The Weak Lefschetz Property for monomial complete intersection in positive characteristic

Abstract

, where k is an infinite field. If k has characteristic zero, then Stanley proved that A has the Weak Lefschetz Property (WLP). Henceforth, k has positive characteristic p. If n = 3, then Brenner and Kaid have identified all d, as a function of p, for which A has the WLP. In the present paper, the analogous project is carried out for 4 ≤ n. If 4 ≤ n and p = 2, then A has the WLP if and only if d = 1. If n = 4 and p is odd, then we prove that A has the WLP if and only if d = kq + r for integers k, q, r with 1 ≤ k ≤ p−1 2 , r ∈ q−1 2 , q+1 2 , and q = p e for some non-negative integer e. If 5 ≤ n, then we prove that A has the WLP if and only if n(d−1)+3 2

Algebraic Stories from One and from the Other Pockets

“…There is a complete classification of the monomial complete intersections of uniform degrees that enjoy the WLP, see Brenner and Kaid (2011) (n = 3) and Kustin and Vraciu (2014) (n ≥ 4). The SLP has been classified also for mixed degrees, see Nicklasson (2017b) (n = 2) and Lundqvist and Nicklasson (2016) (n ≥ 3).…”

On the degrees of relations onx1d1,…,xnVraciu1 2015Journal of AlgebraSelf Cite5060View full textAdd to dashboardCite

“…, k n + n , k n+1 ) = 2q, and the desired conclusion holds. Thus we may assume without loss of generality that the only terms in the sum (12) correspond to M = x r 1 1 · · · x r n n , and (a 1,M,l , . .…”

“…, x k n q n , f k n+1 q of the form given by Eq. (12). Moreover, by Lemma 3.7, we may assume that the relation A is not Koszul.…”

“…We may assume without loss of generality that each entry a n+1,M,l of a non-Koszul relation from Eq. (12) is not divisible by f . Otherwise, the relation (a 1,M,l , .…”

“…Example 1.8. We wish to calculate E 5 (6,7,11,12). We have k 1 = k 2 = 1, k 3 = k 4 = 2, r 1 = r 3 = 1, r 2 = r 4 = 2.…”