Let $X\_1, X\_2$ be two smooth projective connected schemes of the same dimension in $\mathbb{P}^n$ such that $\dim(X\_1)=\dim(X\_2)\le n-2$. Assume futher that $X\_1 \cap X\_2$ is smooth. Is it possible that there exists a smooth projective hypersurface in $\mathbb{P}^n$ containing both $X\_1$ and $X\_2$? If not possible in general, is true if $X\_1, X\_2$ are complete intersection subchemes in $\mathbb{P}^n$?



Embedding a projective scheme is a smooth hypersurface
This is certainly not possible in general, since $X\_1$ itself cannot (in general) be embedded in a smooth hypersurface: for instance $\mathbb{P}^1\times \mathbb{P}^2$, embedded in $\mathbb{P}^5$ by the Segre embedding, is not contained in any smooth hypersurface, because then by Lefschetz theorem it would be a complete intersection.


Even if $X\_1$ and $X\_2$ are complete intersections, this is still not always possible. For instance take for $X\_1$ and $X\_2$ two planes in $\mathbb{P}^4$ in general position, so that they meet along a point. The previous argument tells you again that $X\_1\cup X\_2$ is not contained in any smooth hypersurface.


