Abstract
We find that some union of two star-configurations in ℙ2 has generic Hilbert function. Applying the result, we prove that some Artinian quotients of a coordinate ring of a star-configuration in ℙn satisfy the weak-Lefschetz property. More precisely, let 𝕏 and 𝕐 be star-configurations in ℙ2 of type (2, s) and (2, s + 1) defined by forms F1,…, Fs, and G1,…, Gs, L, respectively, with deg(Fi) = deg(Gi) ≤2 for i = 1,…, s and s ≥ 3. If L is a general linear form in R = 𝕜[x0, x1, x2], then R/(I𝕏 + I𝕐) has the weak-Lefschetz property with a Lefschetz element L, which extends the result of [21].
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