Secant varieties of Segre-Veronese varieties $\mathbb{P}^m\times\mathbb{P}^n$ embedded by $\mathcal{O}(1,2)$ are non-defective for $n\gg m^3$, $m\geq3$

TL;DR

This work proves non-defectivity for secant varieties of Segre–Veronese varieties $X_{m,n}$ embedded by $\mathcal{O}(1,2)$ in a high-noise regime where $n$ grows cubically with $m$ (with explicit parity-based refinements). The authors extend subabundant results by Abo–Brambilla to the superabundant range using a generalized inductant construction inspired by Brambilla–Ottaviani, recasting the problem in terms of coordinate configurations and a virtual dimension $\mathrm{vdim}$ that tracks conditions imposed by double points on $\mathcal{I}(1,2)$. They organize a cascade of inductants, separately handling nice and ugly $(m,n)$ pairs, and confirm the necessary base cases via computer-assisted proofs, thereby reducing the non-defectivity question to finitely many explicit checks. The resulting theorem ensures non-defectivity for all secant varieties in the stated regime, advancing the understanding of border/rank phenomena for two-factor Segre–Veronese varieties and providing a framework that integrates rigorous arithmetic, combinatorial configurations, and computational certificates. These insights deepen the link between geometric properties of Segre–Veronese embeddings and computational verification strategies for high-dimensional tensor decompositions.

Abstract

We prove that for any $m\geq3$, $n\gg m^3$, all secant varieties of the Segre-Veronese variety $\mathbb{P}^m\times\mathbb{P}^n$ have the expected dimension. This was already proved by Abo and Brambilla in the subabundant case, hence we focus on the superabundant case. We generalize an approach due to Brambilla and Ottaviani into a construction we call the inductant. With this, the proof of non-defectivity reduces to checking a finite collection of base cases, which we verify using a computer-assisted proof.

Figures (6)

• Figure 1: The parameters specifying the configuration $Z$ of Example \ref{['ex:basic']}, visualized in Venn diagrams.
• Figure 2: The parameter functions specifying the family $B(m,n)$ in Example \ref{['ex:inductant']}, visualized in Venn diagrams. The newly added subvariety $L_3$ is represented by a dashed circle and we leave cells blank instead of writing zeros. Further, in $p'_I$ we list the "initial" values before erasure of irrelevant points and mark the cells where this erasure takes place with a red background.
• Figure 3: On the left, $(M_1,N_1)$ and $(M_2,N_2)$ commute in $(m,n)$ as functions $\mathbb{Z}^2\to\mathbb{Z}^2$. On the right, arrows indicate inductant constructions, with the inductant family pointing to the family is an inductant of. The three full arrows determine $B_1$, $B_2$, and $C$, we then claim that the inductant relation indicated by the dashed arrow also holds.
• Figure 4: Families of coordinate configurations in the inductive strategy for the nice case. An arrow labeled with $(M,N)$ denotes that a family is an $(M,N)$-inductant of the family being pointed to. We denote $Q(m)=3m^2-6m+2$, $\ell(m)=12m-22$ for short.
• Figure 5: Families of coordinate configurations in an inductive strategy to prove non-defectivity of $A_1$. An arrow labeled with $(M,N)$ denotes that a family is an $(M,N)$-inductant of the family being pointed to. These inductant constructions are sufficient to define all $A_2,\dots, A_6$; we defined $A_1$ explicitly in the paragraph preceding Proposition \ref{['prop:A1']}. The reader may notice that $A_2(m,n)$ is actually the configuration from Example \ref{['ex:basic']}.

Theorems & Definitions (73)

• Theorem : abo-brambilla
• Theorem 1.1
• Definition 2.1
• Lemma 2.2
• proof
• Definition 3.1
• Definition 3.2
• Example 3.3
• Definition 3.4
• Lemma 3.5