Vanishing Hessian, wild forms and their border VSP
- PDF / 468,894 Bytes
- 28 Pages / 439.37 x 666.142 pts Page_size
- 81 Downloads / 200 Views
Mathematische Annalen
Vanishing Hessian, wild forms and their border VSP Hang Huang1 · Mateusz Michałek2 · Emanuele Ventura3 Received: 25 June 2020 / Revised: 23 July 2020 / Published online: 14 September 2020 © The Author(s) 2020
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 ≥ 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 VSP of these forms in the corresponding multigraded Hilbert schemes. Mathematics Subject Classification Primary 14C05 · Secondary 15A69
1 Introduction Notions of ranks abound in the literature, perhaps because of their natural appearance in the realms of algebra and geometry, and in numerous applications thereof; see [18,22] and references therein for an introduction to the subject.
Communicated by Vasudevan Srinivas.
B
Emanuele Ventura [email protected] Hang Huang [email protected] Mateusz Michałek [email protected]
1
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
2
Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
3
Mathematical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
123
1506
H. Huang et al.
These ranks vastly generalize matrix rank and yet they are very classical, dating back to the pioneering work of Sylvester. His work featured Waring ranks of binary forms; see [18] for a historical account on the subject. Since then, ever growing research efforts have been devoted to understanding ranks with respect to some special projective varieties X of interest. Last decades have witnessed steady progress on tensor and Waring ranks, i.e. the cases when the projective varieties are the classical Segre and Veronese varieties. These results have been developed in parallel in their geometric and algebraic aspects. The first are naturally related to secant varieties of X [29, Chapter 1], whereas the second to Macaulay’s theory of apolarity and inverse systems [18, §1.1]. Interestingly, scheme-theoretic versions of X -ranks have been introduced and studied as well. These latter ones take into account more general zero-dimensional schemes, besides the reduced zero-dimensional ones featured in the X -ranks. This more general framework naturally leads to new notions of X -rank: the smoothable X -rank, and the cactus X -rank; the latter was originally called scheme length [18
Data Loading...
