An elementary approach to Stix's proof of the real section conjecture

估值截面猜想、étale同调与Berkovich空间

The Valuative Section Conjecture, Étale Homotopy, and Berkovich Spaces

本文通过Berkovich空间重新解释了Pop和Stix关于p进截面猜想的结果，并推广到更广泛的字段和代数簇类。

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.

An elementary proof that vector bundles on $\mathbb P^1$ split

This paper gives a new elementary proof of the theorem that all vector bundles on $\mathbb P^1$ split into the direct sum of line bundles. The proof is based on the study of divisors associated to germs of sections at the generic point.

On abelian birational sections

For a smooth and geometrically irreducible variety X over a field k, the quotient of the absolute Galois group G_{k(X)} by the commutator subgroup of G_{\bar k(X)} projects onto G_k. We investigate the sections of this projection. We show that such sections correspond to "infinite divisions" of the elementary obstruction of Colliot-Thélène and Sansuc. If k is a number field and the Tate-Shafarevich group of the Picard variety of X is finite, then such sections exist if and only if the elementary obstruction vanishes. For curves this condition also amounts to the existence of divisors of degree…

Maps between moduli spaces of vector bundles and the base locus of the theta divisor

Given a vector bundle $E$ of rank $r$ and degree $d$ on a curve $C$ of genus $g$, one can associate to $E$ in a natural way several other vector bundles. For example, one can take wedge powers of $E$. If $E$ is generated by global sections, the kernel of the evaluation map of sections is again a vector bundle. Also, new vector bundles can be produced by taking elementary transformations centered at a fixed point. Under suitable conditions on degree and rank, these constructions can be carried out globally. While all this processes seem quite elementary, very little is known about the resulting…

A home-made Hartshorne-Serre correspondence

We provide an elementary proof of the Hartshorne-Serre correspondence for constructing vector bundles from local complete intersection subschemes of codimension two. This will be done, as in the correspondence of hypersurfaces and line bundles, by patching together local determinantal equations in order to produce sections of a vector bundle.