The
Basel-Dijon-EPFL Joint Seminars

Towards a purity-for-torsors theorem for $F$-regular singularities

Javier CARVAJAL ROJAS (EPFL)

A classical result (by Zariski--Nagata--Auslander in the étale case and by Nori--Moret-Bailly in the general case) establishes that, over a regular scheme, every finite fppf torsor in codimension-1 is a torsor everywhere. This is referred to as the purity theorem for torsors. In this talk, we'll discuss to what extent this theorem fails and holds for a class of singularities in positive characteristic called (strongly) $F$-regular singularities; as defined by Hochster--Huneke. More concretely, we will show that an $F$-regular singularity admits a finite cover for which purity holds for abelian and étale torsors. Time permitting, we'll discuss possible directions to expand this result beyond the abelian case.

Smooth rational varieties with infinitely many real forms

Lucy MOSER-JAUSLIN (IMB)

In this talk, I will discuss joint work with A. Dubouloz and G. Freudenburg. Given a real variety $X$, a real form of X is a real variety $Y$ such that the complexifications of $X$ and $Y$ are isomorphic as complex varieties. We will show how to construct smooth rational affine algebraic varieties of dimension $4$ or higher which admit infinitely many non-isomorphic real forms.

On compactifications of contractible affine threefolds into del Pezzo fibrations

Masaru NAGAOKA (University of Tokyo)

Coniveau filtrations and applications

Jan NAGEL (IMB)

We show how conditions on the (modified) coniveau filtration on the homology of complete intersections influence the Chow groups. This leads to a generalization of results of Voisin and Vial on injectivity of cycle class and Abel-Jacobi maps. Using a variant involving group actions, we obtain several new examples of complete intersections whose motive is 'finite-dimensional'. This is joint work with Robert Laterveer and Chris Peters.

On rationally connected varieties over $C_1$ fields of characteristic $0$

Marta PIEROPAN (EPFL)

In the 1950s Lang studied the properties of $C_1$ fields, that is, fields over which every hypersurface of degree at most $n$ in a projective space of dimension $n$ has a rational point. Later he conjectured that every smooth proper rationally connected variety over a $C_1$ field has a rational point. The conjecture is proven for finite fields (Esnault) and function fields of curves over algebraically closed fields (Graber-Harris-de Jong-Starr), but it is still open for the maximal unramified extensions of $p$-adic fields. I use birational geometry in characteristic $0$ to reduce the conjecture to the problem of finding rational points on Fano varieties with terminal singularities.

On maximal subrings

Immanuel van SANTEN (University of Basel)

This is joint work with Stefan Maubach. Let $R$ be a ring. A proper subring $A$ of $R$ is called a maximal subring if there is no subring that lies properly between $A$ and $R$. We are interested in the classification of all maximal subrings of $R$. The maximal subrings $A$ of $R$ that induce a surjection on the prime spectra $Spec(R)\rightarrow Spec(A)$ can be fully described. We call this the non-extending case. Much harder is the case, when the map on prime spectra is not surjective, we call this the extending case.
In this talk we focus first on the classification of the maximal subrings of $R$ in the non-extending case. We then investigate the extending case for certain finitely generated domains over an algebraically closed field. Namely, we consider the case, when $R$ is one-dimensional and as a two-dimensional example we consider the case when $R$ is the ring of Laurent polynomials over a polynomial ring.

Hyperkähler manifolds and non-positively curved groups

Egor YASINSKY (University of Basel)

I will describe how some classical results in geometric group theory help to understand the structure of bimeromorrphic and biholomorphic automorphism groups of hyperkähler manifolds. In particular, we will see that these groups are finitely presented and satisfy Tits alternative (first proved by K. Oguiso).