Minimal free resolutions of monomial curves in \({\mathbb{P}}^ 3_ k\)
Let \(0=n_ 0<n_ 1<n_ 2<n_ 3\) be integers with \((n_ 1,n_ 2,n_ 3)=1.\) Let P be the prime ideal of the monomial curve with parametric representation \(x_ i=t_ 0^{n_ B-n_ i}t_ 1^{n_ i}, 0\leq i\leq 3,\) in \({\mathbb{P}}^ 3_ k\) with \(\mu(P)=\mu\) the smallest number of generators of P. Let \(R=k[x_ 0,...,x_ 3]\). It is shown that \(0\to R^{\mu -3}\to^{\phi_ 3}R^{2\mu -4}\to^{\phi_ 2}R^{\mu}\to^{\phi_ 1}R\to R/P\to 0\) is a minimal finite free resolution for \(\mu \geq 3\). The mappings \(\phi_ i\), 1\(\leq i\leq 3\), are determined explicitly.
scientific article; zbMATH DE number 3861327