Summer School: June 22-25, 2016   

All talks will take place in the Amphi Schwartz, located in the main building (1R2) of the Math Department (IMT). Find it on openstreetmap. Talks marked with * will take place in the nearby Amphi Fermat.

Both the school and the conference dinner will take place on the roof terrace of the University restaurant, located across the main square (construction works are in progress there).

Conference: June 27-July 1st, 2016  

Mini-courses

• Ana-Maria CASTRAVET: Birational geometry of moduli spaces of rational curves. The Grothendieck-Knudsen moduli space M_0,n of stable rational curves with n markings is arguably the simplest among the moduli spaces of stable curves: it is a smooth projective variety that can be described explicitly as a blow-up of projective space, with strata corresponding to nodal curves similar to the torus invariant strata of a toric variety. However, in contrast with toric varieties, the cones of effective cycles are much more mysterious and only partial results are known n ≥ 8. For n ≥ 13, there are negative results: in characteristic zero, M_0,n is not a Mori Dream Space. I will start these lectures by discussing more generally Hassett’s moduli spaces of weighted stable rational curves and continue with an introduction to Mori Dream Spaces. The final goal will be to present the proof that M_0,n is not a Mori Dream Space by reducing the question to a study of blow-ups of weighted projective planes.

• Igor DOLGACHEV: Cremona groups and their subgroups. A Cremona transformation in dimension n is a birational automorphism of n-dimensional projective space, or, algebraically, an auto­morphism of the field of rational functions in n variables identical on the coefficients. The group Cr(n) of such transformations is named in honour of Luigi Cremona who initiated a systematic study of this group in the middle of the 19th century. In my introductory lectures I will survey classical results and modern developments using minimal background in algebraic geometry and concentrating more on concrete examples.

• Alexander KUZNETSOV: Categorical resolutions of singularities. I will discuss an approach to resolution of singularities of schemes via derived categories of coherent sheaves. I will define what a categorical resolution of singularities is, and will describe a construction of a categorical resolution for any separable scheme of finite type over a field of characteristic zero (thus showing that categorically any such singularity behaves as a rational singularity). Then I will discuss the question of constructing small categorical resolutions, concentrating on the notions of crepancy and explicit examples.

• Emanuele MACRÌ: The Minimal Model Program for moduli spaces of sheaves on K3 surfaces. In these lectures I will present joint work with Arend Bayer (arXiv:1203.4613 and arXiv:1301.6968). Topics include: (1) The Positivity Lemma. Given a Bridgeland stability condition, I will motivate and explain how to naturally associate a nef divisor on the moduli space of Bridgeland stable objects. (2) Explicit wall-crossing for K3 surfaces. I will explain projectivity of moduli spaces of stable objects for K3s, and relate concrete wall-crossing to concrete geometry (e.g., Brill-Noether loci, relation to existence of g^1_n on curves in the case of the Hilbert scheme). (3) Systematic wall-crossing analysis for K3 surfaces in terms of its lattice and applications. I will describe of the nef cone of the moduli spaces in terms of the K3 lattice. If time permits, I will also talk about Lagrangian fibrations, and more systematic examples.

Talks

• Nicolas ADDINGTON: Some new rational cubic 4-folds. First I will recall the beautiful story of cubic 4-folds containing a plane: the quadric surface fibration over P², the degree-2 K3 surface with the Brauer class of order 2, the countable union of 18-dimensional families of rational cubics where the Brauer class vanishes, etc. Then I will discuss work in progress, joint with Hassett, Tschinkel, and Várilly-Alvarado, which yields a similar story for cubics containing an elliptic ruled surface: there is a sextic del Pezzo fibration over P², and a degree-2 K3 surface with a Brauer class of order 3, and a countable union of 18-dimensional families of rational cubics. These are the first new rational cubic 4-folds to come along in two de­cades.

