Let $\pi:\mathcal{X} \to S$ be a flat family of affine curves contained in $\mathbb{C}^n$ for $n \ge 3$ i.e., $\mathcal{X} \hookrightarrow \mathbb{C}^n_S$ and the inclusion commutes with the natural morphism to $S$. Let $p \in \mathbb{C}^n$ be a general point (not intersecting any $\mathcal{X}_s$) and $\phi:\mathbb{C}^n_S \backslash (p \times S) \to \mathbb{C}^{n-1}_S$ be the trivial deformation of the linear projection from the point $p$. Let $\mathcal{Y} \subset \mathbb{C}^{n-1}_S$ be the image of $\mathcal{X}$ under the morphism $\phi$. Is $\mathcal{Y}$ going to be $S$-flat (under the natural morphism to $S$)?
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$\begingroup$ As I understand it, the linear projection from a point lands into $\mathbb{P}^{n-1}_\mathbb{C}$, not $\mathbb{C}^{n-1}$. $\endgroup$– Laurent Moret-BaillyCommented Feb 12, 2019 at 9:16
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$\begingroup$ Linear projection does not always preserve flatness. Images of morphisms do not always preserve flatness. The reduced image of a linear projection can have strictly lower degree than the domain. This forces the linear projection to have non-reduced structure if you want to preserve flatness. The non-reduced structure depends on more than the fiber, and this quickly leads to examples where the linear projection is not flat. $\endgroup$– Jason StarrCommented Feb 12, 2019 at 11:33
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$\begingroup$ @JasonStarr Can we comment about generic flatness i.e., will there exist a non-empty open subset $U$ of $S$ such that $\mathcal{Y}_U$ (the preimage of $U$ under the natural morphism from $\mathcal{Y}$ to $S$) is flat over $U$? $\endgroup$– ChenCommented Feb 12, 2019 at 13:08
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$\begingroup$ "... will there exist a non-empty open subset $U$ of $S$ such that $\mathcal{Y}_U$ ... is flat over $U$?" I am not certain what you are asking. If $S$ is a reduced Noetherian scheme, that is true by Grothendieck's Generic Flatness Theorem. However, if $S$ is nonreduced, that can fail. $\endgroup$– Jason StarrCommented Feb 12, 2019 at 13:15
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$\begingroup$ @JasonStarr Thanks. I was interested in the case $S$ is non-reduced. $\endgroup$– ChenCommented Feb 12, 2019 at 13:40
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