The minimal free resolution for the first infinitesimal neighborhoods of \(n\) general points in the plane
Let \(Y(n)\) denote the scheme in \({\mathbb{P}}^2\) defined by the union of the first infinitesimal neighborhoods of \(n\) general points. If the points are defined by the prime ideals \({\mathfrak p}_1,\dots,{\mathfrak p}_n\) then \(Y(n)\) has saturated homogeneous ideal \(I(n) = {\mathfrak p}_1^2 \cap \dots \cap {\mathfrak p}_n^2\). One can ask for the Hilbert function of \(Y(n)\), i.e.\ one can ask for the number of conditions imposed on forms of any given degree by requiring that the forms, and also their first partial derivatives, vanish at \(n\) general points. This has been studied by several authors, most notably by Alexander and Hirschowitz in a series of papers (in fact, these authors considered the case of points in \({\mathbb{P}}^r\)). Except for a few exceptions, these ``fat points'' impose independent conditions on forms of degree \(d\), i.e.\ \(3n\) conditions. This paper addresses the next natural question, that of determining the minimal free resolution of \(I(n)\). Since \(Y(n)\) is an arithmetically Cohen-Macaulay subscheme of \({\mathbb{P}}^2\), its minimal free resolution is completely determined from knowledge of the Hilbert function (from above) and knowledge of the degrees of the minimal generators of the ideal \(I(n)\). If we let \(R = K[X_0,X_1,X_2]\), the dimension of the cokernel of the multiplication map \(R_1 \otimes I(n)_k \rightarrow I(n)_{k+1}\) is the number of minimal generators of \(I(n)\) in degree \(k+1\). The ``expected'' number of minimal generators can be computed by asking that this map have maximal rank. The main result of this paper is that except for \(n=2\), the expected number of minimal generators is actually achieved.
scientific article; zbMATH DE number 1340598
scientific article