds, Fall 2014
Syzygies, Spring 2014
Toric varieties, Fall 2013
Derived categories in algebraic geometry, Spring 2013
Deformation theory, Fall 2012
Curves, K3 surfaces, and Fano 3-folds, Spring 2012
Topics in algebraic geometry, Fall 2011
Surfaces of general type, Spring 2011
Algebraic surfaces, Fall 2010.
Stability conditions on derived categories and wall crossing, Spring 2010
Mirror symmetry and tropical geometry, Fall 2009
Deformation theory,
Fall 2008
Green's Conjectures,
Spring 2008
Bridgeland Stability, Fall 2007.
Minimal Model Program,
Spring 2007
Commutative Algebra and Polyhedra Seminar, Spring 2006.
Geometry and Algebra of Polyhedra Seminar, Fall 2005.
Commutative Algebra
Seminar, 2004-2005
This page is maintained by Paul Hacking hacking@math.umass.edu