Let $X\_1,X\_2\subset\mathbb{P}^{n+1}$ be two smooth complex cubic hypesurfaces, then I know the following Torelli theorems:


[$n=1$] In this case, $X\_1\cong X\_2$ if and only if there exists a Hodge isometry $H^1(X\_1,\mathbb{Z})\cong H^1(X\_2,\mathbb{Z})$.


[$n=3$] In this case, $X\_1\cong X\_2$ if and only if there exists a Hodge isometry $H^3(X\_1,\mathbb{Z})\cong H^3(X\_2,\mathbb{Z})$.


[$n\geq 3$] In this case, $X\_1\cong X\_2$ if and only if there exists an isomorphism of their Fano varieties of lines $F(X\_1)\cong F(X\_2)$. This isomorphism is asked to be polarized when $n=4$.


Do we know a similar result for cubic surfaces? Explicitly, I would like to know if there are some invariant that can determine a smooth complex cubic surface.



Torelli theorem for smooth complex cubic surfaces?
The Hodge structures of cubic surfaces do not allow to distinguish them. But there are some ways around. For instance, given a cubic surface $X \subset \mathbb{P}^3$, you can associate with it a cyclic covering $Y(X) \to \mathbb{P}^3$ of degree 3 ramified over $X$ (thus, $Y(X)$ is a cubic threefold) and use the Hodge structure of $Y(X)$.


