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The problem of representing a homogeneous form as a sum of powers of linear forms has been studied since the last decades of the 19th century. This is called the Waring problem for a homogeneous form. We are interested in the study of the global structure of a suitable compactification of the variety parameterizing all such representations of a homogeneous form. A precise definition of the claimed compactification is the following.

Geometries of lines and conics on the quintic del Pezzo 3-fold and its application to varieties of power sums