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

Abstract

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, Miró-Roig and Nagel and, equivalently, characterizes for which characteristics the rank-2 syzygy bundle Syz(X ^d, Y ^d, Z ^d) on ${{\mathbb {P}}^2}$ satisfies the Grauert-Mülich theorem. As a corollary we obtain that for p = 2 the algebra A has the weak Lefschetz property if and only if ${d=\lfloor\frac{2^t+1}{3}\rfloor}$ for some positive integer t. This was recently conjectured by Li and Zanello.