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Most secant varieties of tangential varieties to Veronese varieties are nondefective

Authors: Hirotachi Abo and Nick Vannieuwenhoven
Journal: Trans. Amer. Math. Soc. 370 (2018), 393-420
MSC (2010): Primary 14M99, 14Q15, 14Q20, 15A69, 15A72
Published electronically: August 3, 2017
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Abstract: We prove a conjecture stated by Catalisano, Geramita, and
Gimigliano in 2002, which claims that the secant varieties of tangential varieties to the $ d$th Veronese embedding of the projective $ n$-space $ \mathbb{P}^n$ have the expected dimension, modulo a few well-known exceptions. It is arguably the first complete result on the dimensions of secant varieties of a classically studied variety since the work of Alexander and Hirschowitz in 1995. As Bernardi, Catalisano, Gimigliano, and Idá demonstrated that the proof of this conjecture may be reduced to the case of cubics, i.e., $ d=3$, the main contribution of this work is the resolution of this base case. The proposed proof proceeds by induction on the dimension $ n$ of the projective space via a specialization argument. This reduces the proof to a large number of initial cases for the induction, which were settled using a computer-assisted proof. The individual base cases were computationally challenging problems. Indeed, the largest base case required us to deal with the tangential variety to the third Veronese embedding of $ \mathbb{P}^{79}$ in $ \mathbb{P}^{88559}$.

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Additional Information

Hirotachi Abo
Affiliation: Department of Mathematics, University of Idaho, Moscow, Idaho 83844-1103

Nick Vannieuwenhoven
Affiliation: Department of Computer Science, KU Leuven, B-3000 Leuven, Belgium

Keywords: Chow--Veronese variety, Chow--Waring rank, tangential variety, nondefectivity of secant varieties
Received by editor(s): October 30, 2015
Received by editor(s) in revised form: March 30, 2016
Published electronically: August 3, 2017
Additional Notes: The first author was partly supported by NSF grant DMS-0901816
The second author’s research was partially supported by a Ph.D. Fellowship of the Research Foundation–Flanders (FWO) and partially by a Postdoctoral Fellowship of the Research Foundation–Flanders (FWO)
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