On the section conjecture and Brauer-Severi varieties

J. Stix proved that a curve of positive genus over $\mathbb{Q}$ 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 $R_{k/\mathbb{Q}}X$ 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 $\operatorname{cor}_{k/\mathbb{Q}}([P])\in\operatorname{Br}(\mathbb{Q})$ is non-trivial, then $X$ satisfies the section conjecture.

section conjecture holds for X. Equivalently, if π 1 (X) → Gal(k/k) admits a section, then the map Br(Q) → Br(R k/Q X) is injective.
As a consequence, we get the following corollaries.
Corollary 2. Let X be a smooth projective curve of positive genus over a number field k and P a Brauer-Severi variety such that the corestriction cor k/Q [P] ∈ Br(Q) is non-trivial. If there exists a morphism X → P, then the section conjecture holds for X. Equivalently, if π 1 (X) → Gal(k/k) admits a section, the kernel of Br(k) → Br(X) is contained in the kernel of cor k/Q : Br(k) → Br(Q).
Corollary 3. Let k be a number field and P a Brauer-Severi variety over Q with [k : Q][P] = 0 ∈ Br(Q). If X is a smooth projective curve over Q of positive genus with a morphism X → P, then the section conjecture holds for the base change X k .
Our argument is analogous to Stix's one and we rely heavily on his results. Our contribution consists essentially of two things: we realized that such a generalization was possible and we overcame the lack, for higher dimensional varieties, of a sufficiently strong analogue of Lichtenbaum's theorem about the period and index of a curve over a p-adic field, which is an essential ingredient of Stix's proof.
We mention that it is possible to prove Corollary 2 (and thus Corollary 3) analogously to Stix' theorem over Q [Sti10a, Corollary 18] without using Weil's restriction of scalars. The proof is basically the same plus the observation that, if α ∈ Br(k) is a Brauer class over the number field k, the Hasse invariant of cor k/Q (α) at p is the sum of the Hasse invariants of α at places over p.

WEIL RESTRICTION AND THE SECTION CONJECTURE
The behaviour of the étale fundamental group and the section conjecture with respect to the Weil restriction of scalars has been studied by J. Stix in [Sti10b]. Let k/h be a finite, separable extension of fields and X is a geometrically connected variety over k. Assume either that X is proper or that k has characteristic 0. Stix describes explicitly the étale fundamental group of R k/h X in terms of the one of X, and uses this description to show that the section conjecture holds for X if and only if it holds for R k/h X. We give here an alternative treatment based on étale fundamental gerbes.
Recall that A. Vistoli and N. Borne have introduced the étale fundamental gerbe X → Π X/k of a geometrically connected scheme, see [BV15, Section 9] and [Bre21,Appendix]. The set of Galois sections of the étale fundamental group is in natural bijection with the isomorphism classes of Π X/k (k). We show that the étale fundamental gerbe and the Weil restriction commute.
Proposition 4. Let k/h be a finite separable extension of fields, and X a geometrically connected quasiprojective variety over k. Assume either that X is proper or that char k = 0.
Then R k/h X is geometrically connected and the natural morphism R k/h X → R k/h Π X/k induces a natural isomorphism where the product runs over the h-linear embeddings σ : k ⊂h, see [Wei82, Theorem 1.3.2] (Weil's original work deals only with varieties, but his proof easily generalizes to any fibered category). In particular R k/h X is geometrically connected and R k/h Π X/k is a pro-finite étale gerbe, thus the morphism R k/h X → R k/h Π X/k induces a natural morphism The base change of ϕ toh is an isomorphism since both terms are naturally isomorphic to ∏ σ Π σ * X/k , thus ϕ is an isomorphism too.
Corollary 5. [Sti10b, Theorem 2, Theorem 3] The set of isomorphism classes of Galois sections of π 1 (X) over k is in natural bijection with the one of π 1 (R k/h X) over h.
Theorem 4] Let X be a smooth, projective curve of genus g ≥ 2 over a number field k. The section conjecture holds for X if and only if it holds for R k/Q X.
Corollary 7. Let X be a smooth, projective curve of positive genus over a number field k and s ∈ Π X/k (k) a Galois section. If Y → X is an étale neighbourhood of s, then R k/Q Y → R k/Q X is an étale neighbourhood of s Q . The étale neighbourhoods of this form are cofinal in the system of all étale neighbourhoods of s Q .

