Let X be a regular tame stack. If X is locally of finite type over a field, we prove that the essential dimension of X is equal to its generic essential dimension; this generalizes a previous result of P. Brosnan, Z. Reichstein and the second author. Now suppose that X is locally of finite type over a 1-dimensional noetherian local domain R with fraction field K and residue field k. We prove that ed Introduction, and the statement of the main theoremsLet k be a field, X → Spec k an algebraic stack, an extension of k, ξ ∈ X( ) an object of X over . If k ⊆ L ⊆ is an intermediate extension, we say, very naturally, that L is a field of definition of ξ if ξ descends to L. The essential dimension ed k ξ, which is either a natural number or +∞, is the minimal transcendence degree of a field of definition of ξ. If X is of finite type then ed k ξ is always finite.This number ed k ξ is a very natural invariant, which measures, essentially, the number of independent parameters that are needed for defining ξ. The essential dimension ed k X of X is the supremum of the essential dimension of all objects over all extensions of k (if X is empty then ed k X is −∞). This number is the answer to the question "how many independent parameters

Grothendieck gave two forms of his "main conjecture of anabelian geometry", namely the section conjecture and the hom conjecture. He stated that these two forms are equivalent and that if they hold for hyperbolic curves, then they hold for elementary anabelian varieties too. We state a stronger form of Grothendieck's conjecture (equivalent in the case of curves) and prove that Grothendieck's statements hold for our form of the conjecture. We work with DM stacks, rather than schemes. If X is a DM stack over k ⊆ C, we prove that whether X satisfies the conjecture or not depends only on X C. We prove that the section conjecture for hyperbolic orbicurves stated by Borne and Emsalem follows from the conjecture for hyperbolic curves. Let us recall the two forms of the main conjecture. If G and H are extensions of a group Γ, then Hom-ext Γ (G, H) is the set of homomorphisms G → H which commute with the projection to Γ modulo the natural action of ker(H → Γ) by conjugation. If k is a field, we denote by Γ k = Gal(k s /k) the absolute Galois group. If T and X are geometrically connected over k with etale fundamental groups π 1 (T) and π 1 (X) (we omit base points), there is a natural map Hom k (T, X) → Hom-ext Γ k (π 1 (T), π 1 (X)). Conjecture (Grothendieck, "hom conjecture"). Let k be finitely generated over Q. If T /k is

Let $k$ be a field, $X$ a variety with tame quotient singularities, and $\tilde {X}\to X$ a resolution of singularities. Any smooth rational point $x\in X(k)$ lifts to $\tilde {X}$ by the Lang–Nishimura theorem, but if $x$ is singular this might be false. For certain types of singularities, the rational point is guaranteed to lift, though; these are called singularities of type $\textrm {R}$. This concept has applications in the study of the fields of moduli of varieties and yields an enhanced version of the Lang–Nishimura theorem where the smoothness assumption is relaxed. We classify completely the tame quotient singularities of type $\textrm {R}$ in dimension $2$; in particular, we show that every non-cyclic tame quotient singularity in dimension $2$ is of type $\textrm {R}$, and most cyclic singularities are of type $\textrm {R}$ too.

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J. Stix proved that a curve of positive genus over $$\mathbb {Q}$$ Q which maps to a non-trivial Brauer–Severi variety satisfies the section conjecture. We prove that, if X is a curve of positive genus over a number field k and the Weil restriction $$R_{k/\mathbb {Q}}X$$ R k / Q X admits a rational map to a non-trivial Brauer–Severi variety, then X satisfies the section conjecture. As a consequence, if X maps to a Brauer–Severi variety P such that the corestriction $${\text {cor}}_{k/\mathbb {Q}}([P])\in {\text {Br}}(\mathbb {Q})$$ cor k / Q ( [ P ] ) ∈ Br ( Q ) is non-trivial, then X satisfies the section conjecture.