Fat points, inverse systems, and piecewise polynomial functions
type: Item
instance of: scholarly article
title: Fat points, inverse systems, and piecewise polynomial functions
zbMATH Open document ID: 0934.13013
author: Anthony V. Geramita
author: Hal Schenck
published in: Journal of Algebra
publication date: 1999-09-09T00:00:00Z
review text: The authors review and explore recently discovered, interesting new connections between the three topics mentioned in the title of the paper. In J. Algebra 174, No. 3, 1080-1090 (1995; Zbl 0842.14002), \textit{J. Emsalem} and \textit{A. Iarrobino} observed that there is a relationship between fat points in \(\mathbb{P}^n\) (i.e. zero-dimensional subschemes \(\mathbb{X}\) defined by ideals of the form \(I_{\mathbb{X}}= {\mathfrak p}_1^{\alpha_1} \cap\cdots\cap {\mathfrak p}_s^{\alpha_s}\)) and ideals generated by powers of linear forms. More precisely, for \(j\geq \max\{\alpha_1,\dots, \alpha_s\}\), the \(j\)-th graded piece of the Macaulay inverse system \((I_{\mathbb{X}})^{-1}\) equals the corresponding graded piece of a certain ideal generated by \(s\) powers of linear forms. In the case of linearly independent linear forms \(\{L_1,\dots, L_s\}\) in two indeterminates, the authors compute explicit formulas for the Hilbert function of \(J= (L_1^{\alpha_1},\dots, L_s^{\alpha_s})\), for the socle degree of \(k[y_0,y_1]/J\), and thus for the minimal graded free resolution of \(J\). In the paper: Compos. Math. 108, No. 3, 319-356 (1997; Zbl 0899.13016), \textit{A. Iarrobino} also observed that there is a relationship between splines (i.e. piecewise polynomial functions satisfying certain smoothness conditions) on a \(d\)-dimensional simplicial complex \(\Delta\) and the ideals generated by powers of the linear forms defining hyperplanes incident to the interior faces of \(\Delta\). The authors recall a certain chain complex whose top homology module is precisely the module \(C^\alpha (\widehat{\Delta})\) of mixed splines on the cone \(\widehat{\Delta}\) over \(\Delta\) which are smooth of order \(\alpha\). In the case of planar splines \(\Delta\subset \mathbb{R}^2\), they are then able to provide formulas for the number of splines in \(C^\alpha (\widehat{\Delta})_k\) of sufficiently large degrees \(k\gg 0\). The paper ends with a discussion of the obstacles which have …
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 65D07
zbMATH DE Number: 1218691
zbMATH Keywords: zero-dimensional scheme
zbMATH Keywords: Macaulay inverse system
zbMATH Keywords: mixed spline
zbMATH Keywords: ideal of linear forms
zbMATH Keywords: fat points
zbMATH Keywords: Hilbert function
zbMATH Keywords: number of splines
reviewed by: Martin Kreuzer
MaRDI profile type: Publication
OpenAlex ID: W1984591672
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DOI: 10.1006/JABR.1997.7361
description: scientific article; zbMATH DE number 1218691