I'm having problems understanding one part of the proof of the Residue Theorem, on chapter 8 of Fulton's book Algebraic Curves, section 8.1 (http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf page 98) There is a lot of notation, because it came from another propositions. But the Residue Theorem is the following:
Let $C,E$ be as above. Suppose $D$ and $D'$ are effective divisors on X, with $D' = D$. Suppose $G$ is an adjoint of degree $m$ such that $div(G)=D+E+A$, for some effective divisor $A$. Then there is an adjoint $G'$ of degreem such that $div(G')=D'+ E + A$.
Proof. Let $H,H'$ be curves of the same degree such that $D+div(H)=D'+div(H')$. Then $div(GH)= div(H')+D´+E+ A \geq div(H')+E$. Let $F$ be the form defining $C$. Applying the criterion of Proposition 3 of §7.5 to $F, H'$, and $GH$,we see thatNoether’s conditions are satisfied at all $P\in C$. By Noether’s theorem,$GH = F'F+G'H'$ for some $F',G'$,where $deg(G')=m$. Then $\textbf{div(G')=div(GH)-div(H')}=D'+E+A$, as desired. Is the first equality of the last part, the one I don't understand..
I've understood all of it, but can't figure out why $Div(F)=0$
I'll appreciate any suggestion, thanks a lot!
The equality holds because the divisors are defined in terms of the local rings of the curve $C$. Since $F$ is the form defining $C$, its residue is zero in the coordinate ring and the local rings of the curve. Hence,
$$ \text{div}(F'F + G'H') = \text{div}(G'H') $$