The following should be pretty standard for any algebraic geometer.
Let $X$ be a compact complex variety, and let $L$ be a line bundle on $X$. We say $L$ is 'generated by global sections' if for every point $p$ in $X$, there is a global section of $L$ which doesn't vanish. If this is true, then $L$ determines a map to a projective space in the following way. The global sections of $L$ are finite dimensional, so choose a basis $(a_i)$. Then send a point $p$ in $X$ to the projective point
$$
[a_1(p):a_2(p):…:a_n(p)]
$$
It should be noted that $a_i(p)$ is only a point in the fiber of $L$ over $p$, and not a complex number. By choosing an isomorphism from the fiber over $p$ to $\mathbb{C}$, the $a_i(p)$ can be identified with complex numbers. The ambiguity introduced in choosing this isomorphism dissappears when taking the projective coordinates.
This constructions is remarkably ad hoc for something that ends up being foundational in algebraic geometry. It requires the variety be over $\mathbb{C}$, and requires that some inconsequential choices be made enroute.
My question is, what are nicer, more intrinsically algebraic ways to construct this map? Three ways this construction could be nicer:
- It could work over other fields, or possibly even over $\mathbb{Z}$ (though then its less clear what a line bundle should be).
- It could give a good intuitive justification of why this map is a natural and powerful thing to look at.
- It could lend itself to generalizations in different directions. For instance, a rank n vector bundle V with 'enough global sections' (in some sense) should determine a map from X to $Hom_{\mathbb{C}}(\Gamma(V),\mathbb{C}^n)//GL(n)$ (where this is a GIT quotient).
Aside. Is there even a good name for this construction? The "map to projective space determined by a line bundle" is a bit long-winded.
Best Answer
This is one of the most fundamental questions possible. Hence although it is old and well answered, I venture to add something, hoping to make it seem as transparent as possible.
I would suggest the way to understand this construction is to look at it backwards. I.e. by its very definition, projective space carries a tautological line bundle, whose dual bundle has as sections the linear coordinates. These sections have no common zeroes because the hyperplanes have no common points. Hence any subvariety of projective space also has by restriction a line bundle whose sections have no common zeroes.
Moreover a point of projective space is determined by the set of hyperplanes through it, so any subvariety is determined by the restricted line bundle, since each point is recovered from the set of sections vanishing on it. Moreover the projective space it self is dual to the space of hyperplanes, hence to the space of global sections of the bundle. Hence the bundle on the subvariety determines both the ambient projective space and the embedding.
Now one sees immediately that one can imitate this to give a map, not necessarily an embedding, from any variety with a line bundle whose sections have no common zeroes, to the dual projective space of its space of sections, by sending each point to the subset of its sections vanishing at that point, as Anton said.
In a nutshell, since projective space has a line bundle whose sections have no common zeroes, and line bundles and sections pull back under maps, having such a line bundle is a necessary condition for a map to projective space. Then one asks whether it is also sufficient, and it is, as above.
It is easy also to recover the properties that determine whether the map is an embedding.
Looked at this way, there is nothing mysterious about this construction - it is in fact the defining property of projective space.