--- created: 2023-03-26T11:58 updated: 2024-05-04T13:38 aliases: - Coble - coble surface - coble surfaces links: "[[Enriques surfaces|Enriques surface]], [[moduli of Coble surfaces]]" --- [0000 Formulae](0000%20Formulae.md) # Trivia - $K_X^2 \leq -1$ by [Riemann-Roch](Riemann-Roch.md). - $K_X^2 = -1 \implies X$ is minimal but not conversely. - $p_a = {1\over 2}(D^2 + K_X\cdot D) + h^{0, 0} = h^{0, 1} \da h^1(\OO_X)$. - See [[0000 Formulae]] - $D \in \abs{-2K_X} \implies D = \union C_i$ smooth rational curves, SNC. - Canonical of a degree $d$ branched cover $X\to Y$: ? # [@DZ99] > A Coble surface is a nonsingular projective rational surface $S$ with $|-K_S| = \emptyset$ but $|-2K_S| \neq \emptyset$. > > A classical example: let $C \subseteq \PP^2$ be an irreducible rational curve, $\deg C = 6$ (i.e. a rational plane sextic) with $C^\sing$ comprised of 10 singularities that are ordinary nodes. Take $S = \Bl_{C^\sing} \cong \Bl_{10} \PP^2$, i.e. blow up at the nodes of $C$. > > Coble's original result: the Cremona equivalence class of $C$ can be expressed as $[C] = \Union_{i=1}^N [C_i]$ with each $C_i$ projective. > > Equivalently, $\Aut(S) \cong H \leq \Orth(M_S) \da \Orth( K_S^{\perp \Pic(S)} )$ is a finite-index subgroup. > > Note $\Orth(M_S)/{\pm 1} \cong W(E_{10})$. > > For general $C$, $H \cong W(E_{10})(2)$, the congruence subgroup of level 2. > There is a similar result for [Enriques surfaces](Enriques%20surfaces.md). Relations to [K3 surfaces](K3%20surfaces.md) and [Enriques surfaces](Enriques%20surfaces.md): > Let $S$ be a Coble surface and $C\subseteq S$ a reduced sextic. The branched cover $X$ branched along (the proper transform of) $C$ is a K3 surface. Moreover $X = \mcx_0$ where $\mcx_t$ is the K3 cover of an [Enriques surface](Enriques%20surface.md). A very special property: let $S$ be a nice Coble and $A_1(S)^{\smooth, \mathrm{rat}}$ be the set of smooth rational curves in $S$. Let $A_1(S)^{\smooth, \mathrm{rat}, < 0}$ be the negative cone with respect to self-intersections. Then $A_1(S)^{\smooth, \mathrm{rat}, < 0}/\Aut(S)$ is finite. So up to surface automorphisms, there are only finitely many smooth rational negative curves. #open/problems One can ask for a classification of all such surfaces satisfying this property. > When $|-2K_S|$ contains a reduced divisor $D$, the double cover of $S$ branched in $D$ is a $(X, \iota)$ a K3 with involution. $X$ has at worst [[ordinary double points]]. Two types: - Elliptic: there exists $S\birational Y$ where $|-K_Y|$ is a singleton, and the mobile part of $|-2K_Y|$ is generically a smooth elliptic curve. Splits into two further types: - Halphen type: blow-ups of singular points and their infinitely near points of a non-multiple fibre on a minimal rational elliptic surface with one multiple fibre of multiplicity 2. - Jacobian type: blow-downs of some disjoint sections (and maybe components of one fibre) of a non-minimal rational elliptic surface with a section - Rational: the mobile part of $|-2K_Y|$ consists of divisors $D$ with $p_a(D)$ = 0 (see [arithmetic genus](arithmetic%20genus.md)) where $D$ is not necessarily irreducible. - Blowups of minimal rational surfaces. Form a poset of Coble surfaces ordered by [dominant](dominant%20morphism.md) birational morphisms. This allows defining: - **minimal Coble surface**: does not admit a birational, but not biregular, morphism onto another Coble surface - Since $K_S^2 \leq -1$ for any Coble, if $K_S^2 = -1$ then $S$ is minimal. The converse is not true. - **terminal Coble surface**: not the image of any birational but not biregular morphism of Coble surfaces. **6.2 Lemma.** Let $X$ be a Coble surface. Then the following properties are equivalent: 1. $\left|-2 K_X\right|$ contains a reduced divisor. 2. There exists a double cover $\tilde{X} \rightarrow X$, where $\tilde{X}$ is a K3-surface with at most ordinary double points as singularities. There is a strong relation between Coble surfaces of K3-type and minimal resolutions of rational log Enriques surfaces of index 2. **A rational log Enriques surface $\bar{X}$ of index 2** is a normal rational surface with at worst quotient singularities such that $\mathcal{O}\left(-2 K_{\bar{X}}\right) \cong \mathcal{O}_{\bar{X}}$ A terminal Coble is the minimal resolution of a maximum rational log Enriques surface of index 2. The minimal resolution $X$ of a rational $\log$ Enriques surface $\bar{X}$ of index 2 is a Coble surface such that $h^0\left(-2 K_X\right)=1$ and the only member $D$ in $\left|-2 K_X\right|$ is a reduced divisor whose connected component is either a single $(-4)$-curve or a linear chain with the following dual graph: ![[Drawing 2024-01-10 12.23.07.excalidraw]] The converse is also true. **6.5 Theorem.