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created: 2023-03-26T11:58
updated: 2024-05-03T23:22
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# 000 Enriques Talk Facts 0

# From [Papers COMPACT MODULI OF K3 SURFACES WITH A NONSYMPLECTIC AUTOMORPHISM](Papers%20COMPACT%20MODULI%20OF%20K3%20SURFACES%20WITH%20A%20NONSYMPLECTIC%20AUTOMORPHISM.md)

- [x] What is a stable pair $(S, \eps D)$? Give an example. ✅ 2023-01-17
	- [ ] slc singularities 
	- [ ] $\K_S + \eps D \in \QQ\dash\CDiv(X)^\amp$.
- [x] What is a Kulikov model? ✅ 2023-03-30
	- [ ] $X\to \DD$ an extension of $X^\circ \to \DD^\circ$ where
	- [ ] $X$ is a smooth algebraic space,
	- [ ] $K_X\sim 0$,
	- [ ] $X_0$ is RNC
- [x] What are Types I, II, II of Kulikov models? ✅ 2023-01-17
	- [ ] I: $X_0$ smooth, 
	- [ ] II: $X_0$ has double curves but no triple points, 
	- [ ] III: $X_0$ has triple points 
- [x] What is a non-symplectic automorphism? ✅ 2023-01-07
	- [ ] $\sigma: X\to X$ with finite order $n > 1$ where $\sigma^*(\omega_X) = \zeta_n \omega_X$ for some primitive root of unity $\zeta_n$, where $\omega_X \in H^{2, 0}(X)$ is a nonzero 2-form.
- [x] What is a K3 surface? ✅ 2023-01-17
	- [ ] Complete, smooth, $\dim_\CC X = 2$, $\omega_{X/k}\cong \OO_X$, and $H^1(\OO_X) = 0$.
	- [ ] Interpret $q(X) \da h^{0, 1}(X)$ as the irregularity.

- [x] What is the Picard number $\rho(X)$? ✅ 2023-02-07
	- [ ] For $X$ compact complex, $$\rho(X) \da \rank_\ZZ \im(\Pic(X) \to H^{1, 1}(X))$$
	- [ ] If $X$ is Kahler, $\rho(X) = \rank H^{1,1}(X) = \rank \NS(X)$.


- [ ] What is a Type I domain?
- [ ] What is a Type IV domain?
- [ ] What is a polarizing divisor?
- [ ] What are the invariants $g, k, \delta$?
- [ ] What are canonical singularities?
- [ ] What are slc singularities?      
- [ ] What is a log center?
- [ ] What is a semifan?

- [x] What is the Kahler cone $\mck_X$? ✅ 2023-01-07
	- [ ] $\mck_X \subseteq H^{1, 1}(X; \RR)$ the set of Kahler forms, forms an open convex cone.
- [x] Give an example of a stable slc pair. ✅ 2023-02-14
	- [ ] Any K3 $X$ with ADE singularities and any effective ample divisor $R$, 
	- [ ] CY: $\omega_X\cong \OO_X$ 
	- [ ] Has canonical singularities, better than slc
	- [ ] No log centers.
- [x] What is a stable slc surface pair? ✅ 2023-01-17
	- [ ] A pair $(S, \epsilon D)$, where
		- [ ] $S$ is a connected, reduced, projective Gorenstein surface $S$ with $\omega_S \simeq \mathcal{O}_S$ which has semi log canonical singularities.
			- [ ] CRPG CY SLC
		- [ ] $D$ is an effective ample Cartier divisor on $S$ that does not contain any log canonical centers of $S$.
	- [ ] Thus $(S, \eps D)$ is stable pair for small $\eps \QQ_{\gt 0}$.
- [x] What is a canonical choice of polarizing divisor? ✅ 2023-01-17
	- [ ] An algebraically varying big and nef $R \in \Div(U)$ for $U$ a Zariski dense subset of $F_\rho$.
- [x] What is a Kulikov surface? ✅ 2023-01-17
	- [ ] $X_0$ the central fiber of a Kulikov model.
- [x] What is Kulikov's key result on degenerations of K3s? ✅ 2023-01-17
	- [ ] For $Y^*\to C^*$ a family, there is a finite base change $(C', 0) \to (C, 0)$ and a birational modifcation of its pullback $Y'\rational X$ such that $X$ has smooth total space, $K_X \sim_{C'} 0$, and $X_0$ is reduced normal crossings. 
- [x] What are the log monodromies for Kulikov degenerations. ✅ 2023-01-17
	- [ ] Define $N \da \log T = (T-I) - {1\over 2}(T-I)^2 + \cdots$
		- [ ] Type I $\implies N = 0$.
		- [ ] Type II $\implies N\neq 0, N^2 = 0$.
		- [ ] Type III $\implies N \neq 0, N^2 \neq 0, N^3=0$.
	- [ ] $N$ is always integral and of the form $$Nx = (x\cdot \lambda)\delta - (x\cdot \delta)\lambda,\qquad \delta\in H^2(X_t; \ZZ), \lambda \in \delta^\perp/\delta$$
- [x] What is a **recognizable divisor**? ✅ 2023-01-17
	- [ ] A canonical choice of polarizing divisor $R$ for $U \subset F_\rho$ is **recognizable** if 
		- [ ] For every Kulikov surface $X_0$ of Type I, II, or III which smooths to some $\rho$-markable $\mathrm{K} 3$ surface, 
		- [ ] There is a divisor $R_0 \subset X_0$ such that on any smoothing into $\rho$-markable K3 surfaces $X \rightarrow(C, 0)$ with $C^* \subset U$, 
		- [ ] The divisor $R_0$ is, up to the action of$\Aut^0\left(X_0\right)$, the flat limit of $R_t$ for $t \neq 0 \in C^*$.
- [x] What is the importance of a regognizable divisor? ✅ 2024-01-29
	- [ ] If $R$ is recognizable, then $\bar{F_\rho}^{\slc}$ is a semitoroidal compactification of $F_\rho$ for a unique semifan $\mcf_R$.
- [x] What is the Baily-Borel compactification associated to a lattice $N$ with isometry $\rho$? ✅ 2023-02-14
	- [ ] $\DD_N \da \ts{x\in \PP(N\tensor \CC)\st x^2=0, \abs{x} > 0}$ a type IV domain
	- [ ] $\DD_\rho = \ts{x\in \PP(N\tensor \CC)^{\zeta_n} \st \abs{x} > 0}$ a Type I subdomain
	- [ ] $\tilde \Gamma_\rho \da Z_{\Orth(N)}(\rho)\actson \DD_\rho$, and also for any finite index subgroup $\Gamma$.
	- [ ] Set $\bar{\DD_\rho}$ the closure of the compact dual (dropping $\abs{x} > 0$)
	- [ ] Define $\bar{\DD_\rho}^\mathrm{BB} = \DD_\rho \union \Union_{J} B_J$ by adding *rational boundary components*, usually $\HH \disjoint(-\HH)$ or points.s
	- [ ] BB compactification is $\overline{\DD_\rho/\Gamma}\da \bar{\DD_\rho}^{\mathrm{BB}}/\Gamma$.  



