I am very new to algebraic geometry, and self-studying varieties. I have the following question.
Suppose $Y$ is a variety of dimension $r$ and degree $d>1$ in $\mathbb{P}^n$. Let $P$ be a nonsingular point of $Y$. Define $X$ to be the closure of the union of the lines $PQ$ for $Q\in Y$ and not equal to $P$. Then does there exist a hyperplane $H$ passing through $P$ which does not contain $Y$ such that the intersection multiplicity of each irreducible component of $X\cap H$ is 1?
I think somehow I have to use the nonsingularity of the point $P$. But I am not getting any idea how to use it.
Any kind of help will be appreciated. Thank you. some days ago I posted this on MSE but did not get any response so I am posting this here.
