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…)

Last time, I described the construction which assigns to every compact ${G}$-space ${X}$ (for ${G}$ a compact Lie group) the equivariant K-group ${K_G(X)}$. We saw that this was a functor from the (equivariant) homotopy category to commutative rings, using more or less the same arguments as in ordinary homotopy theory, only with small alterations.

The purpose of this post is to describe more of Segal’s paper. Actually, I won’t be covering any legitimate K-theory in this post; that’ll have to wait for a third. I’ll mostly be describing various classical constructions for vector bundles in the equivariant setting.

In the classical theory of (ordinary) vector bundles on compact spaces, a basic result is the Serre-Swan theorem, which identifies the category ${\mathrm{Vect}(X)}$ of (complex) vector bundles with the category of projective modules over the ring ${C(X; \mathbb{C})}$ of complex-valued continuous functions on ${X}$. This is essentially a reflection of the fact that any vector bundle on ${X}$, say ${E \rightarrow X}$, can be obtained as a retract of some trivial bundle ${\mathbb{C}^n \times X \rightarrow X}$. Taking retracts corresponds to choosing idempotents in the ring of ${n}$-by-${n}$ matrices in ${C(X; \mathbb{C})}$, and this description via idempotents applies as well to projective modules over ${C(X; \mathbb{C})}$ (or, in fact, any commutative ring).

The crucial statement here, that any vector bundle is a retract of a trivial one, fails in the equivariant case, simply because a vector bundle on which ${G}$ acts nontrivially can’t be a retract of a vector bundle with trivial action. But we have something reasonably close to it.

Definition 1 Given a ${G}$-representation ${V}$ and a ${G}$-space ${X}$, we can form a vector bundle ${V \times X \rightarrow X}$, which his naturally ${G}$-equivariant.

This bundle is, equivalently, formed by taking the equivariant map ${X \rightarrow \ast}$. ${G}$-vector bundles on ${\ast}$ are identified with ${G}$-representations, so we just have to pull back.

Anyway, the claim is:

Theorem 2 (Equivariant Serre-Swan (Segal)) Any ${G}$-vector bundle ${E \rightarrow X}$ is a direct summand of a bundle ${V \times X \rightarrow X}$ for some ${G}$-representation ${V}$. (more…)