55 research outputs found

### 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 $2$

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

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