It is known that in general Globally Torelli does not imply the local Torelli theorem, see Is the Torelli map an immersion? Globally Torelli means that the period map $\mathcal{P}$ is injective and Local Torelli means that the differential of period map $d\mathcal{P}$ is injective.
Now let $Y:=Y_d$ be a smooth index 2 prime Fano threefold of degree $d$. It is known that for $d\geq 3$, Global Torelli holds for $Y_d$, i.e, the intermediate Jacobian $J(Y_d)$ as a p.p.a.v. determines $Y_d$ uniquely up to isomorphism. One can show that $d\mathcal{P}: H^1(Y,T_Y)\rightarrow \mathrm{Hom}(H^{1,2}(Y),H^{2,1}(Y))$ is injective via a case by case analysis. What about $d=2?$ It seems that there is a generic Torelli theorem for $Y_2$, in this case is there any result on the injectivity of $d\mathcal{P}$? What about $d=1?$ I am not sure if there is Torelli theorem for $Y_1$.
Now, let $X_{2g-2}$ be smooth index one prime Fano threefold of genus $g$. In most cases, we do not have Global Torelli theorem. But for say $X_{12}, g=7$, the global Torelli holds. And it is also easy to check that $d\mathcal{P}$ becomes a map $d\mathcal{P}_C: H^1(C,T_C)\rightarrow\mathrm{Hom}(H^{1,0}(C), H^{0,1}(C))$, where $C$ is a genus 7 curve associated to $X_{12}$. In this case, $d\mathcal{P}_C=d\mathcal{P}$ is injective since I think $C$ is non-hyperelliptic curve.
Now, I guess Global Torelli holds for $X_2(g=2), X_4(g=3), X_8(g=5)$, but I can not find the proper reference. Is there any reference talking about whether Global Torelli holds in these three cases? If it does hold, what about injectivity of $d\mathcal{P}$? Is there any reference?