MORPHISMS TO BRAUER-SEVERI VARIETIES
If X is a scheme over k, denote by Br(X/k) the kernel of Br(k) → Br(X). If X is a regular variety, the restriction map Br(X) → Br(k(X)) is injective [Mil80, Corollary IV.2.6] and thus Br(X/k) = Br(k(X)/k). In particular, a Brauer class [P] ∈ Br(k) of a Brauer-Severi variety P is in Br(X/k) if and only if there exists a rational map X P. If X is a smooth, projective variety, the Leray spectral sequence in étale cohomology for the map X → Spec k gives a short exact sequence where Pic X is the Picard scheme of X and Pic(X) = H 1 (X, G m ) is the Picard group. Let us call β the homomorphism Pic X (k) → Br(X/k).
Lemma 8. Let X be a smooth, projective variety over a field k of characteristic 0, s ∈ Π X/k (k) a section, b ∈ Br(X/k) a Brauer class split by X. Assume that the second étale homotopy group of X¯k is trivial. For every positive integer n, there exists an étale neighbourhood f : Y → X of s such that f * b ∈ n Br(Y/k).
Proof. Let L ∈ Pic X (k) be such that β(L) = b ∈ Br(X/k), and L¯k ∈ Pic(X¯k) the associated line bundle over X¯k. We have an exact sequence LetÉt(X¯k) be the étale homotopy type of X¯k and cosk 3 (Ét(X¯k)) its third coskeleton, since π´e t 2 (X¯k) is trivial we have cosk 3 (Ét(X¯k)) = K(π´e t 1 (X¯k), 1). Therefore, we have H 2 (X¯k, µ n ) = H 2 (Ét(X¯k), µ n ) = H 2 (cosk 3 (Ét(X¯k)), µ n ) = H 2 (π´e t 1 (X¯k), µ n ), see [AM69, Corollary 9.3] for the first equality. Since the base change tok of the étale neighbourhoods of s are cofinal in all finite étale covers of X¯k, there exists an étale neighbourhood g : X → X of s such that g * k δ(L¯k) = 0 and thus g * k L¯k ∈ Pic(X k ) is divisible by n. Choose M ∈ Pic(X k ) such that M n = g * k L¯k. Since the Picard scheme of X is locally of finite type, the residue field of Speck M − → Pic X is finite over k and thus the Galois orbit of M is finite. If σ ∈ Gal(k/k) is an element, since g * k L¯k is Galois invariant then M ⊗ σM −1 is n-torsion, and thus it comes from H 1 (X k , µ n ) = Hom(π´e t 1 (X k ), µ n ). It follows that there exists an étale neighbourhood h : Y → X of s such that h * k M ∈ Pic(Y¯k) is Galois-invariant and thus descends to an element N ∈ Pic Y (k).
Let f : Y → X be the composition, we have N n = f * L and hence nβ( Lemma 9. Let X be a variety over C. Assume that π top 2 (X an ) is trivial and that π top 1 (X an ) is good in the sense of Serre. Then π´e t 2 (X) is trivial.

