# Cusp Correspondence ## Coble Cusps :::{.remark} We recall the mirror move algorithm from \cite{AE22nonsympinv}. We have Nikulin's 2-elementary diagram: \begin{figure}[H] \centering \input{tikz/nikulin_table.tikz} \caption{White nodes are $\delta=0$, black are $\delta=1$, double circled are $\delta = 1,2$.} \label{fig:nikulin_table} \end{figure} ::: :::{.remark} Having identified the 2-elementary lattice $S_\Co = (11, 11, 0)_1$, one can apply the mirror move algorithm of \cite[Thm. 5.10]{AE22nonsympinv} to determine the 0-cusps and 1-cusps of $F_\Co$. The outcome of the algorithm is summarized by the following tree: \begin{figure}[H] \centering \input{tikz/mirror-moves-coble.tikz} \caption{Blue (resp. red) indicate lattices which are valid (resp. invalid) targets of mirror moves.} \label{fig:unlabeledtwo} \end{figure} Thus $F_{\Co}$ has one 0-cusp corresponding to an isotropic vector $v$ with \[ v_0^{\perp T_{\Co}}/\gens{v_0} \cong (9,9,1)_1 \cong \gens{2} \oplus E_{8}(2) .\] Moreover, this 0-cusp $v_0$ is incident to one 1-cusp $C_0$ corresponding to an isotropic plane $J = \gens{v_0, v_1}$ with \[ J^{\perp T_{\Co}}/\gens{J} \cong (7,7,1)_0 \cong A_1^{\oplus 7} .\] where $v_1 \in v_0^{\perp T_{\Co}}/\gens{v_0}$. In the diagrammatic language of \cite[Fig.\, 1, Thm.\,5.10]{AE22nonsympinv}, this corresponds to a $U^2$ move and can be summarized in the following mirror move diagram as a composition of two even ordinary $U(2)$-type moves: \begin{figure}[H] \centering \input{tikz/mirror-moves-coble-simplified.tikz} \caption{} \label{fig:unlabeledthree} \end{figure} Note that $v_0$ corresponds to a Type $\rm{III}$ boundary, while $C_0$ corresponds to a type $\rm{II}$ boundary. It is easily verified that the Coxeter diagram $G_{(9,9,1)_1}$ at $v_0$ has precisely one maximal parabolic subdiagram, corresponding to a finite-index root lattice of type $A_7$. We note that by \cite[\S 5]{AE22nonsympinv}, such isotropic vectors are unique up to $\Orth(T_\Co)$, and so we can choose representatives: - $v_0 = e'$, - $v_1 = 2h + \alpha_1 + \alpha_2$. Calculations verify that $v_0^2 = v_1^2 = 0$, that $v_1 \in v_0^{T_{\Co}}/\gens{v_0}$, and that $v_0v_1 = 0$, and thus $J \da \gens{v_0, v_1}$ is an admissible choice of an isotropic plane. We further note that $\div_{T_{\Co}}(v_0) = \div_{T_{\Co}}(v_1) = 2$, which will be an important invariant for establishing a correspondence to cusps of other moduli spaces. For an isotropic plane $J$, we denote the divisibilities of the constituent generating vectors as a tuple $(d_1, d_2)$, and in this convention we have $\div_{T_{\Co}}(v_0, v_1) = (2, 2)$. We summarize this in the following boundary cusp diagram: % Cusp diagram F_Co \begin{figure}[H] \centering \input{tikz/coble-cusp-diagram-detailed.tikz} \caption{Cusp diagram for $F_\Co = F_{(11, 11, 1)}$ where $T_\Co = \gens{2} \oplus E_{10}(2)$.} \label{fig:coble-cusp-diagram} \end{figure} ::: :::{.remark} As further proof that this cusp diagram is correct, we can use the theory of Coxeter diagrams. Given an isotropic vector $e\in L$ a lattice of signature $(2, n)$, the lattice $e^{\perp L}/\gens{e}$ is a hyperbolic lattice equipped with a root system $R_e$ with a Coxeter diagram $G_e$. Generally, when $e$ corresponds to a 0-cusp in a Baily-Borel compactification, the adjacent 1-cusps correspond precisely to maximal parabolic subdiagrams of $G_e$. The cusp diagram above suggests that the 0-cusp $v_0$ should have a Coxeter diagram $G_{v_0}$ with precisely one maximal parabolic subdiagram. One can run Vinberg's algorithm to determine the Coxeter diagram for $v_0$, and it is a straightforward check to determine that there is indeed a unique maximal parabolic subdiagram of the form $\tilde B_7(2)$: \begin{figure}[H] \centering \input{tikz/coble-coxeter-diagram-maximal-parabolic.tikz} \caption{The unique maximal parabolic subdiagram $\tilde B_7(2)$ of $(9,9,1)_1$, corresponding to single one-cusp $(7,7,1)_0$ in $F_\Co$.