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.

sighI'm not saying that; what I was trying imply (crudely) is that the construction seemed to be very 'variety-theoretic', and not 'scheme-theoretic', in that you are defining a map on closed points and calling it a day. I know this construction works in scheme-y generality, but to even think about it over things like finite fields, there are better ways to think about the construction. I was certainly not trying to impune arithmetic algebraic geometry. – Greg Muller Nov 8 '09 at 15:30