• Asher AUEL: The torsion order of an algebraic variety. The minimal multiple of the diagonal to admit a decomposition in the sense of Bloch and Srinivas is called the torsion order of a smooth proper variety.  It is bounded above by the greatest common divisor of the degrees of all unirational parameterizations, and is a stable birational invariant. Recently, the degeneration method initiated by Voisin and developed by Colliot-Thélène and Pirutka has led to a breakthrough in establishing lower bounds for the torsion order, hence obstructions to stable rationality.  I will survey the state of the art of this theory, which includes recent work of Chatzistamatiou and Levine, as well as provide some links to the derived categorical perspective and some new examples

• Christian BÖHNING: Obstructions to rationality, with special regard to cubic fourfolds. We will discuss new and old obstructions to rationality, and then report on more recent work (joint with Hans-Christian v. Bothmer and Pawel Sosna), concerning stable non-rationality of very general hypersurfaces of bi-degree (2,n) for n≥2 in ℙ² x ℙ² as well as some quantitative asymptotic features of birational automorphism groups (dynamical degrees) in the context of the irrationality problem for cubic fourfolds.

• Ana-Maria CASTRAVET: Derived category of moduli of pointed stable rational curves. I will report on joint work with Jenia Tevelev on Kuznetsov's conjecture on the derived category of moduli of pointed stable rational curves and related spaces.

• Izzet COSKUN: Birational Geometry of Moduli Spaces of Shaves on Surfaces. In this talk, I will describe recent progress on computing ample and effective cones of moduli spaces of sheaves on surfaces using Bridgeland stability conditions. The talk will be based on joint work with Jack Huizenga.

• Olivier DEBARRE: On moduli spaces and periods of Gushel-Mukai varieties. Beauville and Donagi proved in 1985 that the primitive middle cohomology of a smooth complex cubic fourfold and the primitive second cohomology of its variety of lines, a smooth hyper­kähler fourfold, are isomorphic as integral Hodge structures. We prove an analogous statement for smooth Gushel-Mukai varieties of dimension 4 (resp. 6), i.e., smooth dimensionally transverse intersections of the cone over the Grassmannian Gr(2,5), a quadric, and two hyperplanes (resp. of the cone over Gr(2,5) and a quadric). The associated hyper­kähler fourfold is now a smooth double cover of a sextic fourfold called an EPW sextic. Moduli spaces of smooth Gushel-Mukai varieties are fibered over the moduli space of these “double EPW sextics” and this can be used to investigate their hyperbolicity properties. This is joint work with Alexander Kuznetsov.

• Julie DÉSERTI: Cremona groups in higher dimensions. The group of birational self maps of the n-dimensional projective space is named Cremona groups. We know a lot of properties when n=2 (group properties, dynamical properties, geometric properties ...). In this talk I will give some properties of the Cremona groups when n>2.

• Gerard FREIXAS I MONTPLET: Some thoughts on an analytic approach to the canonical bundle formula. I will report on joint work with Dennis Eriksson (Chalmers) and Christophe Mourougane (Univ. de Rennes). Our collaboration started by exploring the relations between Kodaira's canonical formula for degenerating families of elliptic curves, and the singularities of some hermitian metrics on families of relative canonical differential forms. We then moved on studying families of K3 surfaces. I will describe the conclusions of this study. I will describe the essentially known case of families of L2 (or Hodge) metrics, whose singularities encode eigenvalues and unipotence order of the monodromy operator, or equivalently log canonical thresholds and a related invariant we call arity. While Hodge metrics are well-known to specialists working on birational geometry and degenerations of Hodge structures, the so-called Quillen metrics are less known and interesting as well. I will explain our conclusions for those. Our approach has its roots in work of Yoshikawa, that we simplify and render somewhat more conceptual by using the theory of localized Chern classes. We show the singularities of the Quillen metric encode the (topological) Euler characteristic of the vanishing cycles for Kulikov models of families of K3 surfaces. It follows a similar phenomenon for holomorphic analytic torsion (for an arbitrary choice of Kähler metric). We pose the problem of finding a refined version of this property, in the lines of old results of Marc Burger on estimates of small eigenvalues of Riemann surfaces in terms of dual graphs and geodesic lengths.

• David JENSEN: Gaussian-Wahl Maps. After defining the Gaussian-Wahl map for algebraic curves, we will discuss applications to deformation theory and the birational geometry of moduli spaces. We will then turn to recent joint work with Yoav Len, in which we use nonarchimedean methods to compute the ranks of Gaussian-Wahl maps for general line bundles on general curves.

