---
created: 2024-05-03T00:14
updated: 2024-05-04T21:36
---
# 8-22
## Research meeting

Let $X\mapsvia\pi Y$ be a K3 covering an Enriques surface, so $L \da \lkt \da H^2(X;\ZZ)= U^3 + E_8^2$ and $M\da H^2(Y;\ZZ)/\tors = U + E_8$. Let $\eta_Y \da e+f$ where $e,f$ form a symplectic basis of $U$, then the pushforward satisfies $\pi^* \eta_Y = (e+f, e+f, 0)$ where we've written $L = M + M +U$. Moreover the map is
$$
\pi^*: M \to M + M + U \\\quad
m \mapsto (m, m, 0)
.$$
There is an involution
$$
I: M \to M \\\quad (m, m', h) \mapsto (m', m, h)
,$$
and we let $L_+$ be the $+1$ eigenspace and $L_-$ the $-1$ eigenspace. One can show that $L_+ ^{\perp H_2(X; \ZZ)} = L_-$. Here $L_+$ is the generic Picard lattice of $X$.[^1] There are identifications 
$$
L_+ = \ts{(m, m, 0) \st m\in M} = M(2) = U(2) + E_8(2) = (10, 10, 0)_1
$$
and
$$
L_- = \ts{(m, -m, h) \st m\in M, h\in U} = U + U(2) + E_8(2) = (12, 10, 0)_2
.$$
We recall that for $\Lambda$ any 2-elementary lattice, $\Lambda\dual/\Lambda \cong (\ZZ/2)^a$ for some $a$, and these invariants are $(r, a, \delta)_i$ where $r \da \rank_\ZZ \Lambda$, the integer $a$ is as above, and $\delta$ is the so-called coparity: $\delta = 0$ if the associated quadratic form $q_L$ satisfies $q_L(A_L) \subseteq \ZZ$, so $q_L(x) \equiv 0 \mod \ZZ$ (a co-even lattice), and $\delta = 1$ otherwise (a co-odd lattice). There is a trick to computing the coparity.[^6]

We construct the usual period domain 
$$
\Omega_- \da \Omega^{L_+} = \Omega_{L_+^\perp} = \Omega_{L_-} = \ts{v\in \PP(L_-\tensor_\ZZ \CC) \st v^2 = 0,\, v\bar v > 0}
.$$

These are periods of $X$, which are necessarily orthogonal to $L_+$, the generic Picard lattice. We have $\dim_\CC \Omega_- = 10$, with an associated 10-dimensional moduli space $E_{?} = \dcosetl{\Gamma_?}{\Omega_{L_-}}$ for some discrete group $\Gamma_?$. Note that $\Orth(\lkt) \contains \Stab(L_+) = \Gamma_-$, and $\Gamma_- \actson \Omega_-$.Note also that $\Stab(L_+)\contains \Gamma_h \da \Stab(h)$, where $h\in M \subseteq \lkt$ is a numerical polarization on an Enriques surfaces. 

It is a fact that there are only finitely many such moduli spaces of Enriques surfaces. Letting $\mce_\emptyset$ be the moduli of unpolarized Enriques surfaces and $\mce_h$ be the moduli of Enriques surfaces with polarization $h$, there are finitely many choices up to isomorphism for what $\Stab(h)$ can be. This induces a finite poset of moduli spaces of the form $\mce_h$, whose minimal element is $\mce_\emptyset$ and whose maximal element in $\mce_{\max}$:

[Link to Quiver diagram](https://q.uiver.app/#q=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)

![Pasted image 20230824135448.png](Pasted%20image%2020230824135448.png)

We can form a moduli space[^2] $M_{\bar h}$ with $\bar h\in \Pic(Y)/\tors$, which will either be of the form $M_L$ or $M_L \amalg M_{L'}$. Note that we can also consider the more standard moduli of polarized Enriques surfaces $(Y, L)$ with $L\in \Pic(Y)^\amp$.[^3]

Sterk takes $\Gamma$ to be the image of $\ts{g\in \Orth(L) \st g\circ I = I\circ g\, g(h) = h}$ in $\Orth(L_-)$, which seems to precisely be something like $\Stab_{\Orth(L_-)}(h)$? [^5] here $h = (e+f, e+f, 0)$. These isometries fix a $U(2)$ summand. If we take $U(2) = \gens{E, F}$, we either have $E\mapstofrom E, F\mapstofrom F$, or $E-F \mapstofrom F-E$ and $E\mapstofrom F$.

