?? bump .... section on riemann-roch ...

??? "linear equivalence of divisors as interesting only in complete case" ?????? .....

?? "adele" / "repartition" .... "geometric" case .... ???

?? :

Example 12.22. This example shows that the Hurwitz genus formula fails
when there is wild ramification. Suppose that the characteristic of k is p > 0.

... ??? ....

Conversely, given a linear system L, one may try to construct a projective
embedding of X (or at least a rational map into projective space) which realizes
L as the linear system cut out by hyperplanes. Sometimes this cannot be done,
since a linear system may be empty. However, if L is "large enough," it may
always be done—see Proposition 13.8 for an illustration of how this may be
accomplished. Hence divisors and linear systems are at the heart of the problem
of constructing embeddings of X into projective space. It becomes essential to
understand the projective space |P| more precisely, or equivalently, the vector
space L(D) = {/ £ F\(f) > -D}. When X is a curve, the Riemann-Roch
Theorem is a formula for its dimension.

????? ....

Let us mention that the concepts of divisors and linear equivalence is not
special to curves. Let X be a variety which is nonsingular in codimension
one, and let F be its function field. A Weil divisor on X is defined to be an
element of the free abelian group generated by the irreducible subvarieties of
codimension one.

???? .... hmmmm .... "hartog ..." .... ???