The Chow rank of a homogeneous polynomial \(f\) of degree \(d\) is the minimum \(s\) such that \(f\) can be represented as a sum of \(s\) products of \(d\) linear forms, i.e., \[ f=l_{1,1}\cdot \ldots \cdot l_{1,d}+\ldots + l_{s,1}\cdot \ldots \cdot l_{s,d}, \] where the \(l_{i,j}\) are linear forms.
Thus, the Chow rank of a generic form of degree \(d\) is equal to the smallest \(s\) such that \(\sigma_s(\mathrm{Split}_d(\mathbb P^n))\), the secant variety of the Chow variety fills the ambient space.
The expected dimension of \(\sigma_s(\mathrm{Split}_d(\mathbb P^n))\) is \[ \mathrm{expdim} \sigma_s(\mathrm{Split}_d(\mathbb P^n))=\min \{s(dn+1),\binom{n+d}{d}\}-1. \]
If \(\dim \sigma_s(\mathrm{Split}_d(\mathbb P^n))= \mathrm{expdim} \sigma_s(\mathrm{Split}_d(\mathbb P^n))\) then \(\sigma_s(\mathrm{Split}_d(\mathbb P^n))\) is nondefective.
In [J. Pure Appl. Algebra 215, No. 3, 201–220 (2011; Zbl 1211.14058)] E. Arrondo and A. Bernardi formulated the conjecture that the secant variety \(\sigma_s(\mathrm{Split}_d(\mathbb P^n))\) is nondefective unless \(d=2\) and \(2\leq s \leq \frac n2\).
The author shows [Theorem 1.4] that this conjecture is true for \(s\leq 35\). Moreover he gives an elementary argument [Theorem 2.4] showing that for \(d=2\) and all \(s\) and \(n\) subject to the condition \(2\leq s\leq \frac n2\), the smallest variety \(\sigma_s(\mathrm{Split}_2(\mathbb P^n))\) is defective.