explained by the étale Brauer–Manin obstruction.

Let $f(x,y)$ and $Z^f$ be as in the previous section. The scheme $Z^f$ is the disjoint union of $\operatorname{Spec}(K_i)$ for $i=1,\ldots,n$, where $K_i$ is a finite extension of $k$. We assume $d=\deg(f)\geq 5$. (For $k={\mathbb{Q}}$ one can take $f(x,y)$ of degree $6$ as in (\[e1\]), and in general degree $8$ will suffice.) We choose field extensions $L/k$ and $F/k$ such that $L\otimes_k K_i$ and $F\otimes_kK_i$ are fields for all $i=1,\ldots, n$, and, moreover, $[L:k]=d/2-1$, $[F:k]=d/2$ if $d$ is even, and $[L:k]=(d-1)/2$, $[F:k]=(d+1)/2$ if $d$ is odd. Fix an embedding $$\operatorname{Spec}(L)\coprod\operatorname{Spec}(F)\hookrightarrow{\mathbb{P}}^1_k$$ and let $D$ be the following curve in ${\mathbb{P}}^1_k\times {\mathbb{P}}^1_k$: $$D=\big(Z^f\times {\mathbb{P}}^1_k\big)\cup \big({\mathbb{P}}^1_k\times \operatorname{Spec}(L)\big)\cup
\big({\mathbb{P}}^1_k\times\operatorname{Spec}(F)\big).$$ This is a bipartite conical curve without $k$-points, see Definition \[bi\]. The class of $D$ in $\operatorname{Pic}({\mathbb{P}}^1_k\times {\mathbb{P}}^1_k)$ is $(d,d-1)$ or $(d,d)$ depending on the parity of $d$.

The natural map $\operatorname{Br}(k)\to \operatorname{Br}(D)$ is an isomorphism.

Since $D(L)\not=\emptyset$, the natural map $\operatorname{Br}(L)\to \operatorname{Br}(D\times_k L)$ has a retraction, and so is injective. The composition of the restriction $\operatorname{Br}(k)\to\operatorname{Br}(L)$ and the corestriction $\operatorname{Br}(L)\to\operatorname{Br}(k)$ is the multiplication by $[L:k]$, hence for any $x$ in the kernel of the natural map $\operatorname{Br}(k)\to\operatorname{Br}(D)$ we have $[L:k]x=0$. Similarly, $D(F)\not=\emptyset$ implies that $[F:k]x=0$. But $[F:k]=[L:k]+1$, therefore 