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Vanishing Hessian, wild forms and their border VSP
Hang Huang, Mateusz Michałek and Emanuele Ventura
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≥3 as well as an infinite series of wild cubics. Inspired by recent work on border apolarity of Buczyńska and Buczyński, we study the border varieties of sums of powers of these forms in the corresponding multigraded Hilbert scheme.