# Well-definedness of pull-back of divisor

I read this paragraph from Complex Geometry by Huybrechts:

...The pull-back of a divisor $$D$$ under a morphism $$f:X\rightarrow Y$$ is not always well-defined, one has to assume that the image of $$f$$ is not contained in the support of $$D$$. Thus, one usually considers only dominant morphisms...

So why would the pull-back be not well-defined if the image of $$f$$ is contained in the support of $$D$$? And why does dominant morphisms solve this issue?

• This is almost a duplicate of this other question, though in the algebraic world instead of the analytic one. Do you think the linked question solves your issues? May 26, 2020 at 20:17
• @KReiser Yes that one explains well. Thanks! May 28, 2020 at 3:04

Suppose you are working with weil divisors on a regular variety. Then, effective divisor correspond to codimension 1 subvarieties. In this case, if the image of $$f$$ is contained in the support of $$D$$; then the pullack would be all of $$X$$, which is not a divisor.
Taking a dominant morphism solves the issue, because then the image of $$f$$ will be an open subset, and a codimension 1 subvariety of $$Y$$ cannot contain any open subset.
• +1 for a good explanation. One very minor thing, though: it may not actually be the case that the image of $f$ is an open subset - take $\Bbb A^2\to \Bbb A^2$ by $(x,y)\mapsto (x,xy)$, for instance. Instead, one should say that the image of a dominant map is dense, and then your explanation is completely fine - no dense set is contained in a codimension one subvariety. May 27, 2020 at 4:20