1
$\begingroup$

I am looking for an explaination or an reference for the following fact:

Let $\pi:X\rightarrow Z$ be a contraction from a smooth surface $X$ to a curve $Z$. Assume that the geometric generic fiber of $\pi$ is a rational curve. Then we may run a $K_X$-MMP over $Z$ and reach a minimal ruled surface $\pi^\prime:X^\prime \rightarrow Z$.

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ Isn't "running MMP" a bit overkill for surfaces? Is this not just something you can find in any introductory text on surfaces (e.g. Beauville, Prop. II.16 and Thm. III.10)? Or are you looking for a more specific construction? $\endgroup$
    – R. van Dobben de Bruyn
    Commented Dec 4, 2023 at 16:05
  • $\begingroup$ @R.vanDobbendeBruyn Sorry I have little knowledge about the surface MMP. In other to use [Beauville, Thm. III.10], one needs to show that $X$ is birational to $Z\times \mathbb{P}^1$ and $Z$ is irrational. I tried to use [Beauville, Corollary. VI.18] and conclude that $X$ is ruled, i.e., $X$ is birational to $C\times\mathbb{P}^1$ where $C$ is some smooth curve. But I'm not sure how to show that $C$ is birational to $Z$ and how to rule out the case that $C=\mathbb{P}^1$ $\endgroup$
    – Hobo
    Commented Dec 27, 2023 at 13:13

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .