Let $C$ be a compact smooth complex curve with $H^0(C,\mathcal{O}(p))=2$.
I feel confused with the following words:
Denote by $a$ and $b$ two non-collinear sections in $H^0(C,\mathcal{O}(p))$. Then one can consider the ratio $f=a/b$ as a holomorphic map from $C$ to $\mathbb{C}P^1$.
$\bullet$ If both sections vanish at the same point $q$ then $f$ does not take the value $0$ so is constant. This contradicts the non-collinearity of $a$ and $b$.
$\bullet$ If $a$ but not $b$ vanishes at $q$ then looking at the fiber of $0\in \mathbb{C}P^1$ the degree of the map $f$ is $1$.
I don't understand "If both sections vanish at the same point $q$ then $f$ does not take the value $0$ so is constant." And does non-collinear means $a,b$ are not linear?
I also wonder how to deduce the degree is $1$ from $a(q)=0,b(q)\not=0$?
At last, how is that related to $p$?