We prove that Q-Fano threefolds of Fano index ≥ 8 are rational.

We introduce and study the question how can stable birational types vary in a smooth proper family. Our starting point is the specialization for stable birational types of Nicaise and the author, and our emphasis is on stable birational types of hypersurfaces. Building up on the work of Totaro and Schreieder on stable irrationality of hypersurfaces of high degree, we show that smooth Fano hypersurfaces of large degree over a field of characteristic zero are in general not stably birational to each other. In the appendix, Claire Voisin proves a similar result in a different setting using the Chow decomposition of diagonal and unramified cohomology.

We give a survey on the connections between terminal 3-fold flips and cluster algebras. In particular we observe that Mori’s algorithm for generating the relations defining a type k2A flipping neighbourhood is a form of generalised cluster algebra mutation. We then use the Laurent phenomenon for this cluster algebra structure to give an alternative proof of the existence of Brown and Reid’s diptych varieties.

We present the outlook and some of the main contents of the book, including some basic definitions.

We present several sufficient conditions for the upper bound.

The aim of this note is to discuss the Weil restriction of schemes and algebraic spaces, highlighting pathological phenomena that appear in the theory and are not widely known. It is shown that the Weil restriction of a locally finite algebraic space along a finite flat morphism is an algebraic space.

We explain a proof of the Theorem of the Base: the Neron– Severi group of a proper variety is a finitely generated abelian group. We discuss, quite generally, the Picard functor and its torsion and identity components. We study representability and finiteness properties of the Picard functor, both absolutely and in families. Along the way, we streamline the original proof by using alterations, and we discuss some examples of peculiar Picard schemes.

We present several formulations of the large deviation principle for empirical measures in the V topology, depending on the initial distribution. The case V = B(S) is further studied.

We discuss the projectivity of the moduli space of semistable vector bundles on a curve of genus g ≥ 2. This is a classical result from the 1960s, obtained using geometric invariant theory. We outline a modern approach that combines the recent machinery of good moduli spaces with determinantal line bundle techniques. The crucial step producing an ample line bundle follows an argument by Faltings with improvements by Esteves–Popa. We hope to promote this approach as a blueprint for other projectivity arguments.

We show that the moduli stack of admissible-covers of prestable curves is an algebraic stack, loosely following [1, App. B]. As preparation, we discuss finite group actions on algebraic spaces.