A hyperelliptic curve is defined as the projective compactification of a planar curve of the form $$ \textbf{Spec}\left( \frac{\mathbb{C}[x,y]}{(y^2 - \prod(x - a_i))} \right) $$ Is there a general method for computing the compactifications of hyperelliptic curves? If so, how can I compute the corresponding projective scheme in terms of a graded coordinate ring?
If not, how can I find the compactification of these hyperelliptic curves? $$ \begin{align*} & y^2 - x(x-1)(x-3)(x-5)(x-7) \\ & y^2 - x(x-1)(x-2) \end{align*} $$