On the minimal number of generators of the quotient of a complete intersection by a general linear form (Q2837118)

Let \(R = k[x_1, \ldots, x_r]\) be the polynomial ring in \(r\) variables, where the field \(k\) has characteristic 0. This paper studies quotients of the form \(I + (\ell)/(\ell)\) where \(I = (f_1, \ldots, f_r) \subset R\) is a complete intersection and \(\ell\) is a general linear form. For ease of notation, let \(\deg(f_i) = d_i\) for \(1 \leq i \leq r\) and assume \(2 \leq d_1 \leq d_2 \leq \cdots \leq d_r\). The minimal number of generators of the quotient \(I + (\ell)/(\ell)\) is either \(r\) or \(r-1\). The main result of this paper is that the minimal number of generators is equal to \(r\) if and only if \(d_r \leq d_1 + d_2 + \cdots + d_{r-1} - (r-1)\). That is, the minimal number of generators depends solely on the degrees of the generators of the complete intersection. The proof of this involves five preliminary lemmas. It is noted in the Introduction that this result was proved for \(r=3\) in the papers \textit{T. Harima} et al. [J. Algebra 262, No. 1, 99--126 (2003; Zbl 1018.13001)] and \textit{J. Watanabe} [Proc. Am. Math. Soc. 126, No. 11, 3161--3168 (1998; Zbl 0901.13019)]. The first of these papers uses the result that the ``standard graded Artinian rings of embedding codimension three have the weak Lefschetz property''. The second of these papers replaces \(k\) with a discrete valuation ring, and the proof in the paper under review generalizes this approach. It is also noted in the Introduction that the main result of this paper (for any \(r\)) follows from Proposition 3.11 of \textit{C. Huneke} and \textit{B. Ulrich} [J. Algebr. Geom. 2, No. 3, 487--505 (1993; Zbl 0808.14041)]; the approach in the paper under review is different and self-contained.NEWLINENEWLINEThe paper concludes with an application to the \textit{Weak Lefschetz Condition}. In particular, the author shows that the quotient \(A = R/(f_1, \ldots, f_r)\), with \(\deg(f_i) = d_i\) and \(2 \leq d_1 \leq \cdots \leq d_r\), has the weak Lefschetz condition if \(d_r \geq d_1 + \cdots + d_{r-1} - r\). That is, if \(d_r \geq d_1 + \cdots + d_{r-1} - r\), then there is a linear form \(\ell\) in \(A\) such that the multiplication map \(\cdot \ell: A_i \rightarrow A_{i+1}\) is either injective or surjective for each degree \(i\).