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.



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


