Let $X$ be a smooth projective variety over an algebraically closed field of characteristic $0$ and of dimension $n\geq 10$. Let $(C_i)_{1\leq i\leq N}$ (resp. $(S_i)_{1\leq i\leq N}$) be smooth curves (resp. surfaces) on $X$. Is it always possible to find a smooth surface (resp. threefold) $\Sigma$ containing the curves $(C_i)_{1\leq i\leq N}$ (resp. the surfaces $(S_i)_{1\leq i\leq N}$)?
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2$\begingroup$ As user100824 suggests below: if the tangent space of $\cup C_i$ is more than $2$-dimensional, the curves $C_i$ cannot be all contained in a smooth surface. $\endgroup$– R. van Dobben de BruynCommented Nov 7, 2016 at 20:51
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Take the affine space $\mathbb{A}^{10}$, and let $C_{1}, \ldots, C_{10}$ be the axes. This provides a counterexample, right?
