Let k be a field of characteristic $0$.



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> > Let $X$ be a variety over k which is isomorphic to a smooth cubic threefold over $\bar{k}$. Then is $X$ isomorphic to a smooth cubic threefold over $k$?
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For motivation, let's consider some other cases. 


* For cubic curves, the analogous question has a a negative answer in general, as elliptic curves can have quite complicated twists.
* For cubic surfaces however the answer is yes. This is because cubic surfaces are embedded by the anticanonical divisor, which is already defined over $k$.


This latter argument does not work however for cubic threefolds, as for a cubic threefold, the hyperplane section is half the anticanonical divisor. So the question is whether the anticanonical divisor is always divisible by $2$ in the Picard group of $X$.


Note that the Hochschild-Serre spectral sequence yields the exact sequence
$$0 \to \mathrm{Pic}(X) \to \mathrm{Pic}(X\_\bar{k})^{\mathrm{Gal}(\bar{k}/k)}=\mathbb{Z} \to \mathrm{Br}(k),$$
so the obstruction to $X$ being isomorphic to a cubic threefold over $k$ is given by an element of $\mathrm{Br}(k)[2]$.



Twists of cubic threefolds
As pointed out by Noam Elkies, my comment above can be turned into a positive answer to the question.


In the exact sequence



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> $\DeclareMathOperator{\Pic}{Pic}\DeclareMathOperator{\Br}{Br}0 \to \Pic X \to (\Pic \bar{X})^{G\_k} \to \Br k$,
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the arrow $\delta \colon (\Pic \bar{X})^{G\_k} \to \Br k$ sends the class of an effective Cartier divisor $D$ to the class of the Severi--Brauer variety whose $\bar{k}$-points are isomorphic to the projectivisation of $\mathrm{H}^0(\bar{X},\mathcal{O}(-D))$. See Serre, *Local Fields*, X.6 for a passing mention of this fact, and Grothendieck, *Le groupe de Brauer, III*, 5.4 for more details (though he doesn't actually prove it, saying "on (plus précisément, J. Giraud) vérifie que la classe de cet élément dans $\Br(Y)$ est bien $\delta(\xi)$.")


In the particular example here, $\Pic X$ is of index $2$ in $(\Pic \bar{X})^{G\_k}$, so the image of $\delta$ is killed by $2$. On the other hand, taking $D=-K\_X/2$, the vector space $\mathrm{H}^0(\bar{X},\mathcal{O}(-D))$ has dimension $5$, since we are assuming that $\bar{X}$ is isomorphic to a cubic threefold, on which $D$ is the class of a hyperplane section. It follows (for example Serre, *loc. cit.*) that $\delta(D) \in \Br k$ is killed by $5$. So $\delta(D)=0$, and $D$ lies in $\Pic X$.


