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?

  • 1
    $\begingroup$ You are right the degree and the genus are independant here. However I think it's probably used to show that the canonical map is not an embedding for an hyperelliptic curve. $\endgroup$ – Nicolas Hemelsoet Jul 23 '18 at 23:08
  • $\begingroup$ Thanks, that must be it. I have found the theorem you are referring to - it is stated two chapters later. $\endgroup$ – merle Jul 23 '18 at 23:28

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