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I'd like to know if the following statement is true or false:

Given a non-singular complex projective surface $S$, it has at most a countable number of minimal models (up to isomorphism).

We know that the proposition is true for non-ruled surfaces (here we have uniqueness of the minimal model) and for rational surfaces. But what about ruled irrational surfaces?

I would like to know if the following statement is true or false:

Given a non-singular complex projective surface $S$, it has at most a countable number of minimal models (up to isomorphism).

We know that it is true for non-ruled surfaces (here we have uniqueness of the minimal model) and for rational surfaces. But what about ruled irrational surfaces?

I'd like to know if the following statement is true or false (if there is an answer):

Given a non-singular complex projective surface $S$, it has at most a countable number of minimal models (up to isomorphism).

The proposition is true for non-ruled surfaces (here we have the uniqueness) and for rational sufaces, but what about ruled irrational surfaces?

Many thanks in advance.

I'd like to know if the following statement is true or false:

Given a non-singular complex projective surface $S$, it has at most a countable number of minimal models (up to isomorphism).

We know that the proposition is true for non-ruled surfaces (here we have uniqueness of the minimal model) and for rational surfaces. But what about ruled irrational surfaces?

I'd like to know if the following statement is true or false (if there is an answer):

Given a non-singular complex projective surface $S$, it has at least a countable number of minimal models (up to isomorphism).

The proposition is true for non-ruled surfaces (here we have the uniqueness) and for rational sufaces, but what about ruled irrational surfaces?

Many thanks in advance.

I'd like to know if the following statement is true or false (if there is an answer):

Given a non-singular complex projective surface $S$, it has at most a countable number of minimal models (up to isomorphism).

The proposition is true for non-ruled surfaces (here we have the uniqueness) and for rational sufaces, but what about ruled irrational surfaces?

Many thanks in advance.