(a) 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 $P$.

Define the dimension of $X$ in terms of (i) the tangent spaces of $X$, and (ii) Krull dimension.

(b) Consider the ideal $I$ generated by the set $\left\{y, y^{2}-x^{3}+x y^{3}\right\} \subseteq k[x, y]$. What is $Z(I) \subseteq \mathbb{A}^{2} ?$

Using the generators of the ideal, calculate the tangent space of a point in $Z(I)$. What has gone wrong? [A complete argument is not necessary.]

(c) Calculate the dimension of the tangent space at each point $p \in X$ for $X=$ $Z\left(x-y^{2}, x-z w\right) \subseteq \mathbb{A}^{4}$, and determine the location of the singularities of $X .$