The secant line variety to the varieties of reducible plane curves

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Let λ=[d₁,,dᵣ] be a partition of d. Consider the variety X₂,λ ⊂ Pⁿ, N=d+2 2-1, parameterizing forms F∈ k[x₀,x₁,x₂]d which are the product of r≥ 2 forms F₁,,Fᵣ, with degFᵢ = dᵢ. We study the secant line variety σ₂(X₂,λ), and we determine, for all r and d, whether or not such a secant variety is defective. Defectivity occurs in infinitely many "unbalanced" cases.