The secant line variety to the varieties of reducible plane curves

Abstract

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.