# Misc

## General

- [x] How is a variety defined scheme-theoretically? ✅ 2023-01-10
	- [ ] Separated geometrically integral scheme of finite type over $k$.

- [x] When is a variety smooth? ✅ 2023-01-10
	- [ ] If $\Omega_{X}$ is locally free of rank $\dim X$
	- [ ] Equivalently, finite type schemes are smooth iff geometrically regular.



- [x] What is the lattice associated to a K3? ✅ 2023-01-28
	- [ ] The intersection pairing: $$\begin{align*}H^2(X; \ZZ)\tensorpower{\ZZ}{2} &\to \ZZ \\ \alpha\tensor \beta \mapsto \int_X \alpha\wedge \beta\end{align*}$$
	- [ ] Nondegenerate and unimodular by Poincare duality.


- [x] What type of curve is the resolution of an $A_1$ singularity? ✅ 2023-01-28
	- [ ] $D\cong \PP^1$ with $D^2 = -2$.
- [x] What is the irregularity of $S$? ✅ 2023-01-27
	- [ ] $q(S) \da h^1(\OO_S)$ or $h^0(\Omega^1)$, so $h^{1, 0}$ or $h^{0, 1}$.
- [x] What is the geometric genus of $S$? ✅ 2023-01-27
	- [ ] $h^{0}(\Omega^2)$, so $h^{2, 0}$.
- [x] Describe $\NS(X)$ and $\Pic(X)$ in terms of divisors. ✅ 2023-01-29
	- [ ] $\NS(X) = \CDiv(X)/\sim$ modulo algebraic equivalence.
	- [ ] $\Pic(X) = \CDiv(X)/\sim$ modulo linear equivalence.
	- [ ] $\CH_k(X) = Z_k(X)/\sim$ modulo rational equivalence
- [x] What is a topological interpretation of $K_X$? ✅ 2023-02-07
	- [ ] $-c_1(\T_X)$.
- [x] Give a classification of elliptic fibers. ✅ 2023-02-07
	- [ ] ![](attachments/2023-02-07-ell-fib.png)




- [x] Define the Néron-Severi group. ✅ 2023-01-10
	- [ ] $\NS(X) \da \Pic(X) / \Pic^0(X)$, line bundles algebraically equivalent to zero.

- [x] What does it mean for a line bundle $L$ to be numerically trivial? ✅ 2023-01-10
	- [ ] $L. \mcl = 0$ for all $\mcl\in \Pic(X)$, e.g. any $L\in \Pic^0(X)$.
- [x] What is $\mathrm{Num}(X)$? ✅ 2023-01-10
	- [ ] $\Pic(X)/\Pic(X)^\tau$. quotienting by the subgroup of numerically trivial line bundles.
	- [ ] A $\zmod$ with a nondegenerate symmetric integral pairing.

- [x] Give a bundle-theoretic condition for projectivity of surfaces. ✅ 2023-01-10
	- [ ] If $X$ is a smooth compact complex surface, $X$ is projective iff $\exists L\in \Pic(X)$ with $L^2 > 0$.
- [x] Define the linear system $\abs{L}$ for $L\in \Pic(X)$. ✅ 2023-01-13
	- [ ] $\PP H^0(L)$ or all $L'\in \Div(X)^\eff$ with $L'\sim L$ linearly equivalent.
- [x] What is the adjunction formula for a curve $C$ on a surface $X$? ✅ 2023-01-13
	- [ ] $\omega_C = (\omega_X \tensor \OO(C))\mid_C$.
- [x] Discuss the Hodge conjecture. ✅ 2023-01-19
	- [ ] The even part $\bigoplus H^{2 k}(X, \mathbf{Q})$ contains all algebraic classes, i.e. classes obtained as fundamental classes $[Z]$ of subvarieties $Z \subset X$. It is not difficult to see that $[Z]$ is an integral class, i.e. that it comes from an element in $H^{2 k}(X, \mathbf{Z})$, and that it is contained in $H^{k, k}(X)$. 
	- [ ] The Hodge conjecture asserts that the space spanned by those is determined entirely by the Hodge structure itself.
	- [ ] (Hodge conjecture). For a smooth projective variety $X$ over $\mathbf{C}$ the subspace of $H^{2 k}(X, \mathbf{Q})$ spanned by all algebraic classes $[Z]$ coincides with the space of Hodge classes, i.e. $$H^{2 k}(X; \mathbf{Q}) \cap H^{k, k}(X)=\gens{[Z] \st Z \subset X}_{\mathbf{Q}}$$

- [x] What is an ample bundle? ✅ 2023-02-14
	- [ ] Some power is very ample 
	- [ ] Very ample: bpf/gg and $\phi_{\abs L}: X\to \PP^{h^0(L) - 1}$ is a closed immersion.
	- [ ] Tensoring high powers with a coherent sheaf gives many global sections.
- [x] When is $\OO_{\PP^n}(d)$ ample? Very ample? ✅ 2023-02-14
	- [ ] BPF iff $d\geq 0$
	- [ ] Very ample iff ample iff $d\geq 1$.
- [x] Draw a picture of 3-dimensional vanishing cycles. ✅ 2023-02-12
	- [ ] ![](attachments/2023-02-12-vanishing.png)
- [x] What is a unimodular lattice? ✅ 2023-01-17
	- [ ] Integral and generated by columns of a matrix $M$ with $\det M = \pm 1$.
	- [ ] Volume of fundamental domain is 1.
- [x] What is an even lattice? ✅ 2023-01-17
	- [ ] $\norm{x}_L \in 2\ZZ$ for any $x\in L$.
- [x] What are Type I and Type II lattices? ✅ 2023-01-28
	- [ ] Type II: even unimodular.
	- [ ] Type I: other ("odd") unimodular.
- [x] What are the rank, signature, and discriminant of $U$? ✅ 2023-01-28
	- [ ] $\rank U = 2, \sgn U = (1,1), \disc U = 1$.
- [x] What are the rank, signature, and discriminant of $E_8$? ✅ 2023-01-28
	- [ ] $\rank E_8 = 8, \sgn E_8 = (8, 0), \disc E_8 = 1$.
- [x] What is a spacelike vector? Lightlike? ✅ 2023-03-27
	- [ ] Spacelike: $\norm{v} > 0$ or $\norm{v} \lt 0$.
	- [ ] Lightlike: $v^2 = 0$ (isotropic) with $v\neq 0$; two connected components of such (future/past).

