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$ ?