Quasi-complete intersection ideals of height \(2\) (Q1295506)

Let \(R = k[x_0,\dots,x_n]\), \(n \geq 3\), and let \({\mathfrak m} = (x_0,\dots,x_n)R\). A homogeneous ideal \(I \subset R\) of height \(m\) is a quasi-complete intersection (q.c.i.) on \(m+1\) elements if \(I\) is unmixed and there exist \(m+1\) homogeneous elements \(F_0,\dots,F_m\) in \(I\) such that \((F_0,\dots,F_m) R = I \cap {\mathfrak q}\) for some \({\mathfrak m}\)-primary ideal \(\mathfrak q\). This paper is concerned with the problem of describing minimal generating sets for such an ideal, assuming that it is prime and its height is 2. One of the main results is the following. Assume \({\mathfrak p}\) is a homogeneous prime ideal of height 2 with at least 3 minimal generators, and assume that \(\mathfrak p\) is a q.c.i.\ on \(F_0,F_1,F_2\). If \(\mathfrak B\) is an arbitrary minimal homogeneous generating set of \(\mathfrak p\), then there are elements \(B_0,B_1,B_2\) in \(\mathfrak B\) such that \( {\mathfrak B}/\{B_0,B_1,B_2\} \cup \{F_0,F_1,F_2\} \) is a minimal generating set of \(\mathfrak p\). Another result, slightly more technical, gives a way of obtaining, from a fixed homogeneous minimal generating set for a q.c.i.\ prime ideal \(\mathfrak p\), a set of three polynomials on which \(\mathfrak p\) is a q.c.i. The authors apply these results to monomial curves, by characterizing those monomial curves with 4 minimal generators which are in fact quasi-complete intersections.