- [x] What is the spinor norm of a reflection $\tau_v$? ✅ 2023-03-28
	- [ ] $\tau_v(x) = x - 2xv/v^2$ has spinor norm $\norm{v}_\Spin \da Q(v) = v^2/2\in k\units/(k\units)^\square$.

- [x] Discuss classification of 2-elementary lattices. ✅ 2023-02-07
	- [ ] Indefinite 2-elementary lattices classified by $(r, a,\delta)$: 
		- [ ] Rank, $\ZZ/2\ZZ\dash$rank, and coparity.
	- [ ] Alternatively: $(g,k,\delta)$ where $g= 11 - {r+a\over 2}$ and $k={r-a\over 2}$; $\Fix(\iota)$ has $k+1$ connected components of genera summing to $g$.
	- [ ] Nikulin's classifications: 75 K3 lattices 
		- [ ] K3 = hyperbolic, admitting a primitive embedding into $\Lambda_{\K3}$.
- [ ] What is the degree of a del Pezzo?
## K3s

- [x] What are the two most widely studied classes of K3s? ✅ 2023-01-17
	- [ ] Sextic double planes: birational to $X\to \PP^1$ a degree 2 cover branched over a sextic curve (equivalently a sextic hypersurface in $\WP(1,1,1,3)$.)
	- [ ] Quartic hypersurfaces: 
		- [ ] $X\subseteq \PP^3$ a quartic hypersurface
		- [ ] $X\to \PP^1 \times \PP^1$ a double cover branched over a bidegree $(4, 4)$ curve.
		- [ ] $X\to C$ an elliptic fibration over the twisted cubic.

- [x] Give a birational definition of a K3. ✅ 2023-01-27
	- [ ] A compact complex surface with $K_S \sim 0$ and $q(S) = 0$.
- [x] What is an **algebraic** K3 surface? ✅ 2023-01-10
	- [ ] Smooth complete variety $X$ over a field $k$, $\dim X = 2$, where $\omega_{X} \cong \OO_X$ and $h^1(\OO_X) = 0$.
- [x] When is a K3 surface projective? ✅ 2023-01-10
	- [ ] Always: any smooth complete surface is projective.
- [x] What is a **complex** K3 surface? ✅ 2023-01-10
	- [ ] A compact connected complex manifold, $\dim X = 2$, with $\Omega^2_X \cong \OO_X$ and $h^1(\OO_X) = 0$.
	- [ ] If $X$ is an algebraic K3 then $X^\an$ is a (projective) complex K3.
- [x] Why is $X^\an$ projective when $X$ is an algebraic K3? ✅ 2023-01-10
	- [ ] GAGA: the ideal sheaf $I$ of $X\subseteq \PP^n$ is coherent analytic and thus isomorphic to an algebraic ideal sheaf.
- [x] Are all complex K3s projective? ✅ 2023-01-10
	- [ ] No: most complex tori $X=\CC^2/\Gamma$ are not.
- [x] What is $\kappa(X)$ for $X$ a K3? ✅ 2023-01-17
	- [ ] $\kappa(X) = 0$, one of four such classes of minimal surfaces.
- [x] Where do K3s fit into the classification of surfaces? ✅ 2023-01-17
	- [ ] Positively curved: del Pezzo
	- [ ] Flat: K3s (generally CYs and hyperkahlers)
	- [ ] Negatively curved: general type.
- [x] Give examples of K3s ✅ 2023-01-27
	- [ ] Any smooth quartic in $\PP^3$.
	- [ ] Complete intersections of type (2,3) or (2,2,2)
	- [ ] $S\to \PP^2$ 2x ramified over a sextic curve.
- [x] Discuss the period domain. ✅ 2023-01-28
	- [ ] $\CC\omega_X \in \PP H^2(X;\CC)$. 
	- [ ] Take a marking $\Phi: H^2(X;\ZZ)\to \lkt$ and define $$\Omega = \ts{[\omega]\in \PP(\lkt\tensor\CC) \st \omega^2 = 0, \abs{\omega} > 0}, \quad \abs{\omega} \da \omega.\overline{\omega}$$
	- [ ] Can write as $$\Omega \cong \ts{\tv{z_0:\cdots:z_{21}}\in \PP^{21}\st z_0^2+z_1^2 + z_2^2 - z_3^2 -\cdots - z_{21}^2 = 0,\, \sum_{i=0}^{21} \abs{z_i}^2 > 0}$$ which is an open subset of a quadric in $\PP^{21}$ and thus 20-dimensional.
	- [ ] One has $\Phi(\CC\omega_X) \subseteq \Omega$; define this as the period domain of a marked K3.
	- [ ] Torelli gives surjectivity.
- [x] Discuss $\NS(X)$ for $X$ a K3. ✅ 2023-01-29
	- [ ] $\Pic^0(X) = 0$ so $\Pic(X) \cong \NS(X)$.
	- [ ] The sublattice of $H^{2}(X; \ZZ)$ generated by algebraic cycles.

- [x] What is a symplectic automorphism? ✅ 2023-01-29
	- [ ] $\sigma^* \omega_X = \omega_X$, i.e. preserves the symplectic form.
	- [ ] Possibly forces $\rank_\ZZ \Pic(X) \geq 8$.

- [x] What are symplectic, non-symplectic, and purely non-symplectic automorphisms? ✅ 2023-01-30
	- [ ] Symplectic: $g^*\omega_X = \omega_X$
	- [ ] Non-symplectic: $g^*\omega_X = \alpha(g)\omega_X$ where $\alpha\neq 0,1$.
	- [ ] Purely non-symplectic: $g^* \omega_X = \alpha(g)\omega_X$ where $\alpha(g) = \zeta_m$ a *primitive* root of unity $m=\abs{g}$.
- [x] Why are symplectic automorphisms important? ✅ 2023-01-30
	- [ ] Quotients are again K3.
	- [ ] Not the case for non-symplectic automorphisms!

- [x] What is the Hodge diamond for a K3? ✅ 2023-01-10
	- [ ] ![](attachments/2023-01-10-hodgek3diamond.png)
	- [ ] Use that $h^{p, q}(X) = 1$ for the corners by definition, $h^{0, 1}(X) = 0$, and $$2 h^0\left(\Omega_X\right)-h^1\left(\Omega_X\right)=\operatorname{ch}_2\left(\Omega_X\right)+4=4-\mathrm{c}_2\left(\Omega_X\right)=-20$$ so $h^1(\Omega_X) = 20$ since $h^0(\Omega_X) = 0$ by HRR.


