# Fano varieties of cubic threefolds

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.

• The polarized Albanese of such a surface equals the polarized intermediate Jacobian of the cubic threefold. There is a Torelli theorem for cubic threefolds, hence there is also a Torelli theorem for these surfaces. These polarized Abelian varieties are Pryms, and they can be characterized via the singularity at the origin of the associated theta divisor. Commented Oct 27, 2014 at 12:01
• @Jason: Nice idea! If you would like to put this as an answer, I will accept it. Commented Oct 27, 2014 at 12:44

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.

• Does every non-hyperelliptic curve of genus $4$ arise this way? Commented Oct 25, 2014 at 17:14
• Yes. A nodal cubic threefold can be written as $F_3(x_1,x_2,x_3,x_4)+x_0F_2(x_1,x_2,x_3,x_4)$ where $\deg F_i=i$. The corresponding genus 4 curve is $F_2=F_3=0$ in $\mathbb{P}^3$. Commented Oct 25, 2014 at 18:37
• Nice! (black space) Commented Oct 25, 2014 at 19:00
• the original reference was Clemens-Griffiths. Commented Oct 27, 2014 at 5:49
• A minor point: there are nonhyperelliptic genus-4 curves with a unique $g^{1}_{3}.$ Commented Oct 27, 2014 at 15:20