Waring Rank of Symmetric Tensors, and Singularities of Some Projective Hypersurfaces

We show that if a homogeneous polynomial $f$ in $n$ variables has Waring rank $n+1$, then the corresponding projective hypersurface $f=0$ has at most isolated singularities, and the type of these singularities is completely determined by the combinatorics of a hyperplane arrangement naturally associated with the Waring decomposition of $f$. We also discuss the relation between the Waring rank and the type of singularities on a plane curve, when this curve is defined by the suspension of a binary form, or when the Waring rank is 5.
v4: We have added Lemma 2.2, which is necessary for a clear proof of the unicity of the integer $k$ in Theorem 2.4. We also attribute Proposition 4.1 to Neriman Tokcan, since we were informed that it appears as Theorem 3.1 in her paper 'On the Waring rank of binary forms', arXiv:1610.09065