The secant line variety to the varieties of reducible plane curves

Let \(\lambda =[d_1,\ldots ,d_r]\) be a partition of \(d\). Consider the variety \(\mathbb {X}_{2,\lambda } \subset {\mathbb {P}}^N,\, N={d+2 \atopwithdelims ()2}-1\), parameterizing forms \(F\in k[x_0,x_1,x_2]_d\) which are the product of \(r\ge 2\) forms \(F_1,\ldots ,F_r\), 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.