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created: 2023-03-26T11:58
updated: 2024-04-19T16:18
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References:

- Luca, *The KSBA compactification of the moduli space of D1,6
-polarized Enriques surfaces*  <https://link.springer.com/article/10.1007/s00209-021-02842-3>
- Luca, KSBA COMPACTIFICATION OF THE MODULI SPACE OF K3 SURFACES WITH PURELY NON-SYMPLECTIC AUTOMORPHISM OF ORDER FOUR <https://www.hanbommoon.net/wp-content/uploads/2013/08/Kondo-K3-KSBA-1.pdf#page=1&zoom=140,-146,797>
- Sterk, [PDF File](Ster91.pdf)
- Alexeev-Engel, MIRROR SYMMETRIC COMPACTIFICATIONS OF MODULI SPACES OF K3 SURFACES WITH A NONSYMPLECTIC INVOLUTION <https://arxiv.org/pdf/2208.10383.pdf#page=1&zoom=200,-10,737>

Other references:

- Phil's thesis: <https://academiccommons.columbia.edu/doi/10.7916/D8B282VW/download#page=50&zoom=auto,-200,730>
	- In paper form: <https://arxiv.org/pdf/1409.7676.pdf#page=1&zoom=auto,-200,763>

# Items from meeting with Valery

Meeting notes: 
![](With%20Zack%20Mar%2029,%202023.pdf)

- See Valery's "complete moduli" paper, accepted to Annals?
- Of interest: moduli of Enriques surfaces.
- $h^1(\OO_X) = 0, K_X\neq 0, 2K_X = 0$.
- $Y\to X$ 2-to-1 with $Y = \spec_{\OO_X}(\OO_X \oplus \OO_X(K_X))$ an unramified cover, $Y X = Y/\iota$ a basepoint free involution where $\iota$ is anti-symplectic.
- $\dim \mcm_{\mathrm{Enr}} = 10$, polarized or unpolarized.
- See Horikawa's construction: a span $X\fromvia{\pi} Y\tovia{p} (\PP^1)^2$  where $p$ is ramified along $D \in \abs{-2K_{(\PP^1)^2 }}$ with $\deg D = (4,4)$.
- Theory very similar to degree 2 K3s.


# To discuss with Luca

> 1. In terms of papers: 
> 	1. Sterk's ([PDF File](Ster91.pdf))
> 	2. Valery-Phil's one on non-symplectic involutions, 
> 	   <https://arxiv.org/pdf/2208.10383.pdf#page=8&zoom=180,-45,300>
> 	3. Laza-O'Grady on Baily-Borel for K3 surfaces which are double cover of $\PP^1 \times \PP^1$ ([https://arxiv.org/pdf/1801.04845.pdf](https://arxiv.org/pdf/1801.04845.pdf))
>    
> 2. For each Sterk 0-cusp find a Kulikov model. If we have this everything should follow using Valery-Phil's paper.

- AE find isotropic vectors in 2-elementary lattices: <https://arxiv.org/pdf/2208.10383.pdf#page=21&zoom=200,-10,415>
- Classified by 3 types of mirror moves (Def 5.6) $S\to \bar{T}$, corresponding to 3 types of cusps (Prop 5.5). 
	- A cusp of each type exists iff [the triangle figure](https://arxiv.org/pdf/2208.10383.pdf#page=5&zoom=200,-10,684) allows for such a move. See thm 5.8 <https://arxiv.org/pdf/2208.10383.pdf#page=23&zoom=200,-10,620>
- Problem: for each BB cusp of $F_S$ and monodromy invariant $\lambda$, describe central fibers $\mcx_0$ (adapted to $R$)
	- Read this off of [Coxeter diagram](Coxeter%20diagram.md) of lattices $\bar{T}$ at cusps.
- Vinberg's algorithm: finds fundamental domain of a hyperbolic reflection group.
	- Can be used to produce Coxeter diagrams for a lattice.
- Get a polyhedral fundamental domain $P$, then for $G\leq \Orth_C(\Lambda)$ the isometries fixing the light cone $C \da \ts{x^2 > 0}$, $W(G)$ the group generated by reflections through $v$ with $v^2 < 0$, get $G = W(G) \semidirect S$ where $S \leq \Sym(P)$; construct a Dynkin diagram $\Sigma G$.
	- Isotropic vectors $\mapstofrom$ ideal vertices of $P$ $\mapstofrom$ rank $n-1$ parabolic subdiagrams of $\Sigma G$.

A general IAS: 

- Vertices are triple points
- Edges are double curves
- Faces are anticanonical pairs $(V_i, D_i)$.
- Numbers on each side label self-interesection numbers in the respective surfaces

![](2023-04-04.png)

See [internal blowup](internal%20blowup.md) and [node smoothing](node%20smoothing.md) and [nodal slide](nodal%20slide)