Algebraic Geometry in Hannover

A cornerstone in the Enriques Classification of algebraic surfaces is Castelnuovo's Rationality Criterion. It can be seen as a consequence of Iskovskih's Theorem, which is of arithmetic nature: A minimal smooth surface with \(h^1(\mathcal{O}_S) = h^0(\omega_S^{\otimes 2})=0\) over an arbitrary ground field is either the projective plane or a quadric surface, or admits a fibration where base and generic fiber are Brauer–Severi curves, or the dualizing sheaf generates the Picard group. We generalize this result to regular surfaces over imperfect ground fields. This involves a theory of inseparable pencils, the geometry of non-normal quartic surfaces, blowing-ups centered at twisted cubics, and vector bundles over genus-zero curves. This is joint work with Andrea Fanelli.

Recently, Yobuko introduced the notion of quasi-F-splitting, which can be viewed as a generalization of the Artin–Mazur height for Calabi–Yau varieties. In this talk, I will discuss the quasi-F-splitting of Fano varieties. In particular, I will introduce a criterion for quasi-F-splitting using Cartier operators, which plays an important role in studying the quasi-F-splitting of Fano varieties. This is joint work with Takamatsu, Tanaka, Witaszek, Yobuko, and Yoshikawa.

In characteristic \(0\), singularities from the Minimal Model Program have good cohomological properties, as exemplified by finite quotient singularities. This is not true in positive characteristic: the geometric conditions imposed by the MMP do not guarantee the good behavior of local cohomology, as shown by examples of Bernasconi, Kovács, Totaro and Yasuda. In this talk, I will present a new and simple way of constructing such pathological examples, using actions by infinitesimal unipotent groups.

We start with a double sextic family of K3 surfaces with four parameters with Picard number \(16\). Then by geometric reduction (specializing at fiber) processes, we obtain three, two and one parameter families of K3 surfaces of Picard number \(17\), \(18\) and \(19\) respectively. All these families turn out to be of hypergeometric type in the sense that their Picard–Fuchs differential equations are given by hypergeometric or Heun functions. We will study the geometry of \(2\)-parameter families and related finite hypergeometric series in detail. We will then discuss some relevant modularity questions. This is a joint work with A. Clingher, A. Malmendier, and N. Yui.

In 1896 Enriques constructed an example of a smooth non-rational surface with invariants \(q=p_g=0\). It arises as the minimal desingularization of an Enriques sextic, i.e a sextic surface that is non-normal along the edges of a tetrahedron. In modern terminology, these are examples of Enriques surfaces, and it was already known to the italian geometers that a general complex Enriques surface arises from this construction. I will talk about a joint work with G. Martin and D. Veniani, in which we show that in fact every Enriques surface (in characteristic different from \(2\)) arises as the minimal desingularization of an Enriques sextic. In a similar vein, we show that every Enriques surface containing a smooth rational curve arises as a certain congruence of lines in projective \(3\)-space, called a Reye congruence.

A \(3\)-dimensional del Pezzo fibration \(X \to C\) is one of the possible outcomes of the MMP. One aims to study the birational geometry of \(X\) in terms of the generic fibre \(X_{k(C)}\) and the base curve \(C\). In positive characteristic, an additional difficulty arises: there are examples where all the fibres of a del Pezzo fibration are singular. In this talk, we explain these examples and describe how to bound such pathological cases. We extend two classical results of Enriques and Manin: (1) a \(3\)-dimensional del Pezzo fibration over a curve, defined over an algebraically closed field, always admits a section; and (2) the total space is rational if the base curve is \(\mathbb{P}^1\).

Let \(\mathcal{A}_g\) be the moduli space over \(\mathbf{F}_p\) of \(g\)-dimensional principally polarised abelian varieties (where \(p\) is a prime and \(g \geq 1\)), and let \(\mathcal{S}_g\) be the supersingular locus. Oort conjectured that all generic points of \(\mathcal{S}_g\) have automorphism group \(\{\pm 1\}\). We will present our results that settle Oort's conjecture for \(g=2,3,4\), and for all higher even dimensions when \(p \geq 5\). This talk is based on joint works with R. Pries, F. Yobuko, and C.-F. Yu.

The global Torelli theorem for IHS manifolds plays a crucial role in the study of their automorphisms. Recent algorithmic developments by Brandhorst and Hofmann offer a framework in which one can systematically classify such automorphisms. This is possible, at least, provided one knows three deformation invariants: the BBF-form, the numerical wall divisors, and the monodromy group. All of these are known for the currently known IHS manifolds.

The generalization of the global Torelli theorem to singular IHS varieties opens the way to extend these classification techniques to singular analogues of IHS manifolds. The deformation class of such varieties for which the deformation invariants are best understood are the Nikulin-type orbifolds. Through extensive work by Menet and Rieß, the global Torelli theorem for such orbifolds, their BBF-form, and the associated wall divisors are known. Describing the corresponding monodromy group is the next essential step in this classification program.

In this talk, we will review the construction of Nikulin orbifolds, which serve as fundamental examples for the deformation class of Nikulin-type orbifolds. We will then outline how to prove that the monodromy of such varieties is maximal. This description enables a classification of symplectic automorphisms of Nikulin-type orbifolds via the global Torelli theorem, leading to the first classification of symplectic automorphisms of a singular IHS variety. This is based on joint work with Simon Brandhorst and Grégoire Menet.

We work over the complex number field. A general hypersurface of multi-degree two \((2, 2, \ldots, 2) \subset ({\mathbf P}^1)^{n+1}\) with \(n \ge 3\) is a strict Calabi–Yau manifold of dimension \(n\), called a Calabi–Yau manifold of Wehler type and their birational automorphism groups were intensively studied by my joint work with Professor Serge Cantat about ten years ago. In this talk, after recalling some remarkable properties of Calabi–Yau manifold of Wehler type, I would like to show that any Calabi–Yau manifold of Wehler type has always a primitive birational automorphism, in particular, a birational automorphism of Zariski dense orbit. This is a generalization of my previous work for a Calabi–Yau manifold of Wehler type of odd dimension. It would be very interesting to see if the same holds over \(\overline{\mathbf{Q}}\) (not yet solved).