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We explore the maximality of the Hilbert square of maximal real surfaces, andfind that in many cases the Hilbert square is maximal if and only if thesurface has connected real locus. In particular, the Hilbert square of nomaximal K3-surface is maximal. Nevertheless, we exhibit maximal surfaces withdisconnected real locus whose Hilbert square is maximal.

Unexpected loss of maximality: the case of Hilbert square of real surfaces