Let $p:=[0,0,...,0,1] \in \mathbb{P}^n$ the point whose all the coordinates are zero except for the $n$-th. This defines a linear projection map $\phi:\mathbb{P}^n-p \to \mathbb{P}^{n-1}$, given by $[x_0,...,x_n] \mapsto [x_0,...,x_{n-1}]$. Let $X$ be a local complete intersection subscheme in $\mathbb{P}^{n-1}$. Is it true that the scheme theoretic closure of $\phi^{-1}(X)$ in $\mathbb{P}^n$ is a local complete intersection subscheme in $\mathbb{P}^n$? If not, is there any known condition on $X$ under which we have an affirmitive answer to the question? (Assume the underlying field is the field of complex numbers and $n \ge 3$)