an algebraic set with Zariski tangent space equal to the entire ambient space at any point Let $X$ be an algebraic set of $\mathbb{C}^n$ with vanishing ideal $I_X$. Let $p_1,\cdots,p_m \in \mathbb{C}[x_1,\cdots,x_n]$ be generators of $I_X$ and suppose that for any $i=1,\cdots,m$ and any $\xi \in X$, we have that $\nabla_x p_i|_{\xi}=0$. This means that the Zariski tangent space of $X$ at any point of $X$ is equal to $\mathbb{C}^n$.
Question: what does this imply geometrically for $X$ and algebraically for $I_X$?

Well, as the comment above says, since the nonsingular points form an open dense subset of $X$, then we must conclude that $X$ has dimension $n$ (since all of its points have a tangent plane of dimension $n$). But if $X$ is closed, then it must be equal to $\mathbb{C}^n$ and thus the ideal must be 0 (since if $f$ is a polynomial that vanishes on $X$, then $X\subseteq\{f=0\}\subseteq\mathbb{C}^n$ and the dimension of $\{f=0\}$ is $n-1$ if $f$ is non-zero).
