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A uniform bound on the size of finite subgroups of polynomial automorphisms
Marc ABBOUD (Neuchâtel)
Abstract
In 1887, Minkowski showed that there is a uniform bound $M(d)$ for the size of finite subgroups of $GL_d(\mathbb{Q})$.
The idea of the proof is to reduce the coefficients modulo $p$ for a suitable prime p and then compute the size of $GL_d (\mathbb{F}_p)$.
Schur later extended the result to number fields with a similar idea. We will show in this talk the following result: If $K$ is a finitely generated field over
$\mathbb{Q}$ then there exists a uniform bound $M(K, d)$ such that any finite subgroup of polynomial automorphisms of $K^d$ has size bounded by $M(K,d)$.
If $K$ is a number field, then $M(K,d)$ is exactly the Minkowski-Schur bound for linear groups.
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A signed count to bitangents to a real plane curve
Thomas BLOMME (Neuchâtel)
Abstract
In 1834, Plücker computed the number of flexes and bitangents to a plane complex curve of degree $d$, which do not depend on the choice of the curve provided
it is generic. The situation in the real case, first studied by Zeuthen in the case of quartic curves, is more delicate.
In this talk, we will prove the existence of a signed count of the real bitangents to a plane curve of even degree that only depend on the topology of
the pair $(C(\mathbb{R}),\mathbb{RP}^2)$. This is a joint work with E. Brugallé and Cristhian Garay.
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Multi-centered dilatations of schemes
Arnaud MAYEUX (Einstein Institute of Mathematics, Jerusalem)
Abstract
Localizations of a structure with a composition law inverse some prescribed elements, so that new fractions are well-defined after localizations.
(Multi-centered) dilatations do the same except that they add only some fractions where both numerators and denominators are prescribed. In this talk,
we will introduce dilatations of rings. We will see that this construction can be globalized and makes sense on Grothendieck schemes.
Then, we will collect and discuss several results on multi-centered dilatations of schemes. References will be given during the talk.
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Simplicity of a ring under differential operators
Alapan MUKHOPADHYAY (EPFL)
Abstract
D-simplicity of a ring i.e. simplicity under the action of (Grothendieck) ring of differential operators implies many nice properties of the ring.
In this talk, we first give a criterion for D-simplicity when the ring is local. As a consequence, we obtain that D-simplicity of a
local ring implies D-simplicity of its completion. The converse is however not true: We will construct exotic regular rings which do not admit
any nontrivial differential operators of higher order; in particular no nonzero derivations. We will discuss the consequence of such
constructions on an open question of Hochster and Yao. Part of the talk will report a joint work with Karen Smith.
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On the uniruledness of asymptotic base loci
Nikolaos TSAKANIKAS (EPFL)
Abstract
Given a normal projective variety $X$ and a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $D$ on $X$, one attaches to $D$ various so-called asymptotic base loci,
which measure the failure of positivity properties of $D$. In this talk I will present some results that compare these asymptotic base loci,
focusing on the case when $D$ is the canonical class of a pair $(X,B)$ with mild singularities.
I will also discuss the uniruledness of the irreducible components of those loci in the same context. The talk is based on joint work with Zhixin Xie.