} \label{fig:coble-cusp-9-9-1-parabolics} \end{figure} ::: :::{.remark} \begin{align*} \div_{T_{\dP}}(v_0) = 2 && \div_{T_{\dP}}(v_1) = 1 .\end{align*} The former is clear, since the image of $v_0$ in $T_{\dP}$ is $e'\in U(2)$ and $e'f' = 2$. The latter follows from the fact that $v_1\alpha_3 = 1$. ::: :::{.remark} We note the divisibilities of $v_i$ under various lattice embeddings: \begin{table}[H] \centering \begin{tabular}{|l|l|l|l|l|} \hline Coble Vector & Representative & $\mathrm{div}_{T_{\Co}}$ & $\mathrm{div}_{T_{\En}}$ & $\mathrm{div}_{T_{\dP}}$ \\ \hline $v_0$ & $e'$ & 2 & 2 & 2 \\ \hline $v_1$ & $2h + \alpha_1 + \alpha_2$ & 2 & 2 & 1 \\ \hline \end{tabular} \caption{Isotropic vectors in $F_{\En, 2}$ and their divisibilities.} \label{tab:sterk-cusps-two} \end{table} More concisely: \begin{table}[H] \centering \begin{tabular}{|l|l|l|l|} \hline Lattice & Image of $v_0$ & Image of $v_1$ & Divisibility \\ \hline $T_{\Co}$ & $e'$ & $2h + \alpha_1 + \alpha_2$ & $(2, 2)$ \\ \hline $T_{\En}$ & $e'$ & $2e + 2f + \alpha_1 + \alpha_2$ & $(2, 2)$ \\ \hline $T_{\dP}$ & $e'$ & $2e + 2f + \alpha_1 + \tilde\alpha_1 + \alpha_2 + \tilde \alpha_2$ & $(2, 1)$ \\ \hline \end{tabular} \caption{Isotropic vectors in $F_{\En, 2}$ and their divisibilities.} \label{tab:coble-cusp-divisibilities} \end{table} ::: ## Enriques Cusps :::{.remark} We recall the cusp diagram of $F_{\En}$: % Cusp diagram F_En \begin{figure}[H] \centering \input{tikz/enriques-cusp-diagram-detailed.tikz} \caption{Cusp diagram for $F_{\En} = F_{(10, 10, 0)}$ corresponding to $T_{\En} = U \oplus E_{10}(2)$.} \label{fig:enriques-cusp-diagram} \end{figure} This can be recovered using the mirror move algorithm: \begin{figure}[H] \centering \input{tikz/mirror-moves-enriques-simplified.tikz} \caption{} \label{fig:mirror-moves-enriques-simplified} \end{figure} ::: :::{.remark} We recall the Coxeter diagrams and their maximal parabolic subdiagrams at the 0-cusps of $F_{\En}$: \begin{figure}[H] \centering \caption{} \input{tikz/enriques-cusp-coxeter-diagrams-maximal-parabolics.tikz} \label{fig:enriques-maximal-parabolics-10-10-0} \end{figure} ::: ## Sterk Cusps :::{.remark} We recall Sterk's cusp diagram for $F_{\En, 2}$: \begin{figure}[H] \centering \input{tikz/sterk-cusp-diagram.tikz} \caption{Sterk's cusp diagram} \label{fig:sterk-cusp-diagram} \end{figure} We have the following divisibilities in various lattices: \begin{table}[] \centering \begin{tabular}{llll} \hline Sterk Cusp & Vector & $\mathrm{div}_{T_{\En}}$ & $\mathrm{div}_{T_{\mathrm{K3} } }$ \\ \hline 1 & $e$ & 1 & 1 \\ 2 & $e'$ & 2 & 2 \\ 3 & $e' + f' + \overline{\alpha}_8$ & 2 & 1 \\ 4 & $2e' + f' + \overline{\alpha}_1$ & 2 & 1 \\ 5 & $2e + 2f + \overline{\alpha}_1$ & 2 & 1 \\ \hline \end{tabular} \caption{Isotropic vectors in $F_{\En, 2}$ and their divisibilities.} \label{tab:sterk-cusp-divisibilities} \end{table} ::: ## K3 Cusps :::{.remark} We recall the cusp diagram for $F_{(2,2,0)}$: \begin{figure}[H] \centering \input{tikz/220-cusp-diagram.tikz} \caption{} \label{fig:220-cusp-diagram} \end{figure} :::