# Fortnightly links (50)

Mica Mustaţă, Rationality of algebraic varieties and Takumi Murayama's live-TeX'd version are some really good-looking lecture notes on the rationality of varieties. Takumi has more live-TeX'd notes on his website too by the way.

Kęstutis Česnavičius, Purity for the Brauer group is a preprint for which I claim no understanding whatsoever of the methods being used, but it is really interesting to see how there is now a complete result on purity for the (cohomological) Brauer group: if $X$ is a regular scheme and $Z$ a closed subscheme of codimension at least 2, then $\operatorname{Br}'(X)=\operatorname{Br}'(X\setminus Z)$. If $X$ comes equipped with an ample line bundle, then this really becomes a description of Azumaya algebras (and not étale cohomology groups).

The classification of Fano 3-folds is one of the greatest accomplishments in algebraic geometry. An interesting family in the classification are the ones labelled $\mathrm{V}_{22}$: their derived category has a full exceptional collection of minimal length (being 4, just like $\mathbb{P}^3$, the quadric, and the quintic del Pezzo 3-fold) but unlike these varieties this is an actual family of varieties (a 6-dimensional one at that). Moreover, their automorphism groups vary significantly within this family: up to a discrete group it is trivial, but it's $\mathbb{G}_{\mathrm{m}}$, $\mathbb{G}_{\mathrm{a}}$ for a curve inside the moduli space, and $\mathrm{PGL}_2$ for a unique point of the moduli space. And this one point is also the one with a formal noncommutative deformation. Can you tell I'm excited by this class of varieties?

In any case, Alexander Kuznetsov, Yuri Prokhorov: Prime Fano threefolds of genus 12 with a $\mathbb{G}_{\mathrm{m}}$-action explicitly describes and classifies the ones with the multiplicative group inside their automorphisms.