The variety of minimal power sum presentations of a homogeneous form is a PGL-invariant that may be used to distinguish forms. In special cases it defines surprising maps between moduli spaces of varieties. I will review old and new results on power sum varieties, and report on recent work with G. and M. Kapustka, M. Stilmann, H. Schenk and B. Yuan on quaternary quartics. In particular, a new map between two 18-dimensional families of quartic surfaces.

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