We discuss the naive duality theory of coherent, torsion free, S2 sheaves on schemes.

It is our great pleasure and honour to say Happy Birthday to our friend Miles Reid, for your 70th birthday and indeed a few subsequent ones; it takes a long time to make a big pot, as they say, perhaps especially when there are so many potters.

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

We consider semi-orthogonal decompositions of derived categories for 3-dimensional projective varieties in the case when the varieties have ordinary double points.

We classify minimal projective 3-folds of general type with pg = 2 by studying the birationality of their 6-canonical maps.

We explain a classical construction of a del Pezzo surface of degree d = 4 or 5 as a smooth order 2 congruence of lines in P3 whose focal surface is a quartic surface X20-d with 20-d ordinary double points. We also show that X15 can be realized as a hyperplane section of the Castelnuovo– Richmond–Igusa quartic hypersurface in P4. This leads to the proof of rationality of the moduli space of 15-nodal quartic surfaces. We discuss some other birational models of X15: quartic symmetroids, 5-nodal quartic surfaces, 10-nodal sextic surfaces in P4 and nonsingular surfaces of degree 10 in P6. Finally we study some birational involutions of a 15- nodal quartic surface which, as it is shown in Part II of the paper jointly with I. Shimada [DS20], belong to a finite set of generators of the group of birational automorphisms of a general 15 nodal quartic surface.

We introduce and study the notion of ‘surface decomposable’ variety, and discuss the possibility that any projective hyper-Käahler manifold is surface decomposable, which would produce new evidence for Beauville’s weak splitting conjecture.

A number of years ago, one of us (Mark Gross) was giving a lecture at the University of Warwick on the material on scattering diagrams from [GPS10]. Of course, Miles was in the audience, and he asked (paraphrasing as this was many years ago) whether, at some point, the lecturer would come back down to earth. The goal of this note is to show, in fact, we have not left the planet by considering a particularly beautiful example of the mirror symmetry construction of [GHK15b], namely the mirror to a cubic surface.

We consider that kernels of inflation maps associated with extraspecial p-groups in stable group cohomology are generated by their degree-2 components. This turns out to be true if the prime is large enough compared to the rank of the elementary abelian quotient, but false in general.

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.