What does it mean by the Riemann Surface of a function $y=\sqrt{x^3}$? I saw how to use the cut and glue method to obtain a sphere where $y=\sqrt{x}$ can be defined. But I was not clear in what sense does it mean by a Riemann surface of a function. can anyone help me with the definition?
Using the same method, I found out the Riemann surface of $y=\sqrt{x^3}$ is a sphere, which seems strange since the algebraic curve $\text{Spec}\ k[x,y]/(x^3-y^2)$ is singular.