# Coble surfaces 

## The geometry of Coble surfaces

:::{.remark}
For the general theory of Coble surfaces, we refer to \cite{EnriquesOne,EnriquesTwo}, along with \cite{DM19, DZ99, dolgachev2013the-rationality, CD89, CD12, Dol17}.
Following \cite[\S 5.1]{DM19}, a \textbf{Coble surface} is a smooth projective rational surface $S$ with $\abs{-K_S} = \emptyset$ but $\abs{-2K_S} \neq \emptyset$. 
We say $S$ is of **K3 type** if $\abs{-2K_S}$ consists of a single smooth divisor $C = C_1 + \cdots + C_n$, the union of $n$ disjoint smooth rational curves satisfying $C_i\cdot C_j = -4\delta_{ij}$. For Coble surfaces obtained as the blowup along ten nodes of a nodal plane sextic, this calculation follows from adjunction and the genus formula.
We refer to $C$ as the *anti-bicanonical curve* of $S$, and note that $K_S\cdot C_i = -2$.
The $C_i$ are referred to as the **boundary components** of $S$.

By \cite[\S 5.1]{Dol17}, such a Coble surface is known to be a *basic rational surface*, i.e. there is a birational morphism $\pi: S\to \PP^2$. Writing $-K_S^2 = n$, one can decompose $\pi$ as a blowup of $N = 9+n$ points in $\PP^2$. 
It is known that $n\leq 10$, c.f. \cite[Prop.\,9.1.5]{EnriquesTwo}.
The image of $C$ is contained in $\abs{-2K_{\PP^2}} = \abs{\OO_{\PP^2}(6)}$ and is thus generically a nodal plane sextic where the images of $C_i$ are its irreducible components.

We will primarily be interested in the case $n=1$, whence $S$ is obtained as the blowup of a plane sextic along $N=10$ ordinary double points, some of which may be infinitely near to each other.
Such Coble surfaces are not the image of any birational but not biregular morphism from another Coble surface and are said to be \textbf{terminal}. We say $S$ is \textbf{minimal} if the blowdown of any $(-1)$-curve on $S$ is no longer a Coble surface, or equivalently if $S$ does not admit a birational but not biregular morphism onto another Coble surface.
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:::{.remark}
Let $\cL \da \OO_S(-K_S) \in \Pic(S)$; By \cite[Prop. 9.1.1]{EnriquesTwo}, taking a section $s\in H^0(\cL ^{\otimes 2})$ with $Z(s) = C$ yields a double branched double cover $f: X\to S$ where $X$ is a smooth K3 surface. By \cite[Def. 5.4.3]{EnriquesOne}, the preimages $f^{-1}(C_i)$ are disjoint $(-2)$-curves and $\Pic(X)$ is a 2-elementary lattice with invariants of the form 
\[
(r,a,\delta)_1 = (10+n, 12-n, \delta)_1
.\] 
By \cite[Def. 5.4.3, Eqn. 5.3.1]{EnriquesOne}, the ramification divisor $R$ is explicitly of one of the following forms:

- $R=\emptyset$ if $(r,a,\delta) = (10, 10, 0)$,
- $R$ is a sum of two elliptic curves if $(r,a,\delta) = (10, 8, 0)$,
- $R$ is the sum of a single rational curve and $n-1$ other disjoint $(-2)$-curves otherwise.

It is also known that $\delta=1$ unless $n=8$, c.f. \cite[Table\, 5.1]{EnriquesOne}.
Thus if $S$ is a terminal Coble surface of K3 type with $n=1$, the ramification locus of the K3 cover is a single smooth rational curve, and we obtain a lattice with invariants 
\[
S_{\Co} \da (11, 11, 1)_1 \cong \gens{-2} \oplus E_{10}(2)
\]
with orthogonal complement 
\[
T_{\Co} \da S_{\Co}^{\perp \lkthree} = (11, 11, 1)_2 \cong \rm{I}_{2, 9}(2) \cong \gens{2} \oplus E_{10}(2)
.\]

Explicitly, $S_{\Co}$ is generated by the preimages $E_i$ for $1\leq i \leq 10$ under $f$ of the ten exceptional divisors of the blowup, along with the preimage $E_0$ under $f$ of the pullback of a hyperplane class of $\PP^2$ under $\pi$.
One can show that $E_0^2=2$ and $E_i^2 = -2$ for $i\geq 1$, and thus these divisors form a lattice of the form 
\[
S_{\Co} \cong \gens{2} \oplus \gens{-2}^{\oplus 10} \cong \rm{I}_{1, 10}(2) \cong (11, 11, 1)_1
,\]
and the identification $S_{\Co} \cong \gens{-2} \oplus E_{10}(2)$ follows from the fact that both definitions of $S_{\Co}$ are 2-elementary lattices of signature $(1, 10)$ with discriminant group $(\bZ/2\bZ)^{11}$, which are classified uniquely up to isometry by their invariants $(r,a,\delta)_{n_+}$.
Similarly, the identification of $T_{\Co}$ follows from the fact that it is again a 2-elementary lattice of signature $(2, 9)$ satisfying $q_{T_{\Co}} \cong -q_{S_{\Co}}$, and the isomorphism class of $q_{T_{\Co}}$ determines $T_{\Co}$ up to isometry.
Alternatively, this can be seen directly using the mirror move $S\leadsto T$ of \cite[Thm. 5.10]{AE22nonsympinv} applied to $S_\Co =(11, 11, 1)_1$ in \cite[Fig.\,1]{AE22nonsympinv}, immediately yielding $T_{\Co} = (11, 11, 1)_2$.


The lattices $S_{\Co}$ and $T_{\Co}$ will be of fundamental importance in constructing the Hodge-theoretic period domain for Coble surfaces, yielding a coarse space for the corresponding moduli space.
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:::{.remark}
Following \cite{CD12}, we note that this computation is a special case of a general construction. Let $S$ be any basic rational surface and write $S$ as the blowup of $\PP^2$ at $N$ points $p_1,\cdots, p_N$ with $N\geq 9$.
It is a fact that $\Pic(S) \cong \rm{I}_{1, N}$, since one can construct a **geometric basis** in the following way: let $e_0$ be the class of the total transform of a hyperplane class in $\PP^2$ and for $1\leq i\leq N$, let $e_i$ be the class of the total transform of the exceptional divisor over $p_i$.
Then $\Pic(S) = \gens{e_0,e_1,\cdots, e_N}$ and $\rho(X) = N+1$; one verifies that $e_0^2 = 1$, and for $i\geq 1$, that $e_i^2 = -1$. Moreover $e_ie_j = 0$ for $i\neq j$, making this an orthogonal basis with respect to the intersection pairing, yielding $\rm{I}_{1, N}$.
In the case of Coble surfaces, the effect of taking the K3 double cover is to twist this lattice by 2, yielding $\Pic(X) = \rm{I}_{1, N}(2)$, generated by preimages of the $e_i$.
We remark that
\[
K_S = -3e_0 + e_1 + \cdots + e_N
.\]
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