Constant Brauer groups on higher-degree affine fibers

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Status and scope

The proposition below is proved, with the relevant passages in the supplied sources checked after their digests. It rules out an ordinary Brauer--Manin obstruction on each full affine fixed-core fiber as a new mechanism for higher-exponent gaps. It is not a local-global principle and supplies no rational or integral point.

The calculation

Proposition. Let $r\ge5$, $d_1,\ldots,d_r,n\in\mathbb Q^\times$, and

$$U:\quad \sum_{i=1}^r d_i x_i^r=n.$$

The natural map $\operatorname{Br}(\mathbb Q)\to \operatorname{Br}(U)$ is an isomorphism.

Proof. Compactify to

$$X:\quad \sum_i d_i X_i^r=nZ^r\subset\mathbb P^r,\qquad D=X\cap\{Z=0\}.$$

Since all displayed coefficients are nonzero and the characteristic is zero, $X$ and $D$ are geometrically smooth hypersurfaces. Their dimensions are respectively $r-1\ge4$ and $r-2\ge3$; in particular they are geometrically integral.

Write bars for base extension to $\overline{\mathbb Q}$. Colliot-Thélène--Skorobogatov, Corollary 4.4.5 and its proof give

$$\operatorname{Br}(\bar X)=0,\qquad \operatorname{Pic}(\bar X)=\mathbb Z[H],\qquad \operatorname{Pic}(\bar D)=\mathbb Z.$$

The Picard assertion uses the restriction isomorphism for smooth complete intersections of dimension at least three stated in that proof.

For each $m\ge1$, the Kummer sequence on $\bar D$ gives $H^1_{\mathrm{\acute et}}(\bar D,\mu_m)=0$: its global units are the divisible group $\overline{\mathbb Q}^{\times}$ and its Picard group has no torsion. Identifying $\mu_m$ with the constant group $\mathbb Z/m$ over the algebraic closure and taking direct limits gives $H^1_{\mathrm{\acute et}}(\bar D,\mathbb Q_\ell/\mathbb Z_\ell)=0$ for every prime $\ell$.

Apply the purity sequence, Theorem 3.7.1 to $\bar U=\bar X\setminus\bar D$. It gives

$$0\longrightarrow\operatorname{Br}(\bar X)\{\ell\} \longrightarrow\operatorname{Br}(\bar U)\{\ell\} \longrightarrow H^1(\bar D,\mathbb Q_\ell/\mathbb Z_\ell).$$

Both outside groups vanish. The Brauer group of a smooth integral variety is torsion (also recorded in the supplied chapter), so $\operatorname{Br}(\bar U)=0$.

Since $D\sim H$, divisor localization gives

$$\operatorname{Pic}(\bar U) =\operatorname{Pic}(\bar X)/\mathbb Z[D]=0.$$

Also $\bar U$ has only constant units: the divisor on $\bar X$ of such a unit must be $k\bar D$. Its class is $kH$, which is zero only for $k=0$. The resulting rational function has no zeros or poles on the projective variety $\bar X$ and hence is constant.

We can therefore apply Proposition 4.3.2, sequence (4.9) to $U$. With $\operatorname{Pic}(\bar U)=0$ it identifies $\operatorname{Br}(\mathbb Q)$ with $\operatorname{Br}_1(U)$. The vanishing of $\operatorname{Br}(\bar U)$ makes $\operatorname{Br}_1(U)=\operatorname{Br}(U)$, proving the claim. $\square$

What this rules out, and what it does not

Constant Brauer classes evaluate independently of the chosen local point, and their local invariants sum to zero by reciprocity. Thus an existing adelic collection of local points on $U$ cannot be excluded by its ordinary Brauer--Manin pairing. The same applies after requiring integrality at the relevant places or positivity at the real place: constant evaluations do not detect those choices.

This does not assert that every locally soluble fiber has an integral point, let alone a positive point with roots of size $O(n^{1/r})$. It also does not apply to arbitrary smaller open subsets obtained by removing additional divisors. The attempted coordinate and binary-subsum punctures yielded no new bound over the remaining root variables. Consequently no target exclusion, void inequality, or counting estimate was deduced.

The proposition concerns $n\ne0$, all coefficients nonzero, and $r\ge5$. No assertion about singular zero fibers or lower-degree affine fibers is made here. The infinite canonical-core family still requires its separate tail estimates.