Essential dimension and pro-finite group schemes

A. Vistoli observed that, if Grothendieck's section conjecture is true and $X$ is a smooth hyperbolic curve over a field finitely generated over $\mathbb{Q}$, then $\underline{\pi}_{1}(X)$ should somehow have essential dimension $1$. We prove that an infinite, pro-finite \'etale group scheme always has infinite essential dimension. We introduce a variant of essential dimension, the fce dimension $\operatorname{fced} G$ of a pro-finite group scheme $G$, which naturally coincides with $\operatorname{ed} G$ if $G$ is finite but has a better behaviour in the pro-finite case. Grothendieck's section conjecture implies $\operatorname{fced}\underline{\pi}_{1}(X)=\dim X=1$ for $X$ as above. We prove that, if $A$ is an abelian variety over a field finitely generated over $\mathbb{Q}$, then $\operatorname{fced}\underline{\pi}_{1}(A)=\operatorname{fced} TA=\dim A$.Comment: Simplified proofs and stronger results in the new versio

The field of moduli of plane curves

It is a classical fact going back to F. Klein that an elliptic curve $E$ over $\bar{\mathbb{Q}}$ is defined by a homogeneous polynomial in $3$ variables with coefficients in $\mathbb{Q}(j_{E})$, where $j_{E}$ is the $j$-invariant of $E$, and $\mathbb{Q}(j_{E})$ is the field of moduli of $E$. The general definition of field of moduli goes back to T. Matsusaka and G. Shimura. With few exceptions, it coincides with the intersection of the fields where the curve is defined. We prove that every smooth plane curve of degree prime with $6$ is defined by a homogeneous polynomial with coefficients in the field of moduli. Furthermore, we show that most plane curves in arbitrary degree, and more generally most algebraic cycles in $\mathbb{P}^{2}$ with finite automorphism group, descend to a Brauer-Severi surface over the field of moduli

On the birational section conjecture with strong birationality assumptions

Let $X$ be a curve over a field $k$ finitely generated over $\mathbb{Q}$ and $t$ an indeterminate. We prove that, if $s$ is a section of $\pi_{1}(X)\to\operatorname{Gal}(k)$ such that the base change $s_{k(t)}$ is birationally liftable, then $s$ comes from geometry. As a consequence we prove that the section conjecture is equivalent to the cuspidalization of all sections over all finitely generated fields.Comment: Final versio

The field of moduli of varieties with a structure

If $X$ is a variety with an additional structure $\xi$, such as a marked point, a divisor, a polarization, a group structure and so forth, then it is possible to study whether the pair $(X,\xi)$ is defined over the field of moduli. There exists a precise definition of ``algebraic structures'' which covers essentially all of the obvious concrete examples. We prove several formal results about algebraic structures. There are immediate applications to the study of fields of moduli of curves and finite sets in $\mathbb{P}^{2}$, but the results are completely general. Fix $G$ a finite group of automorphisms of $X$, a $G$-structure is an algebraic structure with automorphism group equal to $G$. First, we prove that $G$-structures on $X$ are in a $1:1$ correspondence with twisted forms of $X/G\dashrightarrow\mathcal{B} G$. Secondly we show that, under some assumptions, every algebraic structure on $X$ is equivalent to the structure given by some $0$-cycle. Third, we give a cohomological criterion for checking the existence of $G$-structures not defined over the field of moduli. Fourth, we identify geometric conditions about the action of $G$ on $X$ which ensure that every $G$-structure is defined over the field of moduli

On the section conjecture over fields of finite type

Assume that the section conjecture holds over number fields. We prove then that it holds for a broad class of curves over finitely generated extensions of $\mathbb{Q}$. This class contains a non-empty open subset of any smooth curve, and all hyperbolic ramified coverings of curves of genus at least $1$ defined over number fields. Our method also gives an independent proof of the recent result by Sa\"idi and Tyler of the fact that the birational section conjecture over number fields implies it over finitely generated extensions of $\mathbb{Q}$

On the section conjecture and Brauer–Severi varieties

J. Stix proved that a curve of positive genus over 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 Rk/QX 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 cork/Q([P])∈Br(Q) is non-trivial, then X satisfies the section conjecture

The arithmetic of tame quotient singularities in dimension 22

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 $\mathrm{R}$. 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 $\mathrm{R}$ in dimension $2$; in particular, we show that every non-cyclic tame quotient singularity in dimension $2$ is of type $\mathrm{R}$, and most cyclic singularities are of type $\mathrm{R}$ too

Some implications between Grothendieck's anabelian conjectures

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 subset of 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

The field of moduli of a divisor on a rational curve

Let $k$ be a field with algebraic closure $\bar{k}$ and $D \subset \mathbb{P}^{1}_{\bar{k}}$ a reduced, effective divisor of degree $n \ge 3$, write $k(D)$ for the field of moduli of $D$. A. Marinatto proved that when $n$ is odd, or $n = 4$, $D$ descends to a divisor on $\mathbb{P}^{1}_{k(D)}$. We analyze completely the problem of when $D$ descends to a divisor on a smooth, projective curve of genus $0$ on $k(D)$, possibly with no rational points. In particular, we study the remaining cases $n \ge 6$ even, and we obtain conceptual proofs of Marinatto's results and of a theorem by B. Huggins about the field of moduli of hyperelliptic curves

An arithmetic valuative criterion for proper maps of tame algebraic stacks

The valuative criterion for proper maps of schemes has many applications in arithmetic, e.g. specializing $\mathbb{Q}_{p}$-points to $\mathbb{F}_{p}$-points. For algebraic stacks, the usual valuative criterion for proper maps is ill-suited for these kind of arguments, since it only gives a specialization point defined over an extension of the residue field, e.g. a $\mathbb{Q}_{p}$-point will specialize to an $\mathbb{F}_{p^{n}}$-point for some $n$. We give a new valuative criterion for proper maps of tame stacks which solves this problem and is well-suited for arithmetic applications. As a consequence, we prove that the Lang-Nishimura theorem holds for tame stacks