We show that forms with vanishing Hessian and of minimal border rank are wild, i.e. their smoothable rank is strictly larger than their border rank. This discrepancy is caused by the difference between the limit of spans of a family of zero-dimensional schemes and the span of their flat limit. We exhibit an infinite series of 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 Buczynska and Buczynski, we study the border varieties of sums of powers $\underline{\mathrm{VSP}}$ of these forms in the corresponding multigraded Hilbert scheme.

中文翻译：

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消失的 Hessian、野生形式及其边界 VSP

我们证明了具有消失 Hessian 和最小边界秩的形式是野生的，即它们的平滑秩严格大于它们的边界秩。这种差异是由零维方案族的跨度极限与其平面极限跨度之间的差异引起的。我们展示了每个度数 $d\geq 3$ 的无限系列野生形式以及无限系列的野生立方体。受 Buczynska 和 Buczynski 最近关于边界非极性的工作的启发，我们研究了相应多级希尔伯特方案中这些形式的幂和 $\underline{\mathrm{VSP}}$ 的边界变体。