Courses

• COURSE 1: New techniques in the foundations of moduli theory

  Jarod Alper (University of Washington)

  Abstract     Lecture Notes 1     Lecture Notes 2     Lecture Notes 3     Lecture Notes 4
  The main goal for these lectures is to give a stack-theoretic treatment for the main theorems in Mumford's geometric invariant theory (GIT) and to explain how it can be used to construct projective moduli spaces. The motivating example will be the moduli space of semistable vector bundles on a smooth curve.

  • Lecture 1. After reviewing the foundations of Mumford's GIT, we will quickly introduce algebraic stacks and the concept of a good moduli space. We will establish basic geometric properties of good moduli spaces and explain how this also implies the basic properties of affine and projective GIT quotients.
  • Lecture 2. After reviewing various Tannaka duality statements in algebraic geometry and representation theory, we will establish a Tannaka duality theorem for algebraic stacks. Namely, we will prove that a morphism of noetherian algebraic stacks with affine diagonal can be recovered by the pullback tensor functor on the tensor categories of coherent sheaves. We will show a few applications of Tannaka duality.
  • Lecture 3. The goal of this lecture is to apply Tannaka duality to establish a local structure theorem for algebraic stacks: every finite type algebraic stack has an étale neighborhood by a quotient stack around any point with a linearly reductive stabilizer. We will begin explaining how this local structure theorem can be used to establish the existence of good moduli spaces.
  • Lecture 4. We will provide necessary and sufficient conditions for a finite type algebraic stack in characteristic $0$ to admit a good moduli space. We will apply this criterion to construct a projective moduli space of semistable vector bundles on a smooth curve.

• COURSE 2: Cox Rings with Applications in Algebraic Geometry

  Ivan Arzhantsev (Higher School of Economics Moscow)

  Abstract     Lecture Notes
  In this course we introduce an important invariant of an algebraic variety - the total coordinate ring, or the Cox ring, - and discuss how to apply it to the study of geometry of algebraic varieties, to classification problems, and to description of automorphisms of a given variety

  • Lecture 1. Definitions, Basic Properties and Examples. We define Cox sheaves and Cox rings of algebraic varieties and discuss their basic properties including finite generation and unique factorization. Examples of explicit computations of Cox rings will be given.
  • Lecture 2. Cox Rings and Geometric Invariant Theory. Constructions from the fisrt lecture provide a canonical presentation of an algebraic variety as quotient of a quasiaffine variety by a quasitorus action. We study how to construct this presentation effectively and to read off properties of the variety from the quotient morphism. Also a criterion for a given quotient presentation to be a canonical one is given.
  • Lecture 3. Cox Rings and Geometry of Algebraic Varieties. Many geometric properties of algebraic varieties allow simple descriptions in terms of Cox rings and related objects. Especially it concerns properties of divisors. We present the corresponding dictionary and discuss relations between Cox rings, Mimimal Model Program, and classifications of varieties of certain types.
  • Lecture 4. Cox rings and Transformation Groups. There is way to lift automorphisms of a variety to the total coordinate space. This allows to describe the automorphism group in terms of homogeneous automorphisms of a graded ring. This approach is particularly fruitful for toric varieties and generalizations. Also the lifting allows to reduce many problems on algebraic transformation groups to the case of actions on affine factorial varieties. We demonstrate applications of this reduction.

  References

  1. Ivan Arzhantsev, Ulrich Derenthal, Juergen Hausen, and Antonio Laface. Cox rings. Cambridge Studies in Advanced Mathematics 144, Cambridge University Press, 2015, 530 pages.
  2. Ivan Arzhantsev and Juergen Hausen. Geometric invariant theory via Cox rings. J. Pure Appl. Algebra 213 (2009), 154–172.
  3. Florian Berchtold and Juergen Hausen. Cox rings and combinatorics. Trans. Amer. Math. Soc. 359 (2007), 1205–1252.
  4. David Cox. The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4 (1995), 17–50.
  5. Yi Hu and Sean Keel. Mori dream spaces and GIT. Michigan Math. J. 48 (2000), 331–348.

