The secant line variety to the varieties of reducible plane curves

Let $\lambda =[d_1,\dots ,d_r]$ be a partition of $d$. Consider the variety ${\mathbb{X}}_{2,\lambda }\subset {\mathbb{P}}^N$, $N=\left(\genfrac{}{}{0px}{}{d+2}{2}\right)-1$, parameterizing forms $F\in k[x_0,x_1,x_2{]}_d$ which are the product of $r\ge 2$ forms $F_1,\dots ,F_r$, with deg$F_i=d_i$. We study the secant line variety ${\sigma }_2({\mathbb{X}}_{2,\lambda })$, and we determine, for all $r$ and $d$, whether or not such a secant variety is defective. Defectivity occurs in infinitely many “unbalanced” cases.