an:06336179
Zbl 1295.11068
Iliev, Atanas; Roulleau, Xavier
On the \(\mathrm{PSL}_2(\mathbb{F}_{19})\)-invariant cubic sevenfold
EN
J. Algebra 413, 1-14 (2014).
00336049
2014
j
11G10 14J50 14J70
cubic sevenfold; automorphism group; intermediate Jacobian; period lattice
Summary: It has been proved by \textit{A. Adler} [J. Algebra 72, 146--165 (1981; Zbl 0479.20020)] that there exists a unique cubic hypersurface \(X^7\) in \(\mathbb{P}^8\) which is invariant under the action of the simple group \(\mathrm{PSL}_2(\mathbb{F}_{19})\). In the present note we study the intermediate Jacobian of \(X^7\) and in particular we prove that the subjacent 85-dimensional torus is an Abelian variety. The symmetry group \(G = \mathrm{PSL}_2(\mathbb{F}_{19})\) defines uniquely a \(G\)-invariant Abelian 9-fold \(A(X^7)\), which we study in detail and describe its period lattice.
Zbl 0479.20020