On the hypersurface of Lüroth quartics

The hypersurface of Luroth quartic curves inside the projective space of plane quartics has degree 54. We give a proof of this fact along the lines outlined in a paper by Morley, published in 1919. Another proof has been given by Le Potier and Tikhomirov in 2001, in the setting of moduli spaces of vector bundles on the projective plane. Morley’s proof uses the description of plane quartics as branch curves of Geiser involutions and gives new geometrical interpretations of the 36 planes associated to the Cremona hexahedral representations of a nonsingular cubic surface