Question

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!

Answer 1

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".

Answer 2

Comment: The family of hyperplanes in $\mathbb P^3$ is $3$-dimensional, because it is isomorphic to another copy of $\mathbb P^3$.

This does not give you all the conics. The reason is simple. Every conic you constructed lies on a plane entirely contained within $G(2,4)$, so it usually lies on a plane not entirely contained in the other quadric hypersurface. But the other quadric hypersurface also contains planes! These generically fail to lie entirely in $G(2,4)$, meaning that the intersection of that plane with $G(2,4)$ is a conic in $X$ that cannot be a conic of the two types you gave.

The full family of conics consists of the intersection of $X$ with all planes that lie in the vanishing set of some nontrivial linear combination of the definition polynomials of $G(2,4)$ and the other quadric hypersurface. Only for the planes lying in the vanishing set of the defining polynomial of $G(2,4)$ can you have a nice geometric description.