- [x] Is the Hodge structure on a K3 polarizable? ✅ 2023-01-19
	- [ ] If algebraic, always. If not, there are counterexamples.

- [x] What are $q(X), p_g(X), \beta^+(X), \chi(X), c_1^2(X), c_2(X)$ for a K3? ✅ 2023-01-17
	- [ ] $q(X) = 0$
	- [ ] $p_g(X) = 1$
	- [ ] $\beta^+(X) = 3$
	- [ ] $\chi(X) = 2$
	- [ ] $c_1^2(X) = 0$
	- [ ] $c^2(X) =24$.

- [x] Define $\NS(X)$. ✅ 2023-01-17
	- [ ] $$\NS(X) \da H^{1, 1}(X) \intersect H^{2}(X; \ZZ)\subseteq H^2(X; \CC)$$

## Moduli of K3s

- [x] What is a Kulikov model? ✅ 2023-01-17
	- [ ] A semistable degeneration $\pi: \mcx\to \Delta$ with $\omega_{\mcx}\cong \OO_{\mcx}$.
- [x] What is a semistable degeneration? ✅ 2023-01-17
	- [ ] $\pi:\mcx \to \Delta$ with $\mcx$ smooth and central fiber reduced normal crossings.
- [x] What is the dual graph of a degeneration? ✅ 2023-01-17
	- [ ] Let $S_0=\bigcup V_i$ be the central fibre in a semistable degeneration. Define the dual graph $\Gamma$ of $S_0$ as follows: $\Gamma$ is a simplicial complex whose vertices $P_1, \ldots, P_r$ correspond to the components $V_1, \ldots, V_r$ of $S_0$; the $k$-simplex $\left\langle P_{i_0}, \ldots, P_{i_k}\right\rangle$ belongs to $\Gamma$ if and only if $V_{i_0} \cap \cdots \cap V_{i_k} \neq \emptyset$.
- [x] What is the classification of Kulikov models? ✅ 2023-01-25
	- [ ] Write $\pi: \mcx \to \Delta$ for a semistable degeneration with $\omega_{\mcx}\cong \OO_{\mcx}$ where all components of $\mcx_0$ are Kahler.
		- [ ] Type I: $\mcx_0$ is a smooth K3.
			- [ ] No double curves, no triple points.
		- [ ] Type II: $\mcx_0$ is a chain of elliptic ruled components with rational surfaces at each end where all double curves are smooth elliptic
			- [ ] Double curves but no triple points
		- [ ] Type III: $\mcx_0$ is rational surfaces meeting along rational curves forming cycles in each component and $\Gamma$ is a triangulation of $S^2$.
			- [ ] Triple points.
## Enriques

- [x] What is an Enriques surface? ✅ 2023-01-27
	- [ ] Exactly quotients of K3s by fixed point free involutions (non-symplectic)
	- [ ] Smooth projective $Y$ with $2K_Y \sim 0$ and $h^1(\OO_Y) = h^2(\OO_Y) = 0$; the canonical induces an unramified 2-to-1 cover $X\to Y$ for $X$ a K3.
- [x] What are the dimensions of moduli of all and just algebraic K3s? Enriques ✅ 2023-02-01
	- [ ] All K3s: 20-dimensional
	- [ ] Algebraic K3s: 19-dimensional
	- [ ] Enriques: 10-dimensional.
- [x] What is the divisibility of a vector? ✅ 2023-03-28
	- [ ] For $v\in T$, $\div(v) = n$ where $v.T = n\ZZ$.
	- [ ] If $0 \neq x \in L$, we let $\operatorname{div}(x)$ denote the positive generator of the ideal $(x, L)$ 

- [x] Describe ADE dynkin diagrams. ✅ 2023-01-28
	- [ ] ![](attachments/2023-01-28-ade-dynkin.png)

