In the paper "On the Moduli of Curves on Rational Ruled Surface" by A. Fauntleroy, during the proof of theorem 4.2, the author states the result that a hyperelliptic curve of genus $g$ sits on Hirzebruch surface $F_{g+1}$. For the proof, he cites section 6.2 of "Weighted Projective Varieties" by I. Dolgachev. However I could not find the proof there (the article that I found has only 4 sections), so I assume it was miscited.
So I wanted to ask how one can prove the statement. Alternatively, an appropriate reference should suffice.
Thanks.
Let $C$ be a hyperelliptic curve of genus $g$ with the hyperelliptic covering $\pi \colon C \to \mathbb{P}^1$. Its branch divisor has degree $2g + 2$, hence $$ \pi_*\mathcal{O}_C \cong \mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(-g-1). $$ By adjunction, there is a surjective morphism $$ \pi^*(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(-g-1)) \to \mathcal{O}_C. $$ By the universal property of the projective bundle this gives the embedding $$ C \to \mathbb{P}_{\mathbb{P}^1}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(-g-1)) = F_{g+1}. $$