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created: 2023-03-26T11:58
updated: 2024-05-03T23:22
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# Horikawa's Model

See [@Hor78a].

> I went back to Hor78a to look at his proof more carefully, and the story seems to be the following: in Lem 4.1, you let ![S](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=S "S") be an Enriques surface and show there is an elliptic fibration ![g: S\to \mathbb{P}^1](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=g:%09S%5Cto%09%5Cmathbb{P}%5E1 "g: S\to \mathbb{P}^1"). Then, up to birational equivalence, you find (say) some ![\eta: S\to \Sigma_0 := \mathbb{P}^1 \times \mathbb{P}^1](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5Ceta:%09S%5Cto%09%5CSigma%5F0%09:=%09%5Cmathbb{P}%5E1%09%5Ctimes%09%5Cmathbb{P}%5E1 "\eta: S\to \Sigma_0 := \mathbb{P}^1 \times \mathbb{P}^1") which is a branched double cover over a curve ![B](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=B "B") such that there is a factorization ![S\to \mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^1](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=S%5Cto%09%5Cmathbb{P}%5E1%09%5Ctimes%09%5Cmathbb{P}%5E1%09%5Cto%09%5Cmathbb{P}%5E1 "S\to \mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^1"), where the latter is a projection ![\psi](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5Cpsi "\psi") onto (say) the first factor. He also says that ![B \in |4 \Delta_0 + 6\Gamma|](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=B%09%5Cin%09|4%09%5CDelta%5F0%09%2B%096%5CGamma| "B \in |4 \Delta_0 + 6\Gamma|") where ![\Delta_0](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5CDelta%5F0 "\Delta_0") is a section of ![\psi](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5Cpsi "\psi") satisfying ![\Delta_0^2 = 0](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5CDelta%5F0%5E2%09=%090 "\Delta_0^2 = 0") and ![\Gamma](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5CGamma "\Gamma") is a fibre of ![\psi](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5Cpsi "\psi").

  

My first difficulty was seeing that ![|4\Delta_0 + 6\Gamma| = | \mathcal{O}(4, 4)|](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=|4%5CDelta%5F0%09%2B%096%5CGamma|%09=%09|%09%5Cmathcal{O}(4,%094)| "|4\Delta_0 + 6\Gamma| = | \mathcal{O}(4, 4)|"), so that our ![B](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=B "B") matches his. But maybe it does not, and this is what Dolgachev was saying. Or perhaps this is the content of Theorem 4.1, but that branch locus has the additional divisors ![\Gamma_1, \Gamma_2](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5CGamma%5F1,%09%5CGamma%5F2 "\Gamma_1, \Gamma_2") which are the double fibres in ![g: S\to \mathbb{P}^1](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=g:%09S%5Cto%09%5Cmathbb{P}%5E1 "g: S\to \mathbb{P}^1"), and I can't tell if those are in our branch locus.

  

Another difficulty is that we say the branched cover is a K3 ![X](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=X "X"), while Horikawa says the branched cover is (birational to) an Enriques surface ![Z](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=Z "Z"). It seems that he constructs ![X](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=X "X") in Theorem 6.1, but it's actually a pullback of ![\eta: S\to \Sigma_0](https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=%5Ceta:%09S%5Cto%09%5CSigma%5F0 "\eta: S\to \Sigma_0") from above by a ramified degree 2 map.