# Linear projection from a point preserves flatness

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 $$S$$)?

• As I understand it, the linear projection from a point lands into $\mathbb{P}^{n-1}_\mathbb{C}$, not $\mathbb{C}^{n-1}$. – Laurent Moret-Bailly Feb 12 at 9:16
• 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. – Jason Starr Feb 12 at 11:33
• @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 $U$? – Chen Feb 12 at 13:08
• "... 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. – Jason Starr Feb 12 at 13:15
• @JasonStarr Thanks. I was interested in the case $S$ is non-reduced. – Chen Feb 12 at 13:40