(Sorry for my bad English) For the proof of Halphen's theorem, in Hartshorne p.349 I need this:
A divisor $D$ on a curve $X$ is nonspecial and very ample if and only if $D-P-Q$ is nonspecial for all $P, Q\in X$.
I try to prove it in this way:
I'm using this proposition (Hartshorne, IV,3.1, (b))"$D$ is very ample if and only if $dim|D-P-Q|=dim|D|-2$ for all $P,Q\in X$"
for the first direction, I have no problems: $D$ very ample and nonspecial implies $dim |D-P-Q|=dim|D|-2=d-g-2=deg (D-P-Q)-g$ and by Riemann-Roch for $D-P-Q$, we see that $D-P-Q$ is nonspecial.
Viceversa, if $D-P-Q$ is nonspecial I would like to have also $D$ nonspecial (here is my problem) so that by R-R $dim|D|=d-g $ and $dim |D-P-Q|=d-2-g$, i.e. $dim|D-P-Q|=dim|D|-2$ and so $D$ is very ample.
So, my problem is why $D-P-Q$ nonspecial implies $D$ nonspecial?
Maybe it's easy but I can't see it...