Hello, I apologize in advance if this question is misguided somehow, since my algebraic geometry is pretty shaky. 


I am wondering if there is a way to understand all the conics in a generic quadric line complex $X$. Remember that $X$ is defined as the intersection of the image of the complex Grassmanian $G(2,4)$ in $P^5$ under the Plucker embedding, with a quadric hypersurface in $P^5$. Since $G(2,4)\subset P^5$ is itself a quadric, $X$ may also be viewed as a complete intersection of two quadrics in $P^5$. By "conic" I mean rational curve of degree 2.


From looking at Griffiths and Harris, I see that two sources of conics in $X$ come from viewing $X\subset G(2,4) = $ {lines in $P^3$}, and considering the subspaces $\sigma(p),\sigma(h)\subset X$, where $p$ is a point in $P^3$, $h$ is a 2-plane in $P^3$, $\sigma(p)$ is the subspace of $X$ consisting of lines in $P^3$ passing through $p$, and $\sigma(h)$ is the the subspace of $X$ consisting of lines in $P^3$ contained in $h$. I may have made a mistake, but it seems to me that the former should contribute a 3 dimensional family of conics, and the latter a 4 dimensional family of conics, in $X$. I believe 4 is the expected dimension, and I think one can show that the expected dimension is achieved in this case (here I mean complex dimensions). 


Therefore, my question is: do these account for all the conics in $X$, and if so, do they fit together into a moduli space in some way? (And is there a nice way to write down the actual curves, given a nice choice of quadrics? I can try to work this out myself). 


If not, is there some way to write down all the conics?


I assume this is a classical subject, and would be delighted with a reference to more information (I suppose it's even possible that I overlooked something in Griffiths and Harris, which someone more familiar with the subject might be able to let me know). 


Thanks!



Conics in the quadric line complex
One interpretation of the quadric line complex $X$ is as the moduli space of stable, rank $2$, degree $1$ vector bundles on a genus $2$ curve $C$ (whose associated Kummer is the Kummer in G&H), cf. Newstead, "Topological Properties of Some Spaces of Stable Bundles". From this point of view, Ana-Maria Castravet has described the spaces of rational curves in $X$ of every degree (not just degree $2$), cf. "Rational families of vector bundles on curves".


