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created: 2023-03-26T11:58
updated: 2024-05-03T23:22
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# Complete Moduli of Enriques Surfaces

- Vocab one needs to know:
	- [numerical polarization](polarization.md)
	- [big line bundle](big.md)
	- [nef](nef%20divisor.md)
	- [basepoint free](ample.md)
	- [linear system](linear%20system.md)
	- [ample](ample.md)
	- [ramification divisor](ramification%20divisor.md)
	- [Q-Cartier divisor](Weil%20divisor.md)
	- [log canonical](log%20discrepancy.md)
	- [KSBA compactification](KSBA%20compactification)
	- [slc](slc.md)
	- [dlt](dlt) and [[dlt model]]
	- [[K-trivial degenerations]]
	- [integral affine manifold](integral%20affine%20manifold.md)
	- [dual complex](dual%20complex.md)
	- [[ADE surface]], see [LitNote-Alexeev and Thompson-2019-ADE surfaces and their moduli-AT17](LitNote-Alexeev%20and%20Thompson-2019-ADE%20surfaces%20and%20their%20moduli-AT17.md).
- Very specific relevant facts for this paper:
	- $(2, 2, 0) = ?$ 
		- Has two zero cusps:  $U\oplus E_8\sumpower{2}$ and  $U(2) \oplus E_8\sumpower{2}$.
	- $(10, 10, 0) = ?$ 
		- Has $n$ zero cusps: 

- Our moduli space is $F_{\mathrm{En}, 2} = \ts{(Z, [\mcl_Z])}\modiso$ where $Z$ is an [Enriques surface](Enriques%20surfaces.md) with at worst [ADE singularities](ADE%20singularities.md) and $[\mcl_Z] \in \Pic(Z)/C_2$ is a degree 2 ample [numerical polarization](polarization.md).
	- Equivalently: $\ts{(Z, \mcl_Z^{\tensor 2} )}$ where $\mcl_Z^{\tensor 2}\in \Pic(Z)$ is 2-divisible and degree 8.
	- This is the moduli space described by Sterk.
	- #todo/questions : what are the other moduli spaces of Enriques surfaces in the literature? How does ours compare to theirs?
- Main theorem: the [KSBA compactification](KSBA%20compactification) is isomorphic to a [semitoroidal compactification](semitoroidal%20compactification) which is dominated by a [toroidal compactification](toroidal%20compactification.md) with respect to the [Coxeter fan](Coxeter%20group.md).
	- Novelty: there appear ABCDE surfaces, generalizing just the ADE surfaces that appear for stable limits of K3s.
	- Approach: use $\mcl_Z^{\tensor 2}$ to obtain a double cover $\rho: Z\to W$ where $W$ is a quartic [del Pezzo](del%20Pezzo.md).
	- Form KSBA compactification using the [ramification divisor](ramification%20divisor.md) $R_Z$ of $\rho$.
- $\Gamma(\mcx_0)/\iota_{\mathrm{En}} \cong \RP^2, \DD^2$, which are the two well-known types of degenerations of Enriques surfaces.
- Our strategy: take a [closed immersion](closed%20immersion.md) $F_{\mathrm{En}, 2} \injects F_{(2, 2, 0)}$ and study $\iota_{\mathrm{En}}$ on the fibres to describe the universal family $U_{\mathrm{En}, 2} \to F_{\mathrm{En}, 2}$.
	- Note that a generic point $[X]\in F_{(2,2,0)}$ is a branched double cover of $\PP^1\times\PP^1$ with branch locus a $(4, 4)$ curve.
		- #todo/questions Explicitly describe what a $(p, q)$ curve is by explicitly describing $\OO_{\PP^1\times \PP^1}$.
- #todo/questions Can we only do numerically polarized degree 4? What about degree 2?