The ideal resolution for generic 3-fat points in \(\mathbb P^2\). (Q1420632)

Let \(P_1,\dots,P_n\) be generic points in \(\mathbb P^2\), denote by \(Y=Y(n)\) the 0-dimensional scheme of \(m\)-fat points with support on the \(P_i\)'s, i.e. the scheme defined by the ideal \(I_Y= {\mathfrak p}_1^m\cap \dots \cap {\mathfrak p}_n^m\) , where \({\mathfrak p}_i\) is the homogeneous ideal of the point \(P_i\) in \(k[X_0, X_1,X_2]\) (\(k\) an algebraically closed field of characteristic zero). The paper deals with the problem of determining the degrees of a minimal set of generators for the ideal \(I_Y\), when \(m=3\) [for the case \(m=2\) see \textit{M. Idà}, J. Algebra 216, No. 2, 741--753 (1999; Zbl 0943.13008)]. Since \(Y\) is 0-dimensional of codimension 2, this is the same as determining the graded Betti numbers of a minimal free resolution of \(I_Y\). The authors prove that if \(n\neq 2,3,5\) the natural maps \(\sigma_k:H^0({\mathcal I}_Y(k)\otimes H^0({\mathcal O}_{\mathbb P^2} (1)) \rightarrow H^0({\mathcal I}_Y(k+1))\) (where \({\mathcal I}_Y\) is the ideal sheaf associated to \(I_Y\)) are of maximal rank for each \(k\). As \(Y\) has maximal Hilbert function, this means that the ideal \(I_Y\) has the same ''good resolution'' as the ideal of a scheme of \(6n\) generic (reduced) points.