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Giulio Bresciani

Publications and source records attributed to Giulio Bresciani.

24 records · Page 2Linked to original sources
arXiv2021

On the section conjecture and Brauer-Severi varieties

J. Stix proved that a curve of positive genus over $\mathbb{Q}$ which maps to a non-trivial Brauer-Severi variety satisfies the section conjecture. We prove that, if $X$ is a curve of positive genus over a number field $k$ and the Weil restriction $R_{k/\mathbb{Q}}X$ admits a rational map to a non-trivial Brauer-Severi variety, then $X$ satisfies the section conjecture. As a consequence, if $X$ maps to a Brauer-Severi variety $P$ such that the corestriction $\operatorname{cor}_{k/\mathbb{Q}}([P])\in\operatorname{Br}(\mathbb{Q})$ is non-trivial, then $X$ satisfies the section conjecture.

math.NT↗
arXiv2021

A higher dimensional Hilbert irreducibility theorem

Assuming the weak Bombieri-Lang conjecture, we prove that a generalization of Hilbert's irreducibility theorem holds for families of geometrically mordellic varieties (for instance, families of hyperbolic curves). As an application we prove that, assuming Bombieri-Lang, there are no polynomial bijections $\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}$.

math.NT↗
arXiv2021

Essential dimension and pro-finite group schemes

A. Vistoli observed that, if Grothendieck's section conjecture is true and $X$ is a smooth hyperbolic curve over a field finitely generated over $\mathbb{Q}$, then $\underlineπ_{1}(X)$ should somehow have essential dimension $1$. We prove that an infinite, pro-finite étale group scheme always has infinite essential dimension. We introduce a variant of essential dimension, the fce dimension $\operatorname{fced} G$ of a pro-finite group scheme $G$, which naturally coincides with $\operatorname{ed} G$ if $G$ is finite but has a better behaviour in the pro-finite case. Grothendieck's section conjecture implies $\operatorname{fced}\underlineπ_{1}(X)=\dim X=1$ for $X$ as above. We prove that, if $A$ is an abelian variety over a field finitely generated over $\mathbb{Q}$, then $\operatorname{fced}\underlineπ_{1}(A)=\operatorname{fced} TA=\dim A$.

math.AG↗
arXiv2020

Some implications between Grothendieck's anabelian conjectures

Grothendieck gave two forms of his "main conjecture of anabelian geometry", i.e. the section conjecture and the hom conjecture. He stated that these two forms are equivalent and that if they hold for hyperbolic curves then they hold for elementary anabelian varieties too. We state a stronger form of Grothendieck's conjecture (equivalent in the case of curves) and prove that Grothendieck's statements hold for our form of the conjecture. We work with DM stacks, rather than schemes. If $X$ is a DM stack over a field $k\subseteq\mathbb{C}$ finitely generated over $\mathbb{Q}$, we prove that whether $X$ satisfies the conjecture or not depends only on $X_{\mathbb{C}}$. We prove that the section conjecture for hyperbolic orbicurves stated by Borne and Emsalem follows from the conjecture for hyperbolic curves.

math.AG↗
arXiv2019

On Grothendieck's section conjecture for orbicurves

As already noted by Niels Borne and Michel Emsalem, there is a natural generalization of the section conjecture for proper orbicurves. Combined with the reformulation by Niels Borne and Angelo Vistoli of the conjecture in terms of the étale fundamental gerbe, this suggests an even stronger conjecture for orbicurves, asking an equivalence of categories instead of a mere bijection. We prove that the three versions of the conjecture are in fact equivalent, and that "injectivity" (i.e. full faithfulness) holds in the case of orbicurves. As a byproduct, we obtain a new proof of the fact that the section conjecture for proper curves implies the section conjecture for open curves.

math.AG↗