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

Let K be an algebraically closed field of characteristic p > 0. We apply a theorem of Han to give an explicit description for the weak Lefschetz property of the monomial Artinian complete intersection A = K[X, Y, Z]/(X(d), Y(d), Z(d)) in terms of d and p. This answers a question of Migliore, Miro-Roig and Nagel and, equivalently, characterizes for which characteristics the rank-2 syzygy bundle Syz(X(d), Y (d), Z(d)) on P(2) satisfies the Grauert-Mulich theorem. As a corollary we obtain that for p = 2 the algebra A has the weak Lefschetz property if and only if d = left perpendicular2(t)+1/3right perpendicular for some positive integer t. This was recently conjectured by Li and Zanello.