1 Affine Morphisms Definition 1. Let f : X −→ Y be a morphism of schemes. Then we say f is an affine morphism or that X is affine over Y , if there is a nonempty open cover {Vα}α∈Λ of Y by open affine subsets Vα such that for every α, fVα is also affine. If X is empty (in particular if Y is empty) then f is affine. Any morphism of affine schemes is affine. Any isomorphism is affine, and the aff...

The purpose of this note is to prove the following “boundedness” stated in [Ol]. Let X and Y be separated Deligne–Mumford stacks of finite presentation over an algebraic space S and define HomS(X ,Y) as in [Ol, 1.1]. Assume that X is flat and proper over S, and that locally in the fppf topology on S, there exists a finite flat surjection Z → X from an algebraic space Z. Let Y → W be a quasi-fin...

We prove that a well-distributed subset of R 2 can have a separated distance set only if the distance is induced by a polygon. Basic definitions Separated sets. We say that S ⊂ R d is separated if there exists c separation > 0 such that ||x − y|| ≥ c separation for every x, y ∈ S, x = y. Here and throughout the paper, ||x|| = x 2 1 + · · · + x 2 d is the standard Euclidean distance. Well-distri...