We’ve now talked about vector bundles and locally free sheaves, we’re going to specify to the nicest case: rank 1. We’re generally going to ignore the distinction between the sheaf and the line bundle.

We start by noting an alternative name for locally free sheaves of rank one: invertible sheaves. The reason for this is because given an invertible sheaf $\mathscr{L}$, we can define $\mathscr{L}^\vee=\mathscr{H}om(\mathscr{L},\mathscr{O}_X)$, the sheaf hom. This turns out to be a locally free sheaf itself, and has rank one. Now, if we look at the sheaf $\mathscr{L}\otimes\mathscr{L}^\vee$ (recall that we do this by taking tensor products over open sets, and then sheafifying) we get $\mathscr{O}_X$. So in a sense, $\mathscr{L}^\vee$ is the inverse for $\mathscr{L}$, as their tensor product is not just locally free, but is in fact free.

More is true, in fact. $\mathscr{L}\otimes\mathscr{M}$ is always invertible if both factors are. Now we make note of the following, assuming $\mathscr{L}, \mathscr{M}, \mathscr{N}$ are all line bundles.

1. $(\mathscr{L}\otimes\mathscr{M})\otimes\mathscr{N}\cong \mathscr{L}\otimes(\mathscr{M}\otimes\mathscr{N})$
2. $\mathscr{O}_X\otimes\mathscr{L}\cong\mathscr{L}\otimes\mathscr{O}_X\cong\mathscr{L}$
3. $\mathscr{L}\otimes\mathscr{L}^\vee\cong \mathscr{L}^\vee\otimes \mathscr{L}\cong \mathscr{O}_X$.
4. $\mathscr{L}\otimes \mathscr{M}\cong \mathscr{M}\otimes \mathscr{L}$

These isomorphisms should look familiar: they are precisely the axioms for an abelian group! This group is called the Picard group of $X$, and denoted $\mathrm{Pic}(X)$. Because of this group structure, we will denote $\mathscr{L}^{\vee}$ by $\mathscr{L}^{-1}$. We will, however, keep the notation $\mathscr{F}^\vee$ for higher rank locally free sheaves.

The fun with the Picard group will REALLY start next week, when I introduce divisors, but there is a bit more we can do before then: like use line bundles to define maps of varieties into projective spaces.

Let $\mathscr{L}$ be a line bundle. If our variety is defined over the field $k$, then $\Gamma(X,\mathscr{L})$ will be a $k$-vector space. Now, choose a basis $s_0,\ldots,s_n\in\Gamma(X,\mathscr{L})$ (if the vector space is trivial, we call the map the empty map to the empty projective space, and so we assume positive dimension $n+1$).

So now we remember that these are not just sheaves, but the sheaves of sections of a geometric vector bundle. We fix a trivialization of the bundle, and on each open set, fix an isomorphism of the fiber with $k$. Then any global section defines a regular function on $X$. So we get a map $(s_0,\ldots,s_n)$ into $\mathbb{A}^{n+1}$. This map, however, depends on the isomorphisms chosen for the fibers, and the transition functions will multiply it by a scalar as you move from one chart to another, but we can get rid of this dependence by projectivizing. So now we have a rational map $X\dashrightarrow \mathbb{P}^n$. It’s only a rational map, because we don’t know in advance if the $s_i$ have any common zeros, and the map can’t be defined on those points. And finally, the choice of basis for $\Gamma(X,\mathscr{L})$ doesn’t matter, because different bases will give maps that differ only by a linear transformation of the projective space, so we won’t worry about that ambiguity.

So now, we have a rational map $X\dashrightarrow \mathbb{P}^n$ for any line bundle. If the map is in fact a morphism, then we call $\mathscr{L}$ very ample. A line bundle is ample if some tensor power is very ample. There are lots of other characterizations of these line bundles. some of which we’ll encounter.

That’s all for now, next time, we’ll construct a couple of specific vector bundles on nonsingular varieties.

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