Vanishing Hessian, wild forms and their border VSP

Abstract

Wild forms are homogeneous polynomials whose smoothable rank is strictly larger than their border rank. The discrepancy between these two ranks is caused by the difference between the limit of spans of a family of zero-dimensional schemes and the span of their flat limit. For concise forms of minimal border rank, we show that the condition of vanishing Hessian is equivalent to being wild. This is proven by making a detour through structure tensors of smoothable and Gorenstein algebras. The equivalence fails in the non-minimal border rank regime. We exhibit an infinite series of minimal border rank wild forms of every degree $d\geq 3$ as well as an infinite series of wild cubics. Inspired by recent work on border apolarity of Buczy\'nska and Buczy\'nski, we study the border varieties of sums of powers $\underline{\mathrm{VSP}}$ of these forms in the corresponding multigraded Hilbert schemes.