An algebraic approach to the regularity index of fat points in \(P^ n\)
This paper is a survey of results about bounds for the regularity index of fat points in \(\mathbb{P}^ n\) [mainly by \textit{M. V. Catalisano}, \textit{G. Valla} and the author, Proc. Am. Math. Soc. 118, No. 3, 717-724 (1993; Zbl 0787.14030), and \textit{G. Valla} and the author, Math. Z. 219, No. 2, 187-201 (1995)]. The question is the following: Let \(X = \{P_ 1, \dots, P_ s\}\) be a set of points in \(\mathbb{P}^ n_ k\), where \(k = \overline k\) and \(\text{char} k = 0\), and let \(p_ i\) be the homogeneous ideal, in \(R = k[x_ 0, \dots, x_ n]\), associated to \(P_ i\); consider for \(m_ 1 \geq \cdots \geq m_ s \geq 0\), the scheme \(Z\) associated to the ideal \(I = {\mathfrak p}_ 1^{m_ 1} \cap \cdots \cap {\mathfrak p}_ s^{m_ s}\); then, for every \(t \geq 0\), the vector space \(I_ t\) represents the space of hypersurfaces of \(\mathbb{P}^ n\) having a point of multiplicity at least \(m_ i\) at each \({\mathfrak p}_ i\). The multiplicity of \(Z\) is \(e(Z) = \sum_ i {m_ i + n - 1 \choose n}\), and its index of regularity \(r(Z)\) is the least integer \(g\) for which \(h_ A(t) = e(Z)\), where \(h_ A(t)\) is the Hilbert function of the coordinate ring \(A = R/I\) of \(Z\); in other words, \(r(Z)\) is the least integer \(t\) for which \(Z\) imposes \(e(Z)\) independent conditions on hypersurfaces of degree \(t\). Even for \(n = 2\), it is quite a hard problem to determine the value of \(r(Z)\) for a generic choice of \(X\) (it will only depend on the \(m_ i\)'s). A (sharp) bound for \(r(Z)\), for any choice of \(X\) in general position (i.e. no \(n + 1\) points on a hyperplane), can be found thanks to an approach described in the paper under review: consider the ideal \(J = {\mathfrak p}_ 1^{m_ 1} \cap \cdots \cap {\mathfrak p}_{s - 1}^{m_{s - 1}}\), then \(r(Z) = \max \{m_ s - 1, r(R/J), r(R/J + {\mathfrak p}_ s^{m_ s})\}\). One can use this in order to work by induction (since \(r(Z)\) is known for small \(s)\), noting that \(r(R/(J + {\mathfrak p}_ s^{m_ s}))\) is the least \(t\) for which \([R/(J + {\mathfrak p}_ s^{m_ s})]_ t = 0\), and constructing a form \(F\) of degree \(t\) as small as possible with \(F \in J_ t\) and \(F \notin I_ t\). The bound which is found is a generalization of B. Segre's bound for \(n = 2\): \(r(Z) \leq \max \{m_ 1 + m_ 2 - 1, [(\sum m_ i + n - 2)/n]\}\). The bound is sharp since it is attained if \(X\) lies on a rational normal curve; this is an ``if and only if'' when \(s \geq 2n + 3\) and \(m_ s\) is large enough. -- When \(X\) has the uniform position property, i.e. every two subsets of \(X\) with same cardinality have the same Hilbert function, better bounds can be found.
bounds for the regularity index of fat points
scientific article; zbMATH DE number 729228
scientific article