Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

Thanks!

share|improve this question

2 Answers 2

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.

share|improve this answer

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).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.