** Suppose $X$ is a Coble surface with $M^2=0$. If $X$ is of Halphen type obtained from a minimal Halphen surface $Y_m$ of index 2 by one blow-up of a singular point on its non-multiple fibre $F$, then it is of K3-type if and only if $F$ is of type $I_n$, II, III or IV. If $X$ is of Jacobian type obtained as in Theorem 2.8 from a minimal Jacobian rational elliptic surface $Y_{\min }$ by blowing up a singular point from one fibre $F$ and singular points (at least one) and their infinitely near points on another fibre $F_1$, then it is of K3-type if and only if each of $F$ and $F_1$ is of type $I_n$, II, III, or IV. # [Dolgachev's monograph on Cobles](http://www.numdam.org/item/AST_1988__165__1_0.pdf) Assume now that $n=2, m=10$. A point set $x \in \mathbb{F}_2^{10}$ (resp. its blowing-up $V(x)$) is said to be a **Coble point set** (resp. a **Coble surface**) if $\left|-2 K_v\right|$ contains an irreducible curve. It is easy to see that this curve is a smooth rational curve $C$ with $C^2=-4$. Its image in $\mathbb{P}_2$ is a sextic with double points at each $\hat{x}^{\prime}$. # Moduli space of Cobles https://arxiv.org/pdf/1201.6093.pdf, [@dolgachev2013the-rationality] - Construction of moduli space: $\mcm_{Co} = (\PP^2)\cartpower{10} / \PGL_3$ - Take the double cover branched over a sextic with ten nodes to describe $\mcm \subseteq \dcosetl{\Gamma}{\Omega}$ where $\Omega$ is a 9-dimensional Hermitian symmetric domain of type IV. ![[Coble Surfaces 2024-01-10 12.28.23.excalidraw]] - Period domain definition: let $X\to S$ be the double cover branched along the proper transform of the plane sextic defining $S$. - $X$ is a K3 with divisors $E_0,\cdots, E_{10}$ where $E_0$ is the proper transform of a line $L\subseteq \PP^2$ and $E_i$ are transforms of exceptional divisors gotten from blowing up $p_i\in C$. - Then $M \da \gens{E_0,\cdots, E_{10}} \cong \gens{2} \oplus \gens{-2}\sumpower{10} \leq \Pic(X)$ is a 2-elementary sublattice of signature $(1, 10)$ with perp $N \cong \gens{2} \oplus (E_8\oplus U)(2)$, 2-elementary of signature $(2, 9)$. - Note the $(E_8\oplus U)(2) \cong \Num(Y) \cong \Pic(Y)/(K_Y)$ is the Enriques lattice for an Enriques surface $Y$. - Construct $$\mcm_{Co} \da \dcosetl{\Orth(N)}{\Omega_N},\qquad N \cong \gens{2} \oplus (E_8\oplus U)(2)$$ - Relating to moduli of Enriques surfaces: - Let $\mch_{-2}$ be the Heegner divisor of in the period domain of Enriques surfaces - Can write $\mcm_{Co} \birational \mch_{-2} / \Orth(U \oplus (E_8 \oplus U)(2))$. - Dolgachev-Kondo form a moduli space of cuspidal plane quintics $\mcm_{\mathrm{cusp}} = U/\PGL_3$ where $U$ is the projective space of cuspidal curves $C\subseteq \PP^2$ with $\deg C = 5$. They show - $\mcm_{\mathrm{cusp}}\birational \mcm_{\mathrm{En}} \birational F_L$, a moduli of K3s which are branched double covers of $\PP^2$ branched along a cuspidal quintic. This is 10-dimensional. - $\mcm'_{\mathrm{cusp}} \cong \mcm_{Co}$,: let $C$ be a plane quintic with cusp $p$, and $L$ a line tangent to $C$ at $p$. This corresponds to the case when $L$ is also tangent to $C$ at a smooth point $q\in C$. - $\mcm_{Co}$ is rational. - A Coble surface is obtained as a degeneration of an Enriques surface when the branch curve $W$ of the map $\phi_h$ passes through a singular point of $D$. - The the degree 2 polarization $h = [F_1 + F_2] \in \Pic(S)$ for a Coble surface - $|h|$ defines $\phi_{\abs h}: S\to \PP^4$ with image $D$, a 4-nodal quartic del Pezzo surface. - Let $\sigma: S\selfmap$ be the deck transformation, then $\Fix(\sigma) = \bar{W}$ a smooth genus 4 curve union 3 isolated points. - Let $C\in |-2K_S|$ be the Coble's bi-anticanonical curve, then $\phi_h(C) = \ts{q} \subset D$ is a singular point of $D$ - $\bar{W} \intersect C = \ts{p_1, p_2}$, and $W \da \phi_h(\bar W)$ is a curve with $p_a(W) = 5$ and a double point at $q$. # Book [@CDL20] ![](2024-01-10-2.png) Let $\pi:X \to V$ be the simple double cover defined by pair $(\mathcal{L}, s)$, where $\mathcal{L}=O_V(-K)$ and $s \in$ $H^0\left(\mathrm{~V}, \mathcal{L}^{\otimes 2}\right)$ with $Z(s)=C$. We call $X$ and $\pi$, the canonical cover of $\mathrm{V}$. Contrary to the case of Enriques surfaces, the surface $X$ is always smooth if $\mathrm{V}$ is terminal.