Non-defectivity of Segre-Veronese varieties

TL;DR

The paper addresses the defectivity of Segre-Veronese varieties by proving that when the number of factors satisfies $k\ge 3$ and every embedding degree satisfies $d_i\ge 3$, the variety $\mathrm{SV}^{\mathbf{d}}_{\mathbf{n}}$ is never $m$-defective for any $m$, using a differential Horace induction that simultaneously handles dimension, degree, and the number of factors. A key technical tool is a splitting lemma (Segre induction) based on Terracini's lemma, which transfers non-defectivity from a factor to a Segre product and enables an inductive chain that proves the main result and related almost-optimal non-defectivity statements when one degree equals $1$. The authors also derive identifiability consequences via MM-identifiability, showing that non-defectivity implies uniqueness of typical decompositions in many cases. The work extends the non-defectivity landscape to arbitrary factor counts and provides a streamlined inductive framework with potential impact on the study of partially symmetric tensor decompositions in mathematics, computer science, and statistics.

Abstract

We prove that Segre-Veronese varieties are never secant defective if each degree is at least three. The proof is by induction on the number of factors, degree and dimension. As a corollary, we give an almost optimal non-defectivity result for Segre-Veronese varieties with one degree equal to one and all the others at least three.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 2.2
• Theorem 2.3: AboBra13
• Remark 2.4
• Theorem 2.5: blomenhofer2023nondefectivity
• Theorem 3.1: ballico
• Corollary 3.2
• proof