Consider a smooth cubic complex hypersurface $X\subset\mathbf{P}^{n+1}$ of dimension $n\geqslant 3$. The associated Fano variety of lines $F(X)$ is a smooth variety of dimension $2n-4$. Can one recover $X$ from $F(X)$? The answer is positive for $n=3$ (due to Clemens-Griffiths, Tjurin) and $n=4$ (due to Voisin, I think). Are there any conjectures for general $n$?



Is a cubic hypersurface determined by its Fano variety of lines?
**Edit.** Regarding the Picard group, since $\rho^{-1}(F(X))$ is a $\mathbb{P}^1$-bundle over $F(X)$, the Picard group $\text{Pic}(\rho^{-1}(F(X)))$ equals $\text{Pic}(F(X))\oplus \mathbb{Z}\cdot [\pi^\*\mathcal{O}(1)]$. The fibers of the projection, $\pi:\rho^{-1}(F(X))\to X$, are complete intersections of dimension $n-3$ over $X^o$. Thus, for $n\geq 6$, by the Grothendieck-Lefschetz theorem on Picard groups from SGA 2, also $\text{Pic}(\pi^{-1}(X^o)\cap \rho^{-1}(F(X)))$ equals $\mathbb{Z}\cdot [\rho^\*\mathcal{O}(1)] \oplus \text{Pic}(X^o)$. Since the complement of $X^o$ in $X$ has codimension $\geq 2$, also $\text{Pic}(X^o)$ equals $\text{Pic}(X)$, and this equals $\mathbb{Z}\cdot[\mathcal{O}(1)|\_X]$. If $n$ equals $5$, consider the threefolds $\pi^{-1}(\ell)\cap \rho^{-1}(F(X))$, where $\ell\subset X$ is a general line. Intersecting with a general hyperplane $H$ gives a birational version of this threefold in the cubic fivefold $X$, and now I believe we can repeat the argument from the Griffiths-Harris proof of the Noether-Lefschetz theorem to conclude that $\pi^{-1}(\ell)\cap \rho^{-1}(F(X))$ has Picard rank $2$ (this involves analyzing the singularities of the birational version of the threefold). 


**Original post.** I am adding some details to my comment, specifically the proof that the restriction homomorphism is injective, $$H^0(\text{Grass}(\mathbb{P}^1,\mathbb{P}^{n+1}),\mathcal{O}(2)) \to H^0(F(X),\mathcal{O}(2)).$$ One proof uses the flag variety $$\text{Flag}\_{0,1}(\mathbb{P}^{n+1}) \subset \mathbb{P}^{n+1}\times\_{\text{Spec}\ k} \text{Grass}(\mathbb{P}^1,\mathbb{P}^{n+1}),$$ with its two projections, $$\pi:\text{Flag}\_{0,1}(\mathbb{P}^{n+1}) \to \mathbb{P}^{n+1}, \ \ \rho:\text{Flag}\_{0,1}(\mathbb{P}^{n+1}) \to \text{Grass}(\mathbb{P}^1,\mathbb{P}^{n+1}).$$ For a hypersurface $X\subset \mathbb{P}^{n+1}$, for the inverse image $\pi^{-1}(X)$, the induced morphism is dominant, $$\rho:\pi^{-1}(X)\to \text{Grass}(\mathbb{P}^1,\mathbb{P}^{n+1}),$$ and it is even flat over the open complement of the Fano scheme $F(X) = \text{Hilb}^{t+1}\_{X/k}.$ Thus, the following pullback map is injective, $$H^0(\text{Grass}(\mathbb{P}^1,\mathbb{P}^{n+1}),\mathcal{O}(2)) \to H^0(\pi^{-1}(X), \rho^\*\mathcal{O}(2)).$$ Thus, it suffices to prove that the following pullback map is injective, $$H^0(\text{Grass}(\mathbb{P}^1,\mathbb{P}^{n+1}),\mathcal{O}(2)) \to H^0(\pi^{-1}(X)\cap \rho^{-1}(F(X)),\rho^\*\mathcal{O}(2)).$$ There is a maximal open subscheme $X^o\subset X$ over which the following morphism is flat of relative dimension $\leq n-d$ (in fact dimension precisely $n-d$), $$\pi:\pi^{-1}(X)\cap \rho^{-1}(F(X)) \to X.$$ It suffices to prove injectivity of the map, $$H^0(\text{Grass}(\mathbb{P}^1,\mathbb{P}^{n+1}),\mathcal{O}(2)) \to H^0(\pi^{-1}(X^o)\cap \rho^{-1}(F(X)),\rho^\*\mathcal{O}(2)).$$ For $n\geq 3$, the Fano scheme $F(X)$ is smooth of dimension $2n-4$ and $X\_o$ is a dense open subscheme whose complement has codimension $\geq 2$ everywhere. 


There is a sequence of Cartier divisors, $$X^o\_0 = \pi^{-1}(X^o)\supseteq X^o\_1 \supseteq X^o\_2 \supseteq X^o\_3 = \pi^{-1}(X^o)\cap \rho^{-1}(F(X)),$$ where the Cartier divisor $X^o\_{e+1}$ in $X^o\_e$ is the zero scheme of a section $s\_{e+1}$ of the restriction to $X^o\_e$ of the invertible sheaf, $$L\_{3-(e+1),e+1} = \pi^\*\mathcal{O}(3-(e+1))\otimes \rho^\*\mathcal{O}(e+1).$$ For the ideal sheaf $\mathcal{I}$ of $\pi^{-1}(X^o)\cap \rho^{-1}(F(X))$ inside $\pi^{-1}(X^o)$, the pushforward locally free sheaf $$I\_2:=\pi\_\*(\mathcal{I}\otimes \rho^\*\mathcal{O}(2))$$ has an associated filtration that gives a short exact sequence, $$ 0\to J\_2 \to I\_2 \to \mathcal{O}(-(3-2))|\_{X^o}, $$ where $J\_2$ is the pushforward of the twist by $\rho^\*\mathcal{O}(2)$ of the ideal sheaf $\mathcal{J}\_1$ of $X\_1^o$. The scheme $X\_1^o$ parameterizes pointed lines that are tangent to $X$ at the marked point. The image of $X\_1^o$ under $\rho$ is a dense open in a hypersurface in the Grassmannian, and the degree of that hypersurface equals $6$ (the degree of the dual curve of a plane cubic). Since $6$ is greater than $2$, there are no global sections of $J\_2$. Also, there are no global sections of $\mathcal{O}(-1)|\_{X^o}$. Thus, there are no global sections of $I\_2$. Therefore, the only section of $\mathcal{O}(2)$ on $\text{Grass}(\mathbb{P}^1,\mathbb{P}^{n+1})$ that vanishes on $F(X)$ is the zero section.


