Hypersurfaces in projective schemes and Bertini's theorem

• Qing Liu

Room: 0-6 Let $X$ be a projective scheme over an affine base. We develop a technique for proving the existence of closed subschemes $H$ with various favorable properties. We offer several applications of this technique, including the existence of hypersurfaces in $X$ containing a given closed subscheme and intersecting properly a given closed set, and the existence of finite quasi-sections. This is joint work with O. Gabber and D. Lorenzini.

Nodal hypersurfaces with defect, Alexander polynomials and Mordell-Weil groups

• Remke Kloosterman

Room: 0-6 In this talk we present a short proof for Cheltsov's result that a nodal hypersurface of degree $d$ in $P^4$ which is not factorial, has at least $(d-1)^2$ nodes. We will discuss how variants of these arguments yields interesting results on the fundamental group of the complement of a singular plane curve and on the Mordell-Weil group of certain abelian varieties over function fields of characteristic zero.

Volume of complex hyperbolic structures on moduli spaces of genus zero curves

• Vincent Koziarz

Room: 0-6 (joint work with D. M. Nguyen): I will show that the complex hyperbolic metrics defined by Deligne-Mostow and Thurston on the moduli space of genus $0$ curves with $n$ marked points $M_{0,n}$ are singular Kaehler-Einstein metrics when $M_{0,n}$ is embedded in its Deligne-Mumford-Knudsen compactification. As a consequence, I will obtain a formula computing the volume of $M_{0,n}$ with respect to these metrics using intersection of boundary divisors of its compactification. In the case when the weights parametrizing the complex hyperbolic structures are rational, following an idea of Y. Kawamata, I will show that the associated metrics actually represent the first Chern class of some line bundles on the compactification of $M_{0,n}$, from which other formulas computing the same volumes will be derived.

Around the geometry of Calogero-Moser spaces

• Cédric Bonnafé

Room: 0-6 Numerical evidences suggest that the representation theory of a finite reductive group should be connected to the geometry of the Calogero-Moser variety associated with its corresponding Weyl group. Despite we have no (serious) clue for what should be the link, pursuing this analogy leads to new questions about the geometry of this variety, which might have an interest by themselves: symplectic resolutions, Poisson structure and symplectic leaves, fixed points, equivariant cohomology.

GIT vs Baily-Borel compactification for the moduli space of Quartic Surfaces

• Radu Laza

Room: 0-6 This is a report on joint work with Kieran O'Grady. The period map from the GIT moduli space of quartic surfaces to the Baily-Borel compactification of the period space is birational but far from regular. New birational models of locally symmetric varieties of Type IV have been introduced by Looijenga, in order to study similar problems. Looijenga's construction does not succeed in “explaining” the period map for quartic surfaces. We discovered that one can (conjecturally) reconcile Looijenga's philosophy with the phenomenology of quartic surfaces, provided one takes into account suitable Borcherd relations between divisor classes on relevant locally symmetric varieties. We work with a tower of locally symmetric varieties, in particular our results should also “explain” the period map for double EPW sextics.

Isogenies and transcendental Hodge structures of K3 surfaces

• Samuel Boissière

Room: 0-6 Every Hodge class on a product of two complex projective K3 surfaces induces a homomorphism of rational Hodge structures between the respective transcendental lattices. Under the hypothesis that this morphism is an isometry of rational quadratic spaces, Mukai, Nikulin and recently Buskin have proven that the corresponding Hodge class is algebraic, confirming the Hodge conjecture in this context. In this talk, I will show that the hypothesis of isometry is too restrictive by constructing geometrically some families of isogenies between K3 surfaces whose transcendental Hodge structures are nonisometric. This is a collaboration with Alessandra Sarti and Davide Cesera Veniani.

Explicit Schoen surfaces

• Alessandra Sarti

Room: 0-6 I will present an explicit geometric construction of some special surfaces of general type described by Schoen in 2007, that occupy an important place in the geography of surfaces of general type. I will show how the construction involves the Segre cubic, the Igusa quartic and K3 quartic surfaces with fifteen nodes. This is a joint work with Carlos Rito and Xavier Roulleau.

Abelian varieties and Minkowski-Hlawka theorem

• Pascal Autissier

Room: 0-6 A classical theorem of Minkowski and Hlawka states that there exists a lattice in $R^n$ with packing density at least $2^{1-n}$. Buser and Sarnak proved the analogue of this result in the context of complex abelian varieties. Here we give an improvement of this analogue; this shows a conjecture of Muetzel.

Degenerations of Nikulin surfaces and moduli of curves

• Andreas Knutsen

Room: 0-6 Nikulin surfaces are surfaces arising as quotiens of K3 surfaces by a symplectic involution. They have eight nodes (arising from the eight fixed points of the involution), and their desingularizations are smooth K3 surfaces with eight $(-2)$-curves whose sum is $2$-divisible in the Picard group. A particular feature is that their smooth hyperplane sections carry a nontrivial $2$-torsion element in their Picard group that is induced from a line bundle on the surface. There is therefore a natural moduli map from the space $P_g$ of pairs $(S,C)$ where $S$ is a Nikulin surface and $C$ is a smooth genus $g$ hyperplane section of it to the moduli space $R_g$ of genus $g$ Prym curves, that is, of pairs $(C,\eta)$, where $\eta$ is a nontrivial $2$-torsion element in $Pic(C)$. I will give an overview of recent results on this map obtained in a work in progress with Margherita Lelli-Chiesa and Alessandro Verra and how degenerations of Nikulin surfaces to surfaces that are birational to unions of rational ruled surfaces are of help.