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Suppose that $X$ and $Y$ are complex algebraic varieties with $Y$ affine. Suppose that $f:X\rightarrow Y$ is a variety morphism. I am interested in finding conditions on the $\mathbb{C}$-algebra morphism $f^*:\mathbb{C}[Y]\rightarrow\mathbb{C}[X]$ under which the image of $f$ is open. I would appreciate any and all references/suggestions.


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up vote 9 down vote accepted

Assuming $\mathbb C[X]$ is the ring of global sections of $X$: This is not possible. The strongest possible condition is that the morphism be an isomorphism. But that is not sufficient. Choose $Y$ to be $\mathbb A^3$. For $X$, first blow up $\mathbb A^3$ at the origin, then remove a line passing through the origin. The first step clearly does not affect the global sections of the structure sheaf, and the second step is the removal of a codimension $2$ subset from a smooth variety, hence does not affect it either. But the image is $\mathbb A^3$ minus a line plus a point, which isn't open.

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Flat morphisms locally of finite type of noetherian schemes are open. So if $X$ is also affine and $f^*$ is flat, then $f$ is open. But if $X$ is not affine, then there is not much you can say (as Will explained).

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