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 $\mathbb{P}^n$?


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.

  • $\begingroup$ Is there any instance/condition under which this is possible? $\endgroup$
    – user46578
    Feb 19 '14 at 10:13
  • 1
    $\begingroup$ Basically, what is proved is that it's almost never possible, by the definition of transversality. If $X_i$ and $X_2$ intersect transversally, their tangent spaces span everything; on the other hand, you want them to lie in a codimension one subspace (the tangent space to the hypothetical hypersurface). $\endgroup$ Feb 19 '14 at 10:18

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