Globally generated vector bundles and the Weak Lefschetz property

Abstract

Artinian ideals \({I\subset R: =k[x_1,..., x_n]}\) generated by m general forms of given degree have attracted a great deal of attention recently, and one of the main challenging problems is to determine its Hilbert function. The Hilbert function of R/I was conjectured by Froberg, and it is well known that the Weak Lefschetz property imposes severe constraints on the possible Hilbert functions. In this short note, we will focus our attention on Artinian ideals \({I \subset R}\) generated by m general forms all of the same degree d and we analyze whether the Weak Lefschetz property is satisfied. More precisely, for m = n or n ≤ 4 the Weak Lefschetz property holds, and our goal will be to prove that for any integers n and d, the Weak Lefschetz property also holds provided m falls into the interval \({[\frac{1}{d+1} \alpha_{n,d}, \alpha_{n,d}]}\) where \({\alpha_{n,d}={n+d-1\choose d}}\).