• COURSE 3: Quadratic invariants for algebraic varieties

  Frédéric Déglise (ENS de Lyon)

  Abstract     Lecture Notes

  Algebraic geometry has always been infused by the invariants from coming from topology and differential geometry. About thirty years ago, Voevodsky started a program that proposes to merge algebraic topology and algebraic geometry in the so-called motivic homotopy theory. The first success of this endeavour was the proof of the Milnor conjecture, via algebraic cobordism, motivic cohomology and motivic Steenrod operations, the motivic version of Thom’s cobordism, singular cohomology and usual Steenrod operations. Since then, many other geometric invariants have been successful implemented in motivic homotopy theory, and new phenomena have been discovered such as Morel’s quadratic degree.

  The course will focus on the notion of quadratic degree. This invariant can be extended into a full-fledged quadratic intersection theory, reminiscent of the intersection number for oriented varieties. The primary aim will be to present this theory, and to define the so-called Chow-Witt groups after Barge, Morel and Fasel. We will then explain several topics such as oriented vector bundles, Euler classes, quadratic enumerative geometry, and finally the algebraic analog of the link of an isolated (surface) singularity.

  
  

  References and suggested readings

  1. Algebraic Geometry and Intersection Theory: Hartshorne. Algebraic geometry. Chapter I, II. ; Fulton, Intersection theory. Chapter 1, 5, 6 (Appendix B).
  2. (Grothendieck-)Witt groups: Scharlau, Quadratic and Hermitian forms. Chap. 1, 2.
  3. Algebraic topology: Milnor-Stasheff, Characteristic classes. Appendix A. §9, 10.
  4. To go further: Asok, Østvær, $\mathbb{A}^1$-homotopy theory and contractible varieties: a survey. ; Fasel, Lectures on Chow-Witt groups. ; Feld, Milnor-Witt cycle modules.

• COURSE 4: Introduction to noncommutative resolutions

  Špela Špenko (Vrije Universiteit Brussel)

  Abstract     Lecture Notes 1     Lecture Notes 2     Lecture Notes 3

  Resolutions of singularities are a fundamental tool in algebraic geometry for dealing with singular spaces. Hironaka proved that a resolution of singularities always exists (in characteristic $0$). In the case of curves and surfaces there always exists a minimal resolution of singularities but in higher dimensions this is no longer true. However, there is a replacement for minimal resolutions, which are so called crepant resolutions.

  The celebrated Bondal-Orlov conjecture, which drives a lot of current research asserts that all crepant resolutions have the same (co)homological structure. The conjecture initiated a reinterpretation of the minimal model program, where it has also become apparent that one needs mildly ``noncommutative spaces''. A space is by the geometry-algebra correspondence replaced by a ring. Noncommutative resolutions (NCRs) are analogues of classical (commutative) resolutions of singularities, while noncommutative crepant resolutions (NCCRs) are analogues of crepant resolutions, and one may think of them as being ``minimal'' noncommutative resolutions.

  We will focus on NC(C)Rs in the setting of invariant theory. After a brief introduction of the (geometric) invariant theory we will introduce the NC(C)Rs and show that they commonly exist for quotient singularities of reductive groups. We will discuss how these NC(C)Rs relate to some canonical commutative Springer resolutions, e.g. the Springer resolution for determinantal and Pfaffian varieties, and the Kirwan resolution.

  
  

  References

  1. A. Bondal and D. Orlov, Derived categories of coherent sheaves, In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 47-56. Higher Ed. Press, Beijing, 2002.
  2. M. Brion, Introduction to actions of algebraic groups, Notes de la rencontre "Actions hamiltoniennes : invariants et classification" (CIRM, Luminy, 2009).
  3. R.-O. Buchweitz, G. J. Leuschke, and M. Van~den Bergh, Non-commutative desingularization of determinantal varieties I, Invent. Math., 182(1):47-115, 2010.
  4. G. J. Leuschke, Non-commutative crepant resolutions: scenes from categorical geometry, In Progress in commutative algebra 1, pages 293-361, de Gruyter, Berlin, 2012.
  5. Š. Špenko and M. Van den Bergh, Comparing the commutative and non-commutative resolutions for determinantal varieties of skew symmetric and symmetric matrices, Adv. Math., 317:350-370, 2017.
  6. Š. Špenko and M. Van den Bergh, Non-commutative resolutions of quotient singularities for reductive groups, Invent. Math., 210(1):3-67, 2017.
  7. Š. Špenko and M. Van den Bergh, Comparing the Kirwan and noncommutative resolutions of quotient varieties, 2019.
  8. M. Van den Bergh, Non-commutative crepant resolutions, In The legacy of Niels Henrik Abel}, pages 749-770, Springer, Berlin, 2004.
  9. M. Van den Bergh, Non-commutative crepant resolutions, an overview, 2022, ICM 2022.
  10. M. Wemyss, Noncommutative resolutions, In Noncommutative algebraic geometry, volume~64 of Math. Sci. Res. Inst. Publ., pages 239-306. Cambridge Univ. Press, New York, 2016.
  11. M. Wemyss, Flops and clusters in the homological minimal model programme, Invent. Math., 211(2):435-521, 2018.

