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On the \(\mathrm{PSL}_2(\mathbb{F}_{19})\)-invariant cubic sevenfold. (English) Zbl 1295.11068
Summary: It has been proved by 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.
11G10 Abelian varieties of dimension \(> 1\)
14J50 Automorphisms of surfaces and higher-dimensional varieties
14J70 Hypersurfaces and algebraic geometry
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