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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?

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  • $\begingroup$ In this case the minimal model is determined by a rank 2 vector bundle on a curve of genus $H^1(\mathcal{O}_X)$, up to possible twist by a line bundle. I don't know off-hand how large this is... I believe for elliptic curves there's only countably many such things (see Atiyah's paper on the subject). $\endgroup$
    – Dylan Wilson
    Oct 9, 2014 at 15:54

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Your statement is actually false for all ruled surfaces over a curve of strictly positive genus.

In fact, let $C$ be a curve of genus $\geq 1$, consider two distinct points $x, \, y \in C$ and take the rank $2$ vector bundles on $C$ defined by $$\mathscr{E}_x = \mathscr{O}_C \oplus \mathscr{O}_C(x), \quad \mathscr{E}_y = \mathscr{O}_C \oplus \mathscr{O}_C(y).$$

I claim that the two projective bundles over $C$ given by $\mathbb{P}(\mathscr{E}_x)$ and $\mathbb{P}(\mathscr{E}_y)$ are not isomorphic, hence the birational class of $C \times \mathbb{P}^1$ contains uncountably many distinct minimal models.

The argument is the following. If the projective bundles were isomorphic, then there would exist a line bundle $\mathscr{L}$ on $C$ such that $\mathscr{E}_x \otimes \mathscr{L}=\mathscr{E}_y$, that is $$\mathscr{L} \oplus \mathscr{L}(x) = \mathscr{O}_C \oplus \mathscr{O}_C(y).$$

Since by the Krull-Schmidt theorem the decomposition of a vector bundle into indecomposable summands is unique up to permutations of the summands, we must have either $\mathscr{L} = \mathscr{O}_C$ or $\mathscr{L}=\mathscr{O}_C(y).$

Now, both cases are impossible: the former since $x \neq y$ and the curve $C$ has strictly positive genus, and the latter by degree reasons.

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  • $\begingroup$ But these surfaces ARE minimal... $\endgroup$
    – Yosemite Stan
    Oct 9, 2014 at 16:49
  • $\begingroup$ Of course: they are all minimal models belonging to the the birational equivalence class of the ruled surface $C \times \mathbb{P}^1$. $\endgroup$
    – Francesco Polizzi
    Oct 9, 2014 at 16:52
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    $\begingroup$ Right, I shouldn't make comments when I wake up... $\endgroup$
    – Yosemite Stan
    Oct 9, 2014 at 17:09

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