Contributed Research Talks

• The moduli space of $p$-Lie algebras

  Alice Bouillet (IRMAR Rennes)

  Abstract     Slides

  Over fields of characteristic $p$, the Lie algebra of an algebraic group does not carry as much information as it does in characteristic $0$. Still, the presence of the "$p$-mapping" on $\mathrm{Lie}(G)$ allows to reconstruct at least the Frobenius kernel of $G$. In this talk, we describe the restricted locus of the universal Lie algebra (i.e. the locus where it admits a $p$-mapping) and the moduli space of $p$-Lie algebras over the flattening stratification of the center. Finally we revisit the classical example of the moduli space $L_3$ of Lie algebras of rank $3$, showing it is representable over the ring of intergers, and it is flat, finitely presented, with reduced Cohen-Macaulay fibers. We also describe the restricted locus when the center is $0$.

• Quotients for non-reductive group actions and possible applications to matrix factorisations

  Lukas Buhr (Johannes Gutenberg Universität-Mainz)

  Abstract

  I will introduce the notion of a graded unipotent group. Following the work of B. Bécrzi, B. Doran, T. Hawes and F. Kirwan on quotients for non-reductive group actions, their exists in some cases an analogue of classical geometric invariant theory for non-reductive group actions. For example, this is so for linearised actions of linear algebraic groups on projective varieties, where the unipotent radical admits a positive grading and certain stabilisers are trivial. I wish to explain this in detail and throughout with a view towards the construction of moduli spaces of matrix factorisations. Everything will be over the complex numbers.

• Quadratic invariants of moduli of elliptic curves

  Andrea Di Lorenzo (Humboldt University Berlin)

  Abstract

  The Chow ring of moduli of curves is an important invariant which is the subject of extensive investigations and conjectures. It is then pretty natural to ask whether one can say something on the Chow-Witt ring of moduli of curves. In this talk, I will present a complete computation of the Chow-Witt ring of the moduli stack of smooth/stable elliptic curves; in particular, I will explain the meaning of the generators of these rings in terms of quadratic invariants of families of elliptic curves. Joint work with Lorenzo Mantovani.

• Twisted Hodge Diamonds give rise to non-Fourier-Mukai functors

  Felix Kueng (Université Libre de Bruxelles)

  Abstract

  We use computations of twisted Hodge Diamonds in order to compute the Hochschild cohomology of smooth degree d Hypersurfaces. Using these computations we can deduce the dimension of the kernel in Hochschild cohomology of the push forward along closed embeddings into projective space. In particular this allows the construction of new non-Fourier-Mukai functors between well behaved target and source spaces.

• Arithmetic Moduli of Weierstrass Fibrations

  June Park (Max Planck Institute for Mathematics Bonn)

  Abstract     Slides

  We will first consider the formulation of the moduli of fibered algebraic surfaces as the Hom space of algebraic curves on moduli stacks of curves. Motive / Cohomology with weights on these moduli naturally allows us to enumerate elliptic and hyperelliptic curves over global function fields ordered by bounded discriminant height. In the end, we formulate analogous heuristics for parallel countings over number fields through the global fields analogy.

