Let $k$ be an algebraically closed field and $n \geqslant 1$. We say that $f \in k\left[x_{1}, \ldots, x_{n}\right]$ is singular at $p \in \mathbf{A}^{n}$ if either $p$ is a singularity of the hypersurface $\{f=0\}$ or $f$ has an irreducible factor $h$ of multiplicity strictly greater than one with $h(p)=0$. Given $d \geqslant 1$, let $X=\left\{f \in k\left[x_{1}, \ldots, x_{n}\right] \mid \operatorname{deg} f \leqslant d\right\}$ and let

$$Y=\left\{(f, p) \in X \times \mathbf{A}^{n} \mid f \text { is singular at } p\right\}$$

(i) Show that $X \simeq \mathbf{A}^{N}$ for some $N$ (you need not determine $N$ ) and that $Y$ is a Zariski closed subvariety of $X \times \mathbf{A}^{n}$.

(ii) Show that the fibres of the projection map $Y \rightarrow \mathbf{A}^{n}$ are linear subspaces of $\operatorname{dim} e n s i o n N-(n+1)$. Conclude that $\operatorname{dim} Y<\operatorname{dim} X$.

(iii) Hence show that $\{f \in X \mid \operatorname{deg} f=d, Z(f)$ smooth $\}$ is dense in $X$.

[You may use standard results from lectures if they are accurately quoted.]