• Mattias JONSSON: Uniform K-stability and Kahler-Einstein metrics. The recently settled Yau-Tian-Donaldson conjecture asserts that, for a complex Fano manifold X, the existence of a Kahler-Einstein metric is equivalent to X being K-(poly)stable, a condition of algebro-geometric nature inspired by considerations from GIT. I will report on joint work with Berman, Boucksom and Hisamoto on a stronger form on K-stability, which turns out also to be equivalent to the existence of a Kahler-Einstein metric, and which has been suggested as the correct condition for the existence of constant scalar curvature metrics on general polarized varieties.

• Alexander KUZNETSOV: Derived categories and birational properties of Gushel-Mukai varieties. I will talk about Gushel-Mukai varieties (intersections of a cone over Gr(2,5) with a linear space and a quadric) and discuss structure of their derived categories. It turns out that every GM variety of even dimension has a noncommutative K3 surface and every GM variety of odd dimension has a noncommutative Enriques surface as the nontrivial pieces of their derived categories. In particular, this makes GM fourfolds very similar to cubic fourfolds, both from derived category and birational points of view. For instance, some GM fourfolds are known to be rational, but a very general one is expected to be irrational. I will discuss what is known in this direction and what is expected.

• John LESIEUTRE: Dynamical Mordell-Lang and automorphisms of higher-dimensional varieties. I will discuss an extension of the Dynamical Mordell-Lang conjecture to the setting of (possibly non-reduced) closed subschemes, and show how this statement can be applied geometrically to obtain constraints on automorphisms of higher-dimensional varieties. This is joint work with Daniel Litt.

• Emanuele MACRÌ: Bridgeland stability for semiorthogonal decompositions. I will present a general method to induce Bridgeland stability conditions on semiorthogonal decompositions. We will show, in particular, the existence of Bridgeland stability conditions on the Kuznetsov component of the derived category of (some) Fano 3folds and of cubic fourfolds. This is joint work in progress with Martí Lahoz and Paolo Stellari.

• Keiji OGUISO: Isomorphic quartic K3 surfaces in the view of Cremona and projective transformations. We show that there is a pair of smooth complex quartic K3 surfaces S₁ and S₂ in ℙ³ such that S₁ and S₂ are isomorphic as abstract varieties but not Cremona isomorphic. We also show, in a geometrically explicit way, that there is a pair of smooth complex quartic K3 surfaces S₁ and S₂ in ℙ³ such that S₁ and S₂ are Cremona isomorphic, but not projectively isomorphic.

• Mihnea POPA: Hodge ideals. I will present joint work with M. Mustata, in which we study a sequence of ideals arising naturally from M. Saito's Hodge filtration on the localization along a hypersurface. The multiplier ideal of the hypersurface appears as the first step in this sequence, which as a whole provides a more refined measure of singularities. We give applications to the comparison between the Hodge filtration and the pole order filtration, adjunction, and the singularities of hypersurfaces in projective space and theta divisors on abelian varieties.

• Paolo STELLARI: A derived category approach to some moduli spaces on cubic threefolds and fourfolds. We exploit the homological properties and the geometric meaning of Kuznetsov's semiorthogonal decomposition of the derived cate­gories of cubic fourfolds and threefolds to study the (birational) geometry of some interesting moduli spaces on such varieties. This is the case of the moduli space of stable aCM bundles of a given rank on a cubic threefold and of generalized twisted cubics on cubic fourfolds not containing a plane. This is joint work (partly in progress) with M. Lahoz, M. Lehn, and E. Macrì.

• Michel VAN DEN BERGH: Non-commutative crepant resolution of determinental varieties.

• Junyi XIE: Groupe de Cremona et la conjecture de Zimmer. Nous étudions les groupes d’automorphismes et de transformations birationnelles des variétés quasi-projectives; pour cela nous développons deux nouvelles méthodes, l’une qui utilise des arguments de base d’analyse p-adique, et la seconde qui combine inégalités isopérimétriques et estimées de Lang-Weil. Nous démontrons par exemple que si SLn(Z) agit fidèlement par transformations birationnelles sur une variété complexe quasi-projective X, alors dim(X) ≥ n−1, et X est rationnelle en cas d’égalité.