Existence of a hyper surface containing the singular locus of projective variety.

Question

Let $V_n$ be an irreducible algebraic variety in projective space $\mathbb{C}P^n$. Denote by $V_{n_1}$ the subvariety cut out on $V_n$ by a hyper surface $W$ containing the singular locus of $V_n$ and not $V_n$. How to show the existence of $W$?

Answer 1

The following more general fact is true.

Proposition: For any closed subset $S \subset \mathbf P^n$ and any point $p \notin S$ there is a hypersurface $W$ which contains $S$ but not $p$.

(Your question is the case where $S$ is the singular locus of your variety $V_n$, and $p$ is any point at which $V_n$ is nonsingular.)

Proof of Proposition: Let $S$ be the vanishing set of the ideal $$I_S = \langle f_1,\ldots,f_n \rangle$$ where $f_i$ are homogeneous polynomials.

Since $p \notin S$, at least one of generators of the ideal does not vanish at $p$; call it $f_1$. Then let $W=V(f_1)$.