## The holomorphic map from a compact smooth curve $C$ to $\mathbb{C}P^1$ when $H^0(C,\mathcal{O}(p))=2$

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 $$p$$?

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