Let $X$ be a smooth, projective $\mathbb{C}$-variety of dimension $r$ containing in $\mathbb{P}^n$. Denote by $I_X \subset \mathbb{C}[X_0,...,X_n]$ the ideal of $X$ defined by some homogeneous polynomials. For $k \gg 0$ and a general choice of $n-r$ homogeneous polynomials $F_1,...,F_{n-r} \in I_X$ of degree $k$, I want to understand the singularities of the complete intersection variety defined by $F_1,...,F_{n-r}$? In particular, is it possible that the complete intersection variety is a union of smooth projective varieties, intersecting each other transversally?

I would imagine this is too much to expect, so is there some criterion on $X$, when the complete intersection variety has such nice singularities? I would very much like to know a reference which studies similar questions.