Recall that a group
Proof. Write π i = π´e t i (X). The hypothesis implies that the natural homomorphism H 2 (π 1 , M) → H 2 (X, M) is bijective for every finite π 1 -module M.
Assume by contradiction that π 2 is not trivial, then there exists a finite homotopy type F with π n (F) = 0 for n ≥ 3 and a mapÉt(X) → F such that π 2 → π 2 (F) is non-trivial. Up to passing to finite étale coverings of X and F, we may assume that π 1 (F) is trivial and hence F = K(M, 2) for some finite abelian group M (the fundamental group of the covering of X is still good thanks to [GJZ08, Lemma 3.2]).
We thus have a map X → K(M, 2) inducing a non-trivial homomorphism π 2 → M. This defines a cohomology class α ∈ H 2 (X, M) not in the image of H 2 (π 1 , M) → H 2 (X, M), and this is absurd.
Lemma 10. Let k/h be a finite separable extension and P/k a Brauer-Severi variety. There exists a Brauer-Severi variety Q/h with [Q] = cor k/h ([P]) ∈ Br(h) and a closed embedding R k/h P → Q.
Proof. Leth/h be a separable closure, then where the product runs over h-linear embeddings σ : k →h. The Galois group Gal(h/h) permutes the factors and the stabilizer Gal(h/σk) of σ * P acts on it. Note that, even though σ * P is a projective space overh, the action of Gal(h/σk) is non-standard.
Using the fact that summation in the Brauer group can be computed using the Segre embedding of Brauer-Severi varieties [Art82,§4] and the fact that the corestriction homomorphism is the derived augmentation homomorphism, it is easy to show that that [Q] = cor k/h ([P]). Moreover, the Segre embedding S descends to a closed embedding R k/h P → Q since it is Gal(h/h)-equivariant.

PROOF OF THE MAIN THEOREM
Let us now prove Theorem 1. Let X be a smooth projective curve over a number field k such that R k/Q X admits a rational map to a non-trivial Brauer-Severi variety, we want to show that Π X/k (k) is empty. Assume by contradiction that there exists a section s ∈ Π X/k (k) and let b ∈ Br(R k/Q X/Q) be a non-trivial Brauer class. Let s Q ∈ Π R k/Q X/Q (Q) be the associated section.
Since (R k/Q X)¯k is a product of curves and the fundamental group of a curve overk is good in the sense of Serre [GJZ08, Proposition 3.6], then Lemma 9 implies that π´e t 2 ((R k/Q X)¯k) is trivial and we may thus apply Lemma 8 to R k/Q X and s Q . If we apply Lemma 8 together with Corollary 7, for every N > 0 we may find an étale neighbourhood X N → X of s and a Brauer class b N ∈ Br(R k/Q X N /Q) such that Nb N = b ∈ Br(Q) is non-trivial.
Let l/k/Q be a Galois closure. Up to replacing X with X 2[l:Q] and b with b 2[l:Q] , we may assume that 2[l : Q]b ∈ Br(Q) is non-trivial.
Fix p a prime number, let us show that the order of the Brauer class 2[l : Q]b Q p is a power of p. Let L be the completion of l at some place over p, we have that L/Q p is a Galois extension such that [L : Q p ] divides [l : Q], it is enough to show that the order of [L : Q p ]b Q p is a power of p. Let Σ be the set of embeddings k → L, we have The section s ∈ Π X/k (k) induces a section σ * s ∈ Π σ * X/L (L) for every embedding σ : k → L. By [Sti13,Theorem 15], this implies that the index of σ * X is a power of p for every σ. Let D σ ∈ Z 0 (X σ ) be a 0-cycle whose degree is a power of p, then ⊗ σ D σ is a 0-cycle on ∏ σ σ * X whose degree is a power of p. It follows that the index of ∏ σ σ * X is a power of p, too.
Since Q(R k/Q X) splits b, there exists a Brauer-Severi variety P with [P] = b and a smooth projective variety Y/Q birational to R k/Q X with a morphism Y → P. Since the index is a birational invariant, the index of Y L is a power of p, it follows that the index of b L = [P L ] is a power of p. This implies that the order (i.e. the period) of b L ∈ Br(L) is a power of p, and finally the same holds for cor L/Q p b L = [L : We thus have that the order of 2[l : Q]b Q p is p-primary for every p, and clearly 2[l : Q]b R = 0 ∈ Br(R) = Z/2Z.
The rest of the argument is analogous to Stix's one. Let α p ∈ Q/Z be the Hasse invariant of 2[l : Q]b Q p , by the Brauer-Hasse-Noether theorem we have ∑ p α p = 0 ∈ Q/Z. Since α p is p-primary for every p, it follows that α p = 0 for every p and thus 2[l : Q]b ∈ Br(Q) is trivial, which is absurd. This concludes the proof of Theorem 1.