I would like to prove Proposition 3.1, Chapter III, from Hartshorne's book for real curves, in particular: Is the following statement true? All Definitions are taken from Hartshorne; by a point I mean a closed point. Let $X$ be a nonsingular projective curve over $\mathbb{R}$.
(a) A complete linear system |D| has no base points if and only if for every point $p \in X$ we have $\dim |D-p| = \dim|D| - \deg(p)$.
(b) D is very ample if and only if for every two points $p,q \in X$ we have $\dim |D-p-q| = \dim|D| - \deg(p) - \deg(q)$.
I know that the needed theorems (Proposition 7.7 and Remark 7.8.2 and Prop. 7.3's "second direction") do not need the assertion that $k$ is algebraically closed (but Hartshorne still assumes that). But in Proposition 3.1 I do not understand the last steps of the proof, so I wonder if this Proposition is known and proved for $k=\mathbb{R}$.