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created: 2024-05-04T13:38
updated: 2024-05-04T13:39
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Define a [coble surface](Coble%20Surfaces.md) as a singular rational surface $S$ such that $|-K_S|=\emptyset$ but $|-2K_S| \neq \emptyset$. If $|-2K_S| = \ts{D}$ is a singleton, we say $S$ is a strong Coble surface or a Coble of K3 type. Writing $D = \sum_{k=1}^n B_k$ as a sum of disjoint smooth rational curve classes, we refer to $\ts{B_k}$ as the boundary components of $S$. It is known that $n\leq 10$, as well as $B_k^2 = -4$ and $K_S \cdot B_k = 2$. Moreover $D$ is SNC. If $S$ is terminal, then $K_S^2 = -n \geq -10$. Finally, $D$ defines a 2-to-1 branched cover $X\to S$ where $X$ is a K3.

Any smooth rational negative curve $E\subseteq S$ satisfies $E^2 \in \ts{-1,-2,-4}$, and moreover if $E^2 = -4$ then $E$ is a component of $D$.

Since Cobles are rational, there is a birational map $S\rational \PP^2$, and one can show that $W = f(D)$ is a sextic with ordinary double points whose irreducible components are rational curves. Moreover, $W\in |-2K_{\PP^2}|$.

Coble's construction yields a case where $n=1$, so $D = C$ is a single curve. In this case, for $f:S\rational \PP^2$ one has that $W = f(C)$ is a rational nodal sextic with 10 ordinary nodes. So one can write $W = V(f_6)$ for $f_6$ a degree 6 homogeneous polynomial.

For Cobles, we consider the Coble-Mukai lattice $\mathrm{CM}(S) = \gens{B_1,\cdots, B_n}_\ZZ^{\perp L}$ where $L \da \gens{\NS(S), {1\over 2}B_1 + \cdots + {1\over 2}B_n}_\ZZ$. It serves as a replacement for $\NS(S)$.

Note that if $X' \rational X$ is a a blowup at a point with $X$ smooth irreducible, then the pullback $\pi^*: \Pic(X) \to \Pic(X')$ induces an isomorphism $\Pic(X') \cong \pi^* \Pic(X) \oplus \ZZ \cong \Pic(X) \oplus \gens{E}$ for $E$ the class of the exceptional divisor.

Defining a geometric basis: let $\pi:X\to \PP^2$ be a blowup of $N$ points, let $e_0 = c_1 \pi^* \OO_{\PP^2}(1)$ and $e_i = [E_i]$ for $i=1,\cdots, N$. Then note that $e_0^2 = 1, e_i^2 = -1$, and $e_i \perp e_j$ for $i\neq j$. Note also that $K_X = -3e_0 + \sum e_i$....but why?

Then
$$
\Pic(X) = \gens{e_0, e_1,\cdots, e_N}_\ZZ \cong \ZZ^{N+1}.
$$
This is in fact the lattice $\rm{I}^{1, N} = \gens{1} \oplus \gens{-1}^N$. This isomorphism is induced by
$$
\begin{align}
\Pic(X') \oplus \ZZ &\to \Pic(X) \\
(\mcl, n) &\mapsto \pi^*(\mcl) \tensor \OO_X(nE)
\end{align}
$$

For $N\geq 10$ points, write $9-N = -n$, then one can show that $F_S^a$ is a moduli of marked Coble surfaces with $n=1$ since $N=10$, where $S = E_{10}(2) \oplus A_1$. So $\mcm_{\mathrm{Co}}^{\mathrm{mark}} \cong F_{E_{10}(2)\oplus A_1}$.

Define $E_{10} = U \oplus E_8$.

More generally, we can define a moduli functor which sends $T$ to algebraic families $\mcx\to T$ such that the fiber $\mcx_{\bar t}$ over every geometric point $\spec k \mapsvia{\bar t} T$ is an Enriques surface. A polarized surface is a pair $(X,\mcl)$ where $X$ is a surface and $\mcl \in \Sh_X$ is a sheaf with $\bar{t}^* \mcl$ ample or nef.
One can show that if $S$ is Enriques, $\Num(S) \cong E_{10}$, and the moduli space of marked Enriques surfaces $\mcm_{\mathrm{En}}^{\mathrm{mark}} \cong F_{E_{10}(2)}$.

If $\signature M = (1, n)$ for $M\leq \lkt$, then $\signature M^{\perp \lkt} = (2, 19-n)$. Let us consider why the period point $[\phi]\in M^\perp$: write $[\phi] = \phi H^{2, 0}(X) \leq \lkt$, then note that $H^{2, 0}(X) \perp H^{1, 1}(X)$, and conclude.

We have
$$
\mcm_{Co}^m = \mch(-2)/\Gamma^\sharp, \qquad \mch(-2)\subseteq \mcd_{Enr} = \Omega_{E_{10}(2)^{\perp\lkt} }
$$

Generally for a lattice $N$, we define $\Orth(N)^\sharp \da \ker\qty{ \Orth(N) \to \Orth(D_N)}$ where $D_N$ is the discriminant group of $N$.

Questions to answer:

- What is the construction of $\mcm_{Co}$ as a GIT quotient, using direct geometric constructions?
- What is the construction of $\mcm_{Co}$ as a period domain?
	- I.e. $\mcd_{Co} = \Omega_L$ for some lattice $L$ -- what lattice? Or maybe $L^{\perp}$, but the perp in what ambient lattice?
	- Differentiate $\mcm_{Co}^m$ and $\mcm_{Co}$ (unmarked)
	- Is $\mcm_{Co}$ a moduli of unpolarized Cobles? Numerically polarized? If so, what degree?
- Who does $\mcm_{Co}$ relate to $\mcf_{En, 2}$?
- Differentiate marked/unmarked and polarized/unpolarized Enriques surfaces.