- [ ] What is the Enriques lattice?
- [x] What moduli space do we construct? ✅ 2024-01-31
	- [ ] $F_{En, 2}$ pairs $(Z, [\mcl])$ where $Z$ is Enriques with ADE singularities and $[\mcl] \in \Pic(Z)/C_2$. BB compactification described by Sterk.
- [x] How do we set up $\overline{ F_{En, 2} }$? ✅ 2024-01-31
	- [ ] Use $\abs {\mcl\tensorpower{}{2}}$ which is bpf to construct $\rho: Z\to W$ with $W$ a quartic del Pezzo with $4A_1$ or $A_3 + 2A_1$ singularities; use the ramification divisor.
- [x] What is our main result? ✅ 2024-01-31
	- [ ] Theorem 1.1. The normalization of $\bar{F}_{\mathrm{En}, 2}$ is a semitoroidal compactification $\bar{F}_{\mathrm{En}, 2}^{\mathfrak{F}}$ corresponding to a collection $\mathfrak{F}=\left\{\mathfrak{F}^k\right\}_{k=1,2,3,4,5}$ of explicit semifans, one for each 0 -cusp of $F_{\mathrm{En}, 2}$, and it is dominated by a toroidal compactification $\bar{F}_{\mathrm{En}, 2}^{\mathrm{cox}}$ for a collection $\mathfrak{F}_{\text {cox }}=\left\{\mathfrak{F}_{\text {cox }}^k\right\}_{k=1,2,3,4,5}$ of Coxeter fans.
	- [ ] Classify stable pairs by ADE + BC diagrams.
- [x] How do we use IASs? ✅ 2024-02-02
	- [ ] Kulikov degenerations (K-trivial, semistable) of K3s are encoded by IASs
	- [ ] Differ on overlaps by $\SL_2(\ZZ)\semidirect \ZZ^2$
	- [ ] Complete lattice triangulations describe dual complexes $\Gamma(\mcx_0)$ of Kulikov degenerations.
	- [ ] We construct such with two commuting involutions (Enriques, del Pezzo)
- [x] Describe $F_{(2, 2, 0)}$. ✅ 2024-02-02
	- [ ] K3s of degree 4 with a del Pezzo involution $\iota_{dP}$ corresponding to branched double covers of $\PP^1\times \PP^1$ branched along a curve $C \in \abs{\OO(4, 4)}$.
	- [ ] We immerse $F_{En, 2} \injects F_{(2,2,0)}$ and use AE's theory.
- [x] Describe $F_{(10, 10, 0)}$. ✅ 2024-02-02
	- [ ] Moduli of unpolarized K3 surfaces.
- [x] How many cusps are in $F_{En, 2}$? ✅ 2024-02-02
	- [ ] Five 0-cusps, nine 1-cusps.
- [x] What are the two lattices that occur at the 0-cusps of $F_{(2, 2, 0)}$? ✅ 2024-02-02
	- [ ] $U\oplus E_8^2$
	- [ ] $U(2) \oplus E_8^2$.
- [x] What is the main geometric diagram? What are $\iota_{En}, \iota_{dP}$? ✅ 2024-02-02
	- [ ] ![](2024-01-31.png)
	- [ ] $X$ is a K3, $Z$ is an Enriques, $\psi = \cdot/\iota_{En}$
	- [ ] $Y = \PP^1\times \PP^1$, $W = Y/\tau$ is a quartic del Pezzo where $\phi = \cdot/\tau$.
	- [ ] $X\to Y$ is the double branched cover branched over $B\in \abs{-2K_Y}$ not passing through $\Fix(\tau)$, with $Y = X/\iota_{dP}$ and $\pi = \cdot/\iota_{dP}$
	- [ ] $\iota_{En}$ is one lift of $\tau$
	- [ ] $\iota_{dP}$ exists on $X$ via the branched cover construction.
- [x] Why use the main diagram at all? ✅ 2024-02-02
	- [ ] Understand $X \to Z$ the Enriques quotient by passing to $Y\to W$ which is entirely toric.
- [x] How do we get equations for the K3 surfaces in the main diagram? ✅ 2024-02-02
	- [ ] Take $f(x, y)$ bidegree $(4, 4)$ in $Y$
	- [ ] Take pyramid $P$ over our polytope $Q$ with vertex at $(2 ,2, 2)$.
	- [ ] Find $X$ as the hypersurface $z^2 + f(x, y)$ in the toric variety for $P$.
- [x] What dimensional family of K3s do we produce? ✅ 2024-02-02
	- [ ] Get equations, find 13 $\tau\dash$invariant monomials in $P$, coefficients give points in $U\subseteq \PP^{12}$ and modding by symmetry yields $U/(D_4\semidirect \GG_m^2)$ which is dimension 10.
- [x] What are our three involutions? Which are (non)symplectic? ✅ 2024-02-02
	- [ ] ![](2024-01-31-1.png)
	- [ ] Nonsymplectic: $\iota_{dP}, \iota_{En}$.
- [x] How do we get equations for the Enriques surfaces in the main diagrams? ✅ 2024-02-02
	- [ ] $Z$ is also a hypersurface in the toric variety for $P$, but with respect to the sublattice $\ZZ^3_{ev} = \ts{a+b+c \in 2\ZZ}$.
	- [ ] Use the same polynomial $z^2 + f(x, y)$ whose monomials lie in this sublattice.
- [x] What is the history of study of these moduli spaces? ✅ 2024-02-02
	- [ ] Horikawa (78) studies possibilities for these equations $f(x, y)$ and maps $U\to D/\Gamma$
	- [ ] Shah (81) describes a GIT compactification. Comparison is hard.
- [ ] Give an example of $f(x, y)$.
- [x] Describe our three period domains ✅ 2024-02-02
	- [ ] ![](2024-01-31-2.png)
	- [ ] $T_{En}= U + E_{10}(2) = (12, 10, 0)_2 \injects T_{dP} = U + U(2) + E_8^2 = (20, 2, 0)_2$.
	- [ ] $\Gamma_{En} = \Orth(T_{En})$
	- [ ] $\Gamma_{En, 2}$ is the image in $\Orth(T_{En})$ of $\ts{g\in \Orth(\Lambda_{K3}) \st g(h) = h, [g, I_{En}] = 0}$ where $h = e+f \in U(2) \leq E_{10}(2) \leq T_{En}$.
- [x] What is the coparity? ✅ 2024-02-02
	- [ ] $\delta = 0 \iff q_{\Lambda}: \Lambda^* / \Lambda \rightarrow \frac{1}{2} \mathbb{Z} / 2 \mathbb{Z}, q_{\Lambda}(x)=x^2 \bmod 2 \mathbb{Z} \text { is } \mathbb{Z}$-valued and $\delta = 1$ otherwise.
- [x] What is the cusp diagram for $F_{(2, 2, 0)}$? Divisibilities? ✅ 2024-02-02
	- [ ] ![](2024-01-31-3.png)
	- [ ] $\div(e) = 2, 1$ respectively.
- [x] What are the lattices at 0-cusps in cusp diagrams? ✅ 2024-02-02
	- [ ] $e^\perp/e$ for the isotropic vector.
- [x] What is the cusp diagram for $F_{(10, 10, 0)}$? Divisibilities? ✅ 2024-02-02
	- [ ] ![](2024-01-31-4.png)
	- [ ] $\div(e)  = 1,2$ respectively, corresponding to $\RP^2$ and $\DD^2$.
	- [ ] 0,1,0,1.
- [x] What is the cusp diagram for $F_{En, 2}$? ✅ 2024-02-02
	- [ ] Computed by Sterk.
	- [ ] ![](2024-01-31-5.png)
	- [ ] Correspondence determined by divisibility.
- [x] What is the hyperbolic space attached to a lattice? ✅ 2024-02-02
	- [ ] Let $\Lambda$ be a hyperbolic lattice. Let $\mathcal{C}$ be the component of the set $\left\{v \in \Lambda_{\mathbb{R}} \mid\right.$ $\left.v^2>0\right\}$, containing a fixed class $h$ with $h^2>0$. Let $\mathcal{H}=\mathbb{P C}$ be the corresponding hyperbolic space. A vector $v \neq 0$ with $v^2=0$ in the closure of $\mathcal{C}$ defines a point on the sphere at infinity of $\mathcal{H}$. Let $\overline{\mathcal{C}}$ denote the closure of $\mathcal{C}$.
- [x] What is the reflection formula? ✅ 2024-02-02
	- [ ] $w_\alpha(v)=v-\frac{2(\alpha \cdot v)}{\alpha \cdot \alpha} \alpha$
- [x] What is a root? ✅ 2024-02-02
	- [ ] A root is a vector $\alpha$ with $\alpha^2<0$ such that $w_\alpha(\Lambda)=\Lambda$, equivalently such that $2 \operatorname{div}(\alpha) \in(\alpha \cdot \alpha) \mathbb{Z}$.
- [x] What is the Coxeter diagram attached to a lattice? ✅ 2024-02-02
	- [ ] Define $\mathfrak{C}=\left\{v \in \mathcal{C} \mid \alpha_i \cdot v \geq 0 \text { for simple roots } \alpha_i\right\}$ a fundamental chamber for $W(\Lambda)$