Note that we can build $F_{4, \he}$ as $F_{U(2)}$, and we have the following diagram.

[Link to Quiver diagram](https://q.uiver.app/#q=WzAsNixbMiwxLCJcXG1hdGhjYWx7RX1faCJdLFsyLDMsIlxcbWF0aGNhbHtFfV9cXGVtcHR5c2V0Il0sWzQsMSwiRl97VSgyfSkiXSxbMiwwLCJcXG92ZXJsaW5le1xcbWF0aGNhbHtFfV9ofV57XFxtYXRocm17S1NCQX19Il0sWzAsMSwiRl97SzMsIFxcaW90YX0iXSxbMCwwLCJcXG92ZXJsaW5leyBGX3tLMywgXFxpb3RhfX1ee1xcbWF0aHJte0tTQkF9fSJdLFswLDFdLFsyLDBdLFswLDMsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzQsNSwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMCw0XV0=)

![Pasted image 20230824143638.png](Pasted%20image%2020230824143638.png)

Here $F_{K3, \iota}$ is a moduli of K3 surfaces with involution.[^4]

Let $T = (12, 10, 0)_2 \ni e$ with $e^2 = 0$; we can identify this with $U + U(2) + E_8(2) = (2,0,0)_1 + (2,2,0)_1 + (8,0,0)_0$. Note that given $(r, a,\delta)_i$, one can generally construct this decomposition into pieces of the following forms:

- $\gens{2} = (1,1,1)_1$
- $U = (2,0,0)_1$
- $U(2) = (2,2,0)_1$
- $E_8 = (8,0,0)_0$
- $E_8(2) = (8, 8, 0)_0$
- $A_1 = \gens{-2} = (1,1,1)_0$
- $E_7 = (7,1,1)_0$
- $D_{4n} = (4n, 2, 0)_0$
- $D_{4n+2} = (4n+2, 2, 1)_0$

We have $\bar{T} = e^\perp/e \ni f$ and $f^\perp/f = \bar{\bar T} = J^\perp/J$.

As a quick aside:

- $A_n$ has $\Lambda\dual/\Lambda = \ZZ_{n+1}$ and $\Lambda\dual(2) = ?$
- $D_n$ has $\Lambda\dual/\Lambda = (\ZZ/2)^2$ or $\ZZ/4$  for $n$ odd and $\Lambda\dual(2) = ?$
- $E_{6, 7, 8}$ has $\Lambda\dual/\Lambda = \ZZ/3$ or $\ZZ/2$? And $\Lambda\dual(2) = ?$

We get a cusp diagram:

![Pasted image 20230824150239.png](Pasted%20image%2020230824150239.png)


Note that $U + E_8(2) \cong U(2) + D_8$.

Something about $(\rm{ADE})^\perp \subseteq \Lambda_{(24, 0, 0)_1}$.

[^1]: Like $\Pic(\mcx_t)$ for for $\mcx_t \in F_{2d}$ a generic point in a moduli space?
[^2]: Missed what this is a moduli space *of*, can't quite remember what $\bar h$ and $L, L'$ were.
[^3]: Why do we not use this moduli space? Seems pretty natural. Maybe it coincides with something we already use? Valery might have said something along these lines that I've forgotten.
[^4]: Not entirely sure which moduli space this is yet. Like moduli of K3s with a nonsymplectic involution, lattice-polarized by a particular $S$? Or are we doing something like the existence of $F_{2d, \iota}$, a moduli of degree $2d$ K3 surfaces equipped with a (possibly symplectic, possibly not) involution?
[^5]: Would help to know this explicitly, since it's a much simpler description than this centralizer description
[^6]: Can't quite remember the trick...for $L$, it's something like take $L^\dagger(2)$ and check if it is even...? Valery knows how to do this easily. For example, for $A_1$, you get $\gens{-1\over 2}(2) = \gens{-1}$ which is odd, and so $A_1$ is co-even and $\delta = 0$? I don't think I did this correctly.