On the minimal free resolution of s points in \({\mathbb{P}}^ 3\)
Let S be a finite set of points in \({\mathbb{P}}_ k^ 3=\Pr oj(R)\), \(R=k[z_ 0,z_ 1,z_ 2,z_ 3]\) with \(card(S)=s\) (k an algebraically closed field). Write \(s=x+(n+2)(n+1)n/6\) with \(0<x<(n+2)(n+1)/2\). (The case \(x=0\) is not interesting.) Let \({\mathcal I}\) be the sheaf of ideals of S, and I the homogeneous ideal of S. Assume that \(h^ 0({\mathbb{P}}^ 3,{\mathcal I}(n-1))=0=h^ 1({\mathbb{P}}^ 3,{\mathcal I}(n))\) (this is equivalent for S to be in generic position in the language of certain authors, (i.e. that R/I has Hilbert function \(H(i)=\min (s,\left( \begin{matrix} i+3\\ 3\end{matrix} \right))\), which is ``as large as possible''). It is known that I has a minimal free resolution \[ 0\to cR(-n-3)\oplus dR(-n-2)\to uR(-n-2)\oplus vR(-n-1)\to bR(-n-1)\oplus aR(-n)\to I\to 0 \] for nonnegative integers a, b, c, d, u, v. The integers a and c are determined by the generic position assumption on S, namely \(a=(n+3)(n+2)(n+1)/6-s\), \(c=(n+2)(n+1)/2-a\). This leaves two degrees of freedom, say b and d. The integers b, and d are determined generically [\textit{E. Ballico} and \textit{A. V. Geramita} in Algebraic geometry, Proc. Conf., Vancouver/B.C. 1984, CMS Conf. Proc. 6, 1-10 (1986; Zbl 0621.14003)] but a range of values for b and d are possible (still with S in generic position). The author shows how to construct sets S with various values of b and d, as well as restrictions on the possible values of b and d.
scientific article; zbMATH DE number 4150369
scientific article