Let $(L_i)_{i \in I}$ be a finite family of (distinct) smooth irreducible embedded algebraic varieties of codimension one in some complex projective space of dimension $n$.
Consider a finite intersection $I_J:=\bigcap_{j \in J} L_j$, for some $J \subset I$.
Then I know that $I_J$ will be a variety of dimension at least $n-\mathrm{card}\, J$.
Question: it is smooth?
I know it will be the case if the $L_j, j \in J$ are in general position. I know it would not be true for, say, $C^1$ manifolds, without this assumption of general position. But I don't have any counter-example for varieties.
The smooth quadric $L_1=V(xw-yz)\subset \mathbb P^3$ and its (smooth!) tangent plane $L_2=V(x)\subset \mathbb P^3$ at $(1:0:0:0)$ have as intersection $L_1\cap L_2=V(x, yz)=V(x, y)\cup V(x, z)$, which is non-smooth since it is the union of two lines in the tangent projective plane $L_2=V(x)$.