• Hyperkähler manifolds of $K3^{[n]}$-type admitting symplectic birational maps

  Yulieth Prieto (University of Bologna)

  Abstract     Slides

  Motivated by the existence of birational involutions on projective hyperkähler manifolds which are deformation equivalent to Hilbert schemes of $n$ points of $K3$ surfaces, we show that such hyperkähler manifolds are always birational to moduli spaces of (twisted) stable coherent sheaves on a $K3$ surface, when they admit a symplectic birational map of finite order with a non—trivial action on its discriminant group. Passing via Bridgeland stability, one can show these hyperkähler manifolds are itself moduli spaces of stable objects on a (possible different) $K3$ surface. In the second part of this talk, we deduce properties regarding the existence of birational involutions via wall—crossing and the birational geometry of these moduli spaces. This is a work in progress with Yajnaseni Dutta and Dominique Mattei.

• Moduli space for n points on a projective line

  Jiayue Qi (Johannes Kepler University Linz)

  Abstract     Slides

  We describe an elementary construction of the Knudsen-Mumford compactification $\overline{M}_{0,n}$ of the moduli space $M_{0,n}$ of $n$ distinct points on the projective line $\mathbb{P}^1$.

• Tame degree functions in arbitrary characteristic

  Sourav Sen (Harish-Chandra Research Institute)

  Abstract     Slides

  Let $A \subseteq B$ be integral domains and $G$ be a totally ordered Abelian group. D. Daigle has formulated certain hypotheses on degree function $\deg : B \to G \cup \{-\infty\}$ so that it is tame in characteristic zero, i.e., $\deg(D)$ is defined for all $A$-derivations $D : B \to B$. This study is important because each $D \in \mathrm{Der}_k(B)$ for which $\deg(D)$ is defined, we can homogenize the derivation which is an important and useful tool in the study of $\mathbb{G}_a$-action on an algebraic variety. In arbitrary characteristic, $\mathbb{G}_a$-actions on an affine scheme $\mathrm{Spec}(B)$ can be interpreted in terms of exponential maps on $B$. In this talk we shall discuss analogous formulations of hypotheses on the degree function so that $\deg(\phi)$ is defined for each $A$-linear exponential map $\phi$ on $B$. This talk is based on a joint work with N. Gupta.

• Unipotent group actions on complete toric varieties

  Kirill Shakhmatov (Higher School of Economics Moscow)

  Abstract

  Let $X$ be a complete toric variety with an acting torus $T$. It is known that the automorphism group $\mathrm{Aut}(X)$ is a linear algebraic group and $T$ is a maximal torus in $\mathrm{Aut}(X)$. We call $X$ \emph{radiant} if a maximal unipotent subgroup of $\mathrm{Aut}(X)$ acts on $X$ with an open orbit. A subgroup $H \subseteq \mathrm{Aut}(X)$ is called \emph{regular} if $H$ is normalized by $T$. Assume that $X$ is radiant. A description of fans corresponding to radiant toric varieties is obtained in [Ivan Arzhantsev and Elena Romaskevich. Additive actions on toric varieties. Proc. Amer. Math. Soc. 145~(2017), no. 5, 1865-1879]. Using this description we construct a graph $C$ on $n$ vertices, where $n = \dim X$. Explicit structure of a maximal unipotent subgroup of $\mathrm{Aut}(X)$ is obtained and described in terms of $C$. We also provide an algorithm that enumerates all regular unipotent subgroups of $\mathrm{Aut}(X)$ which act on $X$ with an open orbit. The talk is based on the joint work with Ivan Arzhantsev and Alexander Perepechko.

• Projectivity of good moduli spaces of quiver representations

  Tuomas Tajakka (Stockholm University)

  Abstract     Slides

  Quiver representations and their moduli spaces are central objects of study in algebraic geometry and representation theory. In 1994, King constructed these moduli spaces as projective varieties using GIT. The goal of this talk is to give a new construction avoiding GIT, instead utilizing the machinery of algebraic stacks and their good moduli spaces. Using results of Alper--Halpern-Leistner--Heinloth, we show that the moduli stack of semistable quiver representations admits a proper good moduli space, on which we exhibit an ample determinantal line bundle. We also obtain effective bounds for global generation of this bundle. Joint work in progress with Pieter Belmans, Chiara Damiolini, Hans Franzen, Victoria Hoskins, and Svetlana Makarova.