Let $V$ be an irreducible variety over an algebraically closed field $k$. Define the tangent space of $V$ at a point $P$. Show that for any integer $r \geqslant 0$, the set $\left\{P \in V \mid \operatorname{dim} T_{V, P} \geqslant r\right\}$ is a closed subvariety of $V$.

Assume that $k$ has characteristic different from 2. Let $V=V(I) \subset \mathbb{P}^{4}$ be the variety given by the ideal $I=(F, G) \subset k\left[X_{0}, \ldots, X_{4}\right]$, where

$$F=X_{1} X_{2}+X_{3} X_{4}, \quad G=X_{0} X_{1}+X_{3}^{2}+X_{4}^{2}$$

Determine the singular subvariety of $V$, and compute $\operatorname{dim} T_{V, P}$ at each singular point $P$. [You may assume that $V$ is irreducible.]