Let $k$ be an algebraically closed field of any characteristic.

(a) Define what it means for a variety $X$ to be non-singular at a point $P \in X$.

(b) Let $X \subseteq \mathbb{P}^{n}$ be a hypersurface $Z(f)$ for $f \in k\left[x_{0}, \ldots, x_{n}\right]$ an irreducible homogeneous polynomial. Show that the set of singular points of $X$ is $Z(I)$, where $I \subseteq$ $k\left[x_{0}, \ldots, x_{n}\right]$ is the ideal generated by $\partial f / \partial x_{0}, \ldots, \partial f / \partial x_{n} .$

(c) Consider the projective plane curve corresponding to the affine curve in $\mathbb{A}^{2}$ given by the equation

$$x^{4}+x^{2} y^{2}+y^{2}+1=0 .$$

Find the singular points of this projective curve if char $k \neq 2$. What goes wrong if char $k=2$ ?