Blowing up at a point

Question

Let $C:=\{y^2=x^3\}$ be a curve in $\mathbb C^2$ and $\pi:X\to \mathbb C^2$ the blowing up of $\mathbb C^2$ at the origin $o:=(0,0)$. The dimension of $X$? Take another blowing $\pi':X'\to X$ at the singular point of the strict transform of $C$. What about $X'$ and the dimension of $X'$?

Answer 1

A blowing-up morphism of integral algebraic varieties (more generally of integral schemes) is birational: it is an isomorphism outside of the center we blow-up (under the obvious condition that the center is nowhere dense). As birational algebraic varieties have the same dimension (equal to the transcendental degree of the function fields), blowing-up doesn't change the dimension.

The strict transform of $C$ in $X$ is regular (straightforward computationsn see e.g. a solution here), so the second part of your question is irrelevant.