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In Huybrecht's Lectures on K3 surfaces, there's an explicit description of an elliptic fibration for the K3 surface $X:=\{(x_0:x_1:x_2:x_3)\in\Bbb{P}_\Bbb{C}^3\mid x_0^4+x_1^4+x_2^4+x_3^4=0\}$ (Fermat quartic), with the following explanation (example 3.11):

I don't get how proposition 3.10 helps us. My questions are:

  1. How does the existence of the elliptic curve $E$ such that $mE\in|L|$ for some $m$ allow us to conclude that there is an elliptic fibration on $X$?

  2. Was the map $X\to\Bbb{P}^1$ somehow suggested by the proposition?

  3. How do I verify that the map is an elliptic fibration?

rmdmc89
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    The proposition isn't really helping here. He's just observing that you can work out this particular case of the proposition by hand. 1) The existence of an elliptic curve moving in a linear system means that there is an elliptic curve in the same linear system passing through a general point of $X$, which implies the existence of a rational map to the base. 3) He's told you; the hyperplane sections are all the union of $\ell$ with a plane cubic. Those plane cubics, which are elliptic curves generically, are the fibers. – Tabes Bridges Jul 26 '20 at 21:28
  • @TabesBridges, I can't see how the fibers of $X\to\Bbb{P}^1$ are contained in a hyperplane section containing $\ell$. In order for that to be the case, it seems to me that the map should look like $\pi:(x_0:x_1:x_2:x_3)\mapsto (x_1-\xi x_0:x_3-\xi x_2)$ – rmdmc89 Jul 26 '20 at 23:27
  • The way I wrote it, the fiber $\pi^{-1}(a:b)$ would be contained in the hyperplane $a(x_3-\xi x_2)-b(x_1-\xi x_0)=0$ (which in particular contains $\ell$) – rmdmc89 Jul 26 '20 at 23:30

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