	- [ ] Walls encoded in Coxeter group/diagram.
- [x] What are the Coxeter diagrams for $F_{(2, 2, 0)}$? ✅ 2024-02-02
	- [ ] ![](2024-01-31-6.png)
- [x] What are the Coxeter diagrams for $F_{(10, 10 , 0)}$? ✅ 2024-02-02
	- [ ] ![](2024-01-31-7.png)
- [x] What is a folded vector? ✅ 2024-02-02
	- [ ] ![](2024-01-31-8.png)
- [ ] What moduli space and cusps does $T_{En}$ correspond to?
- [ ] What moduli space and cusps does $T_{dP}$ correspond to?
- [x] What is the correspondence between points of Coxeter polytopes and subdiagrams? ✅ 2024-02-02
	- [ ] Parabolic: ideal vertex
	- [ ] Elliptic: interior vertex
- [x] What is the motivating semifan story for $\Ag$? ✅ 2024-02-02
	- [ ] For a ppav $(X, L)$, choose the unique theta divisor $\Theta\in \abs{L}$ and take $(X, \eps \Theta)$ for the stable pair. Then $\overline{\Ag}^\nu \cong \Ag^{\mcf}$, a semitoroidal compactification where $\mcf$ is the second Voronoi fan.
- [x] What is the data of a fan for a toroidal compactification of $D_L/\Gamma$? ✅ 2024-02-02
	- [ ] A $\Gamma\dash$invariant polyhedral decomposition of the positive cone in $L$.
- [x] What is Mumford's toric description of degenerations of PPAVs? ✅ 2024-02-02
	- [ ] Let $M$ be a fixed lattice, and $N=M^*$ be its dual. The Voronoi fan $\mathfrak{F}^{\text {vor }}$ is supported on the rational closure $\overline{\mathcal{C}}$ of the cone of positive definite symmetric forms $\mathcal{C}=\{Q: M \times M \rightarrow \mathbb{R}, Q>0\}$, 
	- [ ] equivalently of positive symmetric maps $f_Q: M \rightarrow N_{\mathbb{R}}$. 
	- [ ] Classically, a positive semi-definite quadratic form $Q$ defines two dual polyhedral decompositions of $M_{\mathbb{R}}$, periodic with respect to translation by $M$ : Voronoi and Delaunay.
	- [ ] A one-parameter degeneration $\left(X_t, \epsilon \Theta_t\right)$ of ppavs with an integral monodromy vector $Q \in \mathcal{C}$ can be written as a $\mathbb{Z}^g$-quotient of an infinite toric variety whose fan in $\mathbb{R} \oplus N_{\mathbb{R}}$ is the cone over a shifted Voronoi decomposition $\left(1, \ell+f_Q(\operatorname{Vor} Q)\right)$.
	- [ ] The quotient $N_\RR /f_Q(M)$ is an integral-affine torus $X_{\text {trop }}=$ $N_{\mathbb{R}} / f_Q(M) \simeq\left(S^1\right)^g$ with a tropical divisor $\Theta_{\text {trop }}$ on it. 
	- [ ] Then $\Theta_{\text {trop }}$ induces a cell decomposition of $X_{\text {trop }}$ which is the dual complex of the singular central fiber $\left(X_0, \epsilon \Theta_0\right)$. The normalization of each component of $X_0$ is a toric variety, whose fan is modeled by the corresponding vertex of $\Theta_{\text {trop }}$.
- [x] What is the Gross-Siebert program? ✅ 2024-02-02
	- [ ] Understand mirror symmetry near maximally unipotent degenerations of CY varieties via tropical/integral-affine geometry.
- [x] Why are IAS's more complicated for K3s than PPAVs? ✅ 2024-02-02
	- [ ] An $\IAS^2$ must have singularities, while an $\mathrm{IA}(S^1)^g$ is nonsingular.
- [x] How are $\IAS^2$s related to Voronoi and Delaunay decompositions? ✅ 2024-02-02
	- [ ] For each vector in a connected component $\mathcal{C} \subset\left\{\vec{a} \in N \otimes \mathbb{R} \mid \vec{a}^2>0\right\}$, we construct an $\operatorname{IAS}^2 B(\vec{a})$ with up to 24 singularities, together with an integral-affine divisor $R_{\mathrm{IA}}$. 
	- [ ] As $\vec{a} \in \mathcal{C}$ varies continuously, so does the pair $\left(B(\vec{a}), R_{\mathrm{IA}}\right)$. Dual to the polyhedral decomposition of $B(\vec{a})$ induced by $R_{\mathrm{IA}}$ is a discrete subdivision of $S^2$ with 24 singularities. The set of the dual subdivisions is discrete. 
	- [ ] Thus, the family of $\left(B(\vec{a}), R_{\mathrm{IA}}\right)$ varying continuously over $\mathcal{C}$ are analogues of $\operatorname{Vor} Q$, and the dual subdivisions are the analogues of $\operatorname{Del} Q$.
- [x] What is the Picard-Lefschetz transformation? ✅ 2024-02-02
	- [ ] $T: H^2(\mcx_t; \ZZ)\selfmap$
- [x] Discuss unipotency of the Picard-Lefschetz operator. ✅ 2024-02-02
	- [ ] Landman '03: $T$ is quasi-unipotent with unipotency index at most $2$, so there is some $k$ such that $(T^k -I)^3 = 0$.
	- [ ] If $\mcx \to \DD$ is semistable (e.g. for Kulikov models), then $k=1$ and $T$ is unipotent.
- [x] Describe the monodromy operator. ✅ 2024-02-02
	- [ ] $N:=\log T=(T-I)-\frac{1}{2}(T-I)^2$
	- [ ] Note $N$ is nilpotent, and $T=I \iff N = 0$.
- [x] Describe the KPP theorem on types of degenerations of K3s. ✅ 2024-01-31
	- [ ] Let $N$ be the monodromy operator and $\mcx\to \DD$ a Kulikov model. Then
	- [ ] Type I: $N = 0$ and $\mcx_0$ is a K3.
	- [ ] Type II: $N^2=0$ and $\mcx_0$ is a chain of smooth rational surfaces  $V_i \cong E \times \PP^1$ for a fixed elliptic curve, capped on each end by smooth **elliptic** ruled surfaces, and $\Gamma(\mcx_0) = [0, 1]$.
	- [ ] Type III: $N^3 = 0$ and $\mcx_0$ is a union of smooth **rational** surfaces, all double curves are cycles of rational curves, and $\Gamma(\mcx_0) = \IAS^2$.
	- [ ] Proof: uses the Clemens-Schmidt exact sequence
- [x] State the KPP theorem precisely and cite. ✅ 2024-01-31
	- [ ] Kul77 and PP81: Let $\mathcal{X} \rightarrow(C, 0)$ be a flat proper family over a germ of a curve such that the fibers of $\mathcal{X}^* \rightarrow C^*=C \backslash 0$ are projective $K 3$ surfaces. 
	- [ ] Then there is a finite ramified base change $\left(C^{\prime}, 0\right) \rightarrow(C, 0)$ and a birational modification $\mathcal{X}^{\prime} \rightarrow \mathcal{X} \times{ }_C C^{\prime}$ such that $\pi: \mathcal{X}^{\prime} \rightarrow C^{\prime}$ is semistable (a smooth threefold with $\mathcal{X}_0^{\prime}$ a reduced normal crossing divisor) with $\omega_{\mathcal{X}^{\prime} / C^{\prime}} \simeq \mathcal{O}_{\mathcal{X}^{\prime}}$.
- [x] What is the charge of an anticanonical pair? ✅ 2024-01-31
	- [ ] $Q(V, D):=12-\sum\left(D_i^2+3\right)$
- [x] What is an anticanonical pair? ✅ 2024-01-31
	- [ ] $(V, D)$ with $D \in \abs{-K_V}$.
- [x] State the conservation of charge theorem. ✅ 2024-01-31
	- [ ] Friedman-Miranda 83: Let $\mathcal{X} \rightarrow(C, 0)$ be a Type III Kulikov degeneration. Then $\sum_{i=1}^n Q\left(V_i, \sum_j D_{i j}\right)=24$. In particular, at most 24 components of $\mathcal{X}_0$ are non-toric.
- [x] Define the vanishing cycle and monodromy invariant. ✅ 2024-01-31
