Trying to visualize in 3D the normalized model $y^2=f(x)$ of a hyper-elliptic curve AND the degree two ramified cover $(X,Y)\mapsto X$, $\mathbb{C}\times \mathbb{C}\to \mathbb{C}$. Is this projection amenable to viewing the RS with the $g$ holes stacked, like a multi-doughnut with a honey flow on it (like in an article by Witten; forgot which), with the above projection x factoring through $(X=x+iy,Y)\to (x+iy, |Y|)$, to bring it in $\mathbb{R}^3$?
I would like to see the ramification points (zeros of $f$) as in a Morse Theory analysis, related to the above standard projection of its normal model; something like in the sketch below:
