5
$\begingroup$

Let $X$ be a smooth projective complex algebraic variety. I believe the answer to the following question should be well-known. It is an instance of the Iitaka conjecture.

There are only finitely many isomorphism classes of smooth projective varieties of general type $Y$ such that there exists a surjective proper birational morphism from $X$ to $Y$.

$\endgroup$

1 Answer 1

8
$\begingroup$

The answer is yes. You can look at this paper: Guerra, Pirola, On the finiteness theorem for rational maps on a variety of general type, Collect. Math. 60 (2009), no. 3, 261–276.

It contains an overview of the topic, and besides it gives some effective bounds for the number of such maps.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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