On the singular loci of higher secant varieties of Veronese embeddings
Katsuhisa Furukawa, Kangjin Han

TL;DR
This paper investigates the singular loci of higher secant varieties of Veronese embeddings, introducing the concept of m-subsecant loci to understand their geometric and singularity properties across various parameters.
Contribution
It defines m-subsecant loci for secant varieties of Veronese embeddings and analyzes their relation to singularities, extending known results to arbitrary k, d, n, and exploring new sources of singularities.
Findings
m-subsecant loci can be contained in the singular locus
Trichotomy between smoothness and singularity depending on parameters
Application of Young flattening to compute conormal spaces
Abstract
The -th secant variety of a projective variety , denoted by , is defined to be the closure of the union of -planes spanned by points on . In this paper, we examine the -th secant variety of the image of the -uple Veronese embedding of to with , and focus on the singular locus of , which is only known for . To study the singularity for arbitrary , we define \emph{the -subsecant locus} of to be the union of with any -plane . By investigating the projective geometry of moving embedded tangent spaces along subvarieties and using known results on the secant defectivity and the…
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Taxonomy
TopicsTensor decomposition and applications · Coding theory and cryptography · Advanced Differential Equations and Dynamical Systems