	- [ ] By FM83, in Types II, III, the log monodromy takes the form $N(x) = (x \cdot \delta) \lambda-(x \cdot \lambda) \delta$ where $\delta, \lambda \in H^2(\mcx_t; \ZZ)$.
	- [ ] $\delta$ is the vanishing cycle, and $\lambda \in\delta^\perp/\delta$ is the monodromy invariant.
- [x] Describe how to obtain a primitive isotropic lattice from a Kulikov degeneration. ✅ 2024-01-31
	- [ ] Type III: $I = \gens{\delta}_\ZZ$.
	- [ ] Type II: $J = \gens{\delta, \lambda}^{\mathrm{sat}}_\ZZ$
- [x] What is an $\IAS$ for $S$ a surface? ✅ 2024-01-31
	- [ ] An integral-affine structure on a real surface $S$ is a collection of charts from $S$ to $\mathbb{R}^2$ such that the transition functions lie in $\mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{R}^2$.
- [x] Give a pictorial example of an $I_1$ singularity. ✅ 2024-01-31
	- [ ] ![](2024-01-31-10.png)
	- [ ] The dashed line is a monodromy invariant ray.
	- [ ] The matrix in this basis is $\matt {1}{-1}{0}{1}$.
- [x] Give a precise definition of an $\IAS^2$. ✅ 2024-01-31
	- [ ] An integral-affine sphere $B$, or IAS ${ }^2$ for short, is a sphere $B=S^2$ and a finite set $\left\{p_1, \ldots, p_n\right\} \in B$ such that $B \backslash\left\{p_1, \ldots, p_n\right\}$ has a non-singular integral-affine structure, and a neighborhood of each $p_i$ is modeled by some integralaffine singularity $I\left(n_1 \vec{v}_1, \ldots, n_k \vec{v}_k\right)$.
- [x] What is a generic $\IAS$? ✅ 2024-01-31
	- [ ] An IAS ${ }^2$ is generic if it has 24 distinct $I_1$ singularities.
- [x] How are $\IAS$ related to symplectic geometry? ✅ 2024-01-31
	- [ ] Sym 03: Let $(S, \omega)$ be a smooth symplectic 4-manifold. Given a Lagrangian torus fibration $\mu:(S, \omega) \rightarrow B$ with only nodal singularities, the base $B$ inherits a natural integral-affine structure with an $I_n$ singularity under a necklace of $n$ two-spheres:
- [x] State the monodromy theorem. ✅ 2024-01-31
	- [ ] Theorem (AET23). Let $B$ be a triangulated generic $\mathrm{IAS}^2$.
	- [ ] Let $\mathcal{X} \rightarrow C$ be a type III Kulikov degeneration such that $\Gamma\left(\mathcal{X}_0\right)=B$.
	- [ ] Let $\mu:(S, \omega) \rightarrow B$ be a Lagrangian torus fibration over the same $B$. 
	- [ ] Then there exists a diffeomorphism $\phi: S \rightarrow \mathcal{X}_t$ to a nearby fiber $t \neq 0$ such that
		- [ ] $\phi_* f=\delta$, where $f = [\mu\inv(p)]$ is a fiber class over an $I_1$ singularity.
		- [ ] $\phi_*[\omega]=\lambda$ in $\delta^{\perp} / \delta \otimes \mathbb{R}$.
- [x] What is the importance of the monodromy theorem? ✅ 2024-02-02
	- [ ] Verifies predictions of SYZ mirror symmetry: the Picard-Lefschetz transformation is given by $(\wait)\cup s$ where $s$ is a section of the SYZ fibration. 
	- [ ] GS10 computes the monodromy of the Picard-Lefschetz transform of a toric degeneration of Calabi-Yau varieties in terms of cup product with the **radiance obstruction**, a cohomology class (in a local system) canonically associated with an $\IAS$. 
	- [ ] Radiance obstruction first studied in GH84.
- [x] What does the monodromy invariant measure? ✅ 2024-02-02
	- [ ] $\lambda^2$ is the number of triple points in $\mcx_0$.
- [x] Write the K3 period domain in terms of $h$. ✅ 2024-02-02
	- [ ] $\mathbb{D}=\mathbb{D}_{2 d}:=\mathbb{P}\left\{x \in h^{\perp} \otimes \mathbb{C} \mid x \cdot x=0, x \cdot \bar{x}>0\right\}$
- [x] Consider a toroidal compactification $\overline{F_{2d}}^{\mcf}$. Describe what the data of $\mcf_e$ is at a cusp correspond to an isotropic vector $e \leq h^{\perp \lkt}$. ✅ 2024-02-02
	- [ ] Take $L_{2d, e} \da e^{\perp} / e$, note $\signature L_{2d, e} = (1, 18)$.
	- [ ] Set $\Gamma_e \da \Stab_\Gamma(e)/U_e$ where $U_e$ is its unipotent subgroup
	- [ ] Set $\mathfrak{C}_e \da L_{2d, e}^{\geq 0}$ to be the positive cone.
	- [ ] Set $\overline{\mathfrak{C}_e}$ to be its rational closure.
	- [ ] Then $\mcf_e$ is a collection of closed, convex, rational polyhedral cones in $\overline{\mathfrak{C}_e}$ closed under taking intersections and faces, such that:
		- [ ] $\supp \mcf_e = \overline{\mathfrak{C}_e}$ and $\mcf_e$ is locally finite in $\mathfrak{C}_e$,
		- [ ] $\mcf_e$ is $\Gamma_e\dash$invariant with only finitely many orbits.
	- [ ] These cones define a (not necessarily projective) toric variety.
- [x] What is the rational closure of the positive cone? ✅ 2024-02-02
	- [ ] Add rational null rays on the boundary.
- [x] Characterize the fundamental chamber of $W(L)$ in the positive cone using inequalities. ✅ 2024-02-01
	- [ ] $P = \ts{\lambda \in \mathfrak{C}^{rc} \st \lambda \cdot \alpha_i \geq 0 \,\,\forall i}$.
- [x] What does (quasi)unipotency mean? ✅ 2024-02-01
	- [ ] For rings: $r-1$ is nilpotent.
	- [ ] For matrices $M$: $\chi_M(t) = (t-1)^N$ for some $N$, so all eigenvalues are 1.
	- [ ] For matrix groups: all elements are unipotent.
	- [ ] Quasi-unipotent: some power is unipotent, so all eigenvalues are roots of unity.
- [x] What is an elliptic subdiagram? A parabolic? For $G$ a Coxeter diagram corresponding to a lattice $L$. ✅ 2024-02-01
	- [ ] Let $G' \subseteq G$ be a subset of roots and take the vector subspace $\RR G' \subseteq L_\RR$. Let $\beta$ be the quadratic form on $L$.
	- [ ] Elliptic: $\ro{\beta}{\RR G'} < 0$ is negative definite.
	- [ ] Parabolic: $\ro{\beta}{\RR G'} \leq 0$ is negative semi-definite.
- [x] Discuss maximal parabolics. ✅ 2024-02-01
	- [ ] Maximal in the inclusion lattice of parabolic subdiagrams.
- [x] What kind of degenerations do maximal parabolics correspond to? Elliptics? ✅ 2024-02-01
	- [ ] Parabolics: Type II, $ADE$ subdiagrams
	- [ ] Elliptics: Type III, $\tilde A\tilde D\tilde E$ subdiagrams.
- [x] Why do we care about subdiagrams of Coxeter diagrams? ✅ 2024-02-01
	- [ ] Vin75 shows there is a contravariant correspondence between faces of $P$ and elliptic & parabolic subdiagrams.
- [ ] Define log canonical
- [ ] Define slc
- [ ] Define a stable pair
- [ ] Define a K-trivial semistable degeneration
- [ ] Define dlt
- [x] Define del Pezzo ✅ 2024-02-02
	- [ ] Ample of anticanonical
- [x] Geometrically describe a generic element of $F_{(2,2,0)}$. ✅ 2024-02-01
	- [ ] Degree 4 K3s with a del Pezzo involution, corresponding to double covers of $\PP^1\times\PP^1$ branched in $C\in \abs{\OO(4, 4)}$.
- [ ] What is $K_{\PP^1\times \PP^1}$?
	- [ ] $\OO_{\PP^1\times \PP^1}(-2, -2)$? Double check.
- [ ] 
# Example questions

