Two characterizations of topological spaces with no infinite discrete subspace

We give two characteristic properties of topological spaces with no infinite discrete subspaces. The first one was obtained recently by the first author. The full result extends well-known characterizations of posets with no infinite antichain.Comment: 10 pages, no figures Replace second author by first author in the abstract and in the first line after Theorem

Finiteness for the k-factor model and chirality varieties

This paper deals with two families of algebraic varieties arising from applications. First, the k-factor model in statistics, consisting of n-times-n covariance matrices of n observed Gaussian variables that are pairwise independent given k hidden Gaussian variables. Second, chirality varieties inspired by applications in chemistry. A point in such a chirality variety records chirality measurements of all k-subsets among an n-set of ligands. Both classes of varieties are given by a parameterisation, while for applications having polynomial equations would be desirable. For instance, such equations could be used to test whether a given point lies in the variety. We prove that in a precise sense, which is different for the two classes of varieties, these equations are finitely characterisable when k is fixed and n grows.Comment: 13 page

Nagata compactification for algebraic spaces

We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian approximation for such algebraic spaces, generalizing earlier results in the case of schemes.Comment: 49 pages, various clarifications and bugfixe

Recasting results in equivariant geometry: affine cosets, observable subgroups and existence of good quotients

Using the language of stacks, we recast and generalize a selection of results in equivariant geometry