In the previous post, we introduced the Fano scheme of a subscheme of projective space, as the Hilbert scheme of planes of a certain dimension on that subscheme. In this post, I’d like to work out an explicit example, of the 27 lines on a smooth cubic surface in \mathbb{P}^3; as we’ll see, the Fano scheme is 27 reduced points, and the count can be made with a little calculation on the Grassmannian. Although the calculation is elementary, I found it worthwhile to work carefully through it, not only for its intrinsic interest but also as motivation for the study of intersection theory on moduli spaces in general. Once again, most of this material is from Eisenbud-Harris’s draft book 3264 and All That.

1. The normal bundle as self-intersection

Suppose {X = S} is a smooth surface, imbedded in some projective space, and consider the scheme {F_1 S} of lines in {S}.

Fix a line {L} in S. In this case, the normal sheaf {N_{S/L}} is actually a vector bundle of normal vector fields, given by the adjunction formula

\displaystyle N_{S/L} = \left(\mathcal{I}_L/\mathcal{I}_L^2\right)^{\vee} = \left(\mathcal{O}_S(-L)/\mathcal{O}_S(-2L)\right)^{\vee} = \mathcal{O}_L(L).

In particular, {N_{S/L}} is a line bundle on {L \simeq \mathbb{P}^1} and has a well-defined degree. This degree is in fact the self-intersection {L.L} of {L}, considered as a divisor on the smooth surface {S}. (more…)