A note on the weak Lefschetz property of monomial complete intersections in positive characteristic (Q2430587)

Let \(R = k[x_1,\dots,x_r]\) be a polynomial ring over a field \(k\). A graded algebra \(A = R/I\) is said to have the Weak Lefschetz Property (WLP) if multiplication by a general linear form, from any component to the next, always has maximal rank. The case where \(I\) is an artinian monomial ideal has been studied recently by many authors. There are some monomial ideals for which \(A\) fails to have the WLP no matter what the characteristic of \(k\) is, but there are also monomial ideals for which the WLP fails for finitely many characteristics \(p\) and holds for all other characteristics. This question turns out to have surprising connections to seemingly unrelated questions in combinatorics, as well as questions concerning the Grauert-Mülich theorem for certain vector bundles (specifically syzygy bundles). The reviewer together with \textit{R. M. Miró-Roig} and \textit{U. Nagel} [Trans. Am. Math. Soc. 363, No. 1, 229--257 (2011; Zbl 1210.13019)] asked, specifically, for which \(d\) and \(p\) the algebra \(A = k[x,y,z]/(x^d,y^d,z^d)\) fails to have the WLP (it is a classical fact that it does hold in characteristic zero). This paper gives an explicit answer to this question, and as a corollary the authors prove a conjecture of \textit{J. Li} and \textit{F. Zanello} [Discrete Math. 310, No. 24, 3558--3570 (2010; Zbl 1202.13013)] that when \(p=2\), \(A\) has the WLP if and only if \(d=\lfloor\frac{2^t+1}{3}\rfloor\) for some positive integer \(t\). The approach is via restriction of the syzygy bundle to a general line (an approach to the WLP that was introduced by \textit{T. Harima}, the reviewer, \textit{U. Nagel} and \textit{J. Watanabe} [J. Algebra 262, No. 1, 99--126 (2003; Zbl 1018.13001)]), and a key ingredient is a theorem of \textit{C. Han} [Ph.D.\ thesis, Brandeis University (1991)], which computes the syzygy gap for certain ideals in \(k[x,y]\).