- [ ] Give an example of an element in $F_{4, h.e.} = F_{(2,2,0)}$.
- [ ] Give an example of a line bundle on a quartic hyperelliptic K3.
- [ ] Give an explicit example of an Enriques surface.
- [ ] Give an example of a type B $ADE$ surface
- [ ] Give an example of type $C$ ADE surface.
- [ ] Give an example of a quartic del Pezzo surface.
- [ ] Give an example of a bidegree $(3,5)$ curve.
	- [ ] $f(x_0, x_1, y_0, y_1) = x_0^3y_1^5 + x_0^3y_0^2y_1^3 + x_0x_1^2 y_0 y_1 ^4$.
	- [ ] Double check!
- [ ] Draw the full square of the main diagram in terms of lattice polytopes.
- [ ] Give an example of an $A_1$ singularity.

# Attribution questions

- [x] Attribute: Kulikov degenerations are encoded by $\IAS^2$s. ✅ 2024-02-02
	- [ ] GHK15, Eng18, EF21
- [x] Attribute: the linear system $\mcm$ defines a double cover of a quartic del Pezzo ✅ 2024-02-02
	- [ ] CD89, CDL24
- [x] Attribute: the general theory of KSBA compactifications. ✅ 2024-02-02
	- [ ] Kol23
- [x] Attribute: the theory of $ADE$ surfaces. ✅ 2024-02-02
	- [ ] AT21
- [x] Attribute: general compactifications of K3 moduli spaces ✅ 2024-02-02
	- [ ] AE22, AE23
- [x] Attribute: 75 moduli spaces ✅ 2024-02-02
	- [ ] AE22, lattices classifed by Nikulin.
- [x] Attribute: the original study of $F_{En, 2}$. ✅ 2024-02-02
	- [ ] Ste91
- [x] Attribute: one can read off degenerations of K3s from Coxeter diagrams ✅ 2024-02-02
	- [ ] AET23, AE22
- [ ] Past work: classifying all KSBA stable limits
	- [ ] ABE22: what is the specific moduli space? Review
	- [ ] AE22: again, speciic moduli space?
- [x] Explain: Horikawa's contribution ✅ 2024-02-02
	- [ ] Hor78, analyzed sets of equations $f(x, y)$ and maps from $\PP^{12}$ to $D/\Gamma$ and its BB compactifcation
- [x] Attribute: GIT compactification ✅ 2024-02-02
	- [ ] Sha81: GIT compactification of $\PP^{12}\modmod D_4 \times \GG_m^2$
- [x] Attribute: comparing GIT to BB (for K3s) ✅ 2024-02-02
	- [ ] Loo86: $F_2$
	- [ ] LO21: $U\subseteq F_{4}$ K3s double-covering $\PP^1\times \PP^1$.
- [x] Attribute: the main square diagram ✅ 2024-02-02
	- [ ] Cos83, birational to Hor78, origin Enr1906.
- [x] Attribute: $F_{(2,2,0)}$ ✅ 2024-02-02
	- [ ] AEH21, AE22 (K3s with a nonsymplectic involution)
- [x] Attribute: $D_{T_{En}}/\Gamma_{En}$ ✅ 2024-02-02
	- [ ] Nam85: unique -2 root, two -4 roots, yielding divisors of Cobles, nodal Enriques, and Enriques which double cover $\PP(1, 1, 2)$.
- [x] Attribute: $F_{En, 2}$ ✅ 2024-02-02
	- [ ] Ste91: complement of discriminant is a coarse space of degree 2 Enriques.