TITLE: Fano varieties of cubic threefolds
QUESTION [5 upvotes]: Let $X$ be a smooth cubic threefold over $\mathbb{C}$. Let $F(X)$ denote the Fano variety of lines in $X$, which is a smooth surface of general type.


Is this class of surfaces distingushed amoungst surfaces of general type?


I appreciate that this question is slightly vague. What I am looking for is something like a geometrical characterisation of such surfaces, say in terms of certain geometric invariants (i.e. hodge numbers, chern classes,...?). If this is too naive to hope for, then perhaps they have distinguished moduli, for example by forming a connected component of the moduli of surfaces of general type.

REPLY [5 votes]: So, not what you're looking for, but closely related.  A nodal cubic threefold's Fano surface is isomorphic to $S^2B/\sim$ where $B\in\mathcal{M}_4$ is nonhyperelliptic, and $\sim$ is constructed as follows:
Let $L,L'$ be the two $g^1_3$'s on $B$, then they give maps $B\to S^2B$ by $r\mapsto p+q$ if $p+q+r$ is in the $g^1_3$.  Then identify these two copies of $B$ in the natural way.
So, not numerical invariants, but $F(X)$ deforms to this singular surface, so you can read off quite a few pieces of data from it.
A reference to this is Donagi's "Fibers of the Prym Map" section 5.4 but I'm sure it appeared somewhere else earlier, I just don't have that reference on hand.