Some algebraic consequences of Green's hyperplane restriction theorems (Q2267435)

The authors study graded level and Gorenstein algebras, looking in particular at the Hilbert function and the weak Lefschetz property. The starting point is the idea of restating some results of \textit{M. Green} [Lect. Notes Math. 1389, 76--86 (1989; Zbl 0717.14002)]. More precisely they restate in an algebraic language Theorem 3 and Theorem 4, in which Green studies some linear systems on projective spaces. In the first of the two main results of the paper the authors prove that, assuming that the characteristic of the base field \(k\) is different from 2, for any integer \(m\geq 2\) \((1,\binom{m+3}{3},(m+1)^2,\binom{m+3}{3},1)\) is not a Gorenstein h-vector. In particular this implies that \((1,10,9,10,1)\) is not a Gorenstein h-vector, which answers a question posed by \textit{J. Migliore, U. Nagel} and \textit{F. Zanello} [Proc. Am. Math. Soc. 136, No. 8, 2755--2762 (2008; Zbl 1148.13011)] and shows that the bound they give in Theorem 4 is not sharp. In the other main result, again under the hypothesis that \(\text{char}\,k\neq 2\), they prove that, if \(h=(1,e,h_2,\dots,h_e)\) is a Gorenstein h-vector and \(e\geq 2\), then \(h_2\geq e\). Moreover, if \(h_2=e\), then \(h_i=e\) for any \(i=1,\dots,e-1\) and any Gorenstein algebra with h-vector \(h\) has the weak Lefschetz property. In the paper the authors provide other results that follow from Green's theorems, giving some conditions on the Hilbert function in such a way that an algebra \(A\), which can be either a level algebra or a Gorenstein artinian algebra, has the weak Lefschetz property. In the appendix of the paper the authors give some sketches of new proofs of Theorem 3 and Theorem 4 of Green's paper [loc. cit.], communicated to the authors by Green himself, since they found some gaps in the original proofs.