Compactification of y squared equals x to the eighth minus one
= Compactification of y squared equals x to the eighth minus one
The affine germ surface over the plane with the eight roots of unity removed is compactified by adding the eight simple branch points above those roots and two unbranched points above infinity. The degree-two map to the Riemann sphere has total ramification eight, so the <Riemann-Hurwitz formula> gives genus three.