On cubic planes and groups connected with cubic varieties (Q800436)

A cubic plane consists of a set P of points and a set L of 3-subsets of P called lines such that any two lines intersect in at most one point. If \(S=\{P,L\}\) is a cubic plane, \(G_ S\) denotes a group generated by \(t_ p\), \(p\in P\), subject to the relations \(t^ 2_ p=1\) and \((t_{p_ 1}t_{p_ 2}t_{p_ 3})^ 2=1\) if \(\ell =\{p_ 1,p_ 2,p_ 3\}\in L\). Groups of such kind appeared in Manin's theory of cubic surfaces [\textit{Yu. J. Manin}, ''Cubic forms, Algebra, geometry, arithmetic'' (1974); translation from the Russian (1972; Zbl 0255.14002)]. The author proves that a cubic plane S is uniquely determined by the group \(G_ S\) and that, under some assumptions on S, the group Aut \(G_ S\) is a semi- direct product of \(G_ S\) and Aut S.