On the minimal free resolution of finite sets in \(\mathbb{P}^ N\)
\textit{A. Lorenzini} [J. Pure Appl. Algebra 60, No. 3, 273-288 (1989; Zbl 0702.13013)] has given a conjecture which describes the expected graded Betti numbers in the resolution of the homogeneous ideal of a generic set of \(s\) points in \(\mathbb{P}^ N\). In the paper under review the following result is proved: Let \(m \geq 3\); then the set of integers \(1 \leq s \leq N\) for which \(s\) generic points in \(\mathbb{P}^ m\) does not have the expected resolution is at most a \(O(N^{1 - (1/m)})\). -- This comes as a corollary of a result which gives algorithms able to compute two constants \(a(m,i,t)\), \(b(m,i,t)\) such that for all \(t,m \in \mathbb{N}\) with \(t \geq m \geq 2\) and for any 0-dimensional scheme \(Z \subseteq \mathbb{P}^ m\), we have: (1) for a generic set \(Y\) of \(a(m,i,t)\) points in \(\mathbb{P}^ m\): \[ (*)\;h^ 1(\mathbb{P}^ m, \Omega^ i(t) \otimes {\mathcal J}_{Z \cup Y}) \leq h^ 1(\mathbb{P}^ m, \Omega^ i(t - 2) \otimes {\mathcal J}_ Z); \] (2) there are two sets \(S \subseteq S' \subseteq \mathbb{P}^ m\), of cardinality \(a(m,i,t)\) and \(b(m,i,t)\) respectively, such that they both satisfy inequality \((*)\). Bounds for \(b(m,i,t) - a(m,i,t)\) are found (in the order of \(t^{m - 3})\) and this allows to prove the previous result. As an application, examples of ways to find sets of points with prescribed deviation from the expected resolution are given.
scientific article; zbMATH DE number 604870
scientific article