Bertini type result for complete intersection varieties containg a non-singular variety

Question

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.

Answer 1

The general such intersection is the union $X \cup Y$, where $Y$ is smooth away from $X$. On the other hand, in general $Y$ has singularities in codimension 4. Indeed, if $\mathcal{I}_X$ is the ideal sheaf of $X$, the polynomials $F_i$ induce a morphism $$ \mathcal{O}^{\oplus (n-r)} \to \mathcal{I}_X(k). $$ When restricted to $X$ it is a morphism $$ \mathcal{O}_X^{\oplus (n-r)} \to \mathcal{I}_X/\mathcal{I}_X^2(k) \cong \mathcal{N}_X^\vee(k) $$ of two locally free sheaves of rank $k$. Its degeneracy locus is the intersection $X \cap Y$, and its second degeneracy locus (the corank-2 locus) is the singular locus of $Y$.