(i) Let $X$ be an affine variety. Define the tangent space of $X$ at a point $P$. Say what it means for the variety to be singular at $P$.

(ii) Find the singularities of the surface in $\mathbb{P}^{3}$ given by the equation

$$x y z+y z w+z w x+w x y=0 .$$

(iii) Consider $C=Z\left(x^{2}-y^{3}\right) \subseteq \mathbb{A}^{2}$. Let $X \rightarrow \mathbb{A}^{2}$ be the blowup of the origin. Compute the proper transform of $C$ in $X$, and show it is non-singular.