My question concerns the following problem from Rick Miranda's Algebraic Curves and Riemann Surfaces (p. 167):
[S]how that if $v^2 = h(u)$ defines a hyperelliptic curve of genus $g$, then $\phi = [1 \colon u \colon u^2 \colon \cdots \colon u^{g-1}]$ defines a degree 2 map onto a rational normal curve of degree $g-1$ in $\mathbb{P}^{g-1}$, and that the hyperplane divisors of $\phi$ have degree $2g-2$.
Constructing $\phi$ is easy: just compose the projection of the hyperelliptic curve onto $u$ with the standard map from $\mathbb{P}^1$ onto the rational normal curve of degree $g-1$.
What I don't understand is why the genus is of importance. Can't we simply obtain a degree 2 map from any hyperelliptic curve onto any rational normal curve in exactly the same way, even if the genus and degree don't match up?