Let $L=\mathbb{P}^l\subset\mathbb{P} ^ N _ {\mathbb{C}}$ be a linear space and let $M=\mathbb{P}^{N-l-1}$ be a linear space skew to $L$, i.e. $L\cap M=\emptyset$. Let $X\subseteq\mathbb{P}^N_{\mathbb{C}}$ be a closed irreducible variety not contained in $L$ and let $$ \pi_L:X\dashrightarrow\mathbb{P}^{N-l-1}=M $$ be the linear projection, i.e. the rational map defined on $X\setminus L$ by $$ \pi_L(x)=\langle L,x\rangle\cap M.$$ Let $x\in X\setminus L$ a point.

Statement: If $\overline{\pi_L(X)}$ and $\overline{\pi_L^{-1}(\pi_L(x))}$ are smooth (at any point), then  $X$ is smooth at $x$.

Thanks.

Let $L=\mathbb{P}^l\subset\mathbb{P} ^ N _ {\mathbb{C}}$ be a linear space and let $M=\mathbb{P}^{N-l-1}$ be a linear space skew to $L$, i.e. $L\cap M=\emptyset$. Let $X\subseteq\mathbb{P}^N_{\mathbb{C}}$ be a closed irreducible variety not contained in $L$ and let $$ \pi_L:X\dashrightarrow\mathbb{P}^{N-l-1}=M $$ be the linear projection, i.e. the rational map defined on $X\setminus L$ by $$ \pi_L(x)=\langle L,x\rangle\cap M.$$ Let $x\in X\setminus L$ a point.

Is true that if $\overline{\pi_L(X)}$ and $\overline{\pi_L^{-1}(\pi_L(x))}$ are smooth varieties, then  $X$ is smooth at $x$?

Thanks.