From the introduction: [For part I of this paper see
J. Sekiguchi, Kyushu J. Math. 48, 123–168 (1994;
Zbl 0841.14009).] The author introduced the notion of cross ratio varieties for root systems in part I, generalizing the construction of Naruki’s cross ratio variety \(C\) [
I. Naruki, Proc. Lond. Math. Soc. (3) 45, 1–30 (1982;
Zbl 0508.14005)], which is a smooth compactification of the moduli space of cubic surfaces. \(C\) can be regarded as one of the cross ratio varieties related to the root system of type \(E_6\). In the case of the root system of type \(E_7\), we obtain three kinds of cross ratio varieties. We focus our attention on one, denoted \(C_{E_7}\) in this article. This variety \(C_{E_7}\) can be regarded as an analogue of Naruki’s cross ratio variety in the sense that it is regarded as a moduli space of marked Del Pezzo surfaces of degree two. Our subject is related to studies of 27 lines on cubic surfaces and 28 bitangents to plane quartic curves, classical and modern.
We review the definition of \(C_{E_7}\) briefly. Let \(\mathbb{P}(2,7)\) be the configuration space of seven points in the complex projective plane \(\mathbb{P}^2\). We introduce an open subset \(\mathbb{P}_0(2,7)\) of \(\mathbb{P}(2,7)\) defined as the quotient space of \(\widetilde\mathbb{P}_0(2,7)\) under the PGL(3)-action, where \(\widetilde{\mathbb P}(2,7) = \{(P_1.\ldots,P_7) | P_i\in\mathbb P^2\); \(P_i\overline P_j\) (\(i\neq j\)); no line passes through three of the \(P_i\); no conic passes through six of the \(P_i\}\). Note that \(\mathbb{P}_0(2,7)\) admits a biregular \(W(E_7)\)-action, though \(\mathbb{P}(2,7)\) does not, where \(W(E_7)\) is the Weyl group of type \(E_7\). The variety \(C_{E_7}\) is defined as a compactification of \(\mathbb{P}_0(2,7)\) obtained by cross ratios for the root system. The \(W(E_7)\)-action on \(\mathbb{P}_0(2,7)\) extends to \(C_{E_7}\) in a natural manner.
We now recall basic properties of Naruki’s cross ratio variety \(C\).
(i) \(C\) is a compactification of \(\mathbb{P}_0(2,6)\), which is the configuration space of six points defined similarly to \(\mathbb{P}_0(2,7)\),
(ii) \(C\) is non-singular.
(iii) \(C\) admits a biregular \(W(E_6)\)-action,
(iv) The complement of \(\mathbb{P}_0(2,6)\) in \(C\) is a divisor with normal crossings only. It is the union of seventy six non-singular hypersurfaces of \(C\) and they are divided into two \(W(E_6)\)-orbits. Moreover, such a hypersurface is described in terms of a subset of the root system of type \(E_6\).
Since the construction of \(C_{E_7}\) is similar to that of \(C\), we are led to the following problem:
(i) Is the variety \(C_{E_7}\) non-singular?
(ii) Is the \(W(E_7)\)-action on \(C_{E_7}\) biregular?
(iii) Is \(D_{E_7}-\mathbb{P}_0(2,7)\) a divisor of \(C_{E_7}\) with normal crossings only? Determine the irreducible components of \(C_{E_7}-\mathbb{P}_0(2,7)\) and describe them in terms of subsets of the root system \(\Delta= \Delta (E_7)\) of type \(E_7\).
The author introduced in part I the subvariety \(Y_{\Delta,D_4} (\Delta_1)\) of \(C_{E_7}\) corresponding to each irreducible subroot system \(\Delta_1\) of \(\Delta\). The main purpose of this article is to study the structure of hypersurfaces in \(C_{E_7}\) of the forms \(Y_{\Delta, D_4}(\Delta_1)\) in the case where the type of \(\Delta_1\) is one of the following: \(A_1,A_2,A_3,A_6,E_6\). There are two reasons why we treat the case \(E_6\) in addition to the cases \(A_1, A_2,A_3,A_6\). One is that if \(\Delta_1\) is an irreducible proper subroot system of \(\Delta\) such that \(\Delta_1\) is not of type \(A_7\), then \(Y_{\Delta,D_4}(\Delta_1)\) coincides with or is contained in the subvariety \(Y_{\Delta,D_4}(\Delta_1')\) corresponding to a subroot system \(\Delta_1'\) whose type is one of \(A_1,A_2,A_3,A_6,E_6\) and that if \(\Delta_l\) is of type \(A^7,Y_{\Delta,D_4}(\Delta_1)\) coincides with \(C_{E_7}\) itself. The other is that the subvariety \(Y_{\Delta,D_4} (\Delta_1)\) corresponding to a root system of type \(E_6\) is interesting in its own sake; each point represents a seven points set in \(\mathbb{P}^2\) related with a special flex of a non-singular plane quartic curve.
