A note on the weak Lefschetz property of monomial complete intersections in positive characteristic

الأصل · EN

Let K be an algebraically closed field of characteristic p > 0. We apply a theorem of C. Han to give an explicit description for the weak Lefschetz property of the monomial Artinian complete intersection A = K[X,Y,Z]/(Xᵈ,Yᵈ,Zᵈ) in terms of d and p. This answers a question of J. Migliore, R. M. Miro-Roig and U. Nagel and, equivalently, characterizes for which characteristics the rank-2 syzygy bundle Syz(Xᵈ,Yᵈ,Zᵈ) on PP² satisfies the Grauert-Muelich theorem. As a corollary we obtain that for p=2 the algebra A has the weak Lefschetz property if and only if d=(2ᵗ+1)/3 or d=(2ᵗ-1)/3 for some positive integer t. This was recently conjectured by J. Li and F. Zanello.