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 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.

We provide an exposition of the canonical self-duality associated to a presentation of a finite, flat, complete intersection over a Noetherian ring, following work of Scheja and Storch.

In this expository paper, we show that the Deligne–Mumford moduli space of stable curves is projective over Spec (Ζ). The proof we exposit is due to Kollár. Ampleness of a line bundle is deduced from nefness of a related vector bundle via the Ampleness Lemma, a classifying map construction. The main positivity result concerns the pushforward of relative dualizing sheaves on families of stable curves over a smooth projective curve.

This is an expository account about height functions and Arakelov theory in arithmetic geometry. We recall Conrad’s description of generalized global fields in order to describe heights over function fields of higher transcendence degree. We then give a brief overview of Arakelov theory and arithmetic intersection theory. Our exposition culminates in a description of Moriwaki’s Arakelov-theoretic formulation of heights, as well as a comparison of Moriwaki’s construction to various versions of heights.

We explain a theorem of D. Schäppi on the reconstruction of an affine category scheme (dually, a coalgebroid) over a general commutative ring from its category of finite-rank representations.

In this expository article, we follow Langer’s work in [5] to prove the boundedness of the moduli space of semistable torsion-free sheaves over a projective variety, in any characteristic.