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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.
##### MSC:
 11G10 Abelian varieties of dimension $$> 1$$ 14J50 Automorphisms of surfaces and higher-dimensional varieties 14J70 Hypersurfaces and algebraic geometry
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