The union of two linear star configurations in \(\mathbb{P}^2\) all have generic Hilbert function
type: Item
instance of: scholarly article
title: The union of two linear star configurations in \(\mathbb{P}^2\) all have generic Hilbert function
zbMATH Open document ID: 1506.13025
author: Yong Su Shin
published in: Journal of Pure and Applied Algebra
publication date: 2022-12-08T00:00:00Z
review text: Let \(\{ F_i \}, 1 \leq i \leq t\) be general forms of respective degrees \(d_1,\dots,d_t\) in \(k[x_0,\dots,x_n]\), where \(k\) is a field. The author calls the codimension \(r \geq 2\) variety \(X\) defined by the ideal \(\bigcap_{1 \leq i_1 < \cdots < i_r \leq s} (F_{i_1}, \dots, F_{i_r})\) a star configuration in \(\mathbb{P}^n\), and if the \(H_i\) are all linear then he calls \(X\) a linear star configuration. (In the literature there is some variation on different aspects of these definitions.) When \(n=2\) we call \(t\) the type of the star configuration. In this paper the author resolves a question he had posed in an earlier paper (2012). Specifically, if \(X\) and \(Y\) are linear star configurations in \(\mathbb{P}^2\) of types \(s\) and \(t\) respectively (note that they are chosen generally by definition), then he asked if they have generic Hilbert function. That is, do they impose the maximum possible number of conditions on forms of every degree? The author gives an affirmative answer to this question. To achieve this, he specializes to the case of the union of a linear star configuration of type \(s\) and a fat point scheme of multiplicity \((t-1)\) in \(\mathbb{P}^2\), proves this case, and then applies it to the case of two linear star configurations. He gives an application to the case of codimension 2 linear star configurations in \(\mathbb{P}^3\). That is, if \(X\) and \(Y\) are such configurations with \(s,t \geq 3\), the author shows that the artinian algebra \(R/(I_X + I_Y)\) has the Weak Lefschetz Property.
reviewed by: Juan C. Migliore
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 7628606
zbMATH Keywords: Hilbert functions
zbMATH Keywords: generic Hilbert function
zbMATH Keywords: star configurations
zbMATH Keywords: linear star configurations
zbMATH Keywords: the weak Lefschetz property
zbMATH Keywords: a fat point scheme
MaRDI profile type: Publication
full work available at URL: j.jpaa.2022.107270
OpenAlex ID: W4309776186
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DOI: 10.1016/J.JPAA.2022.107270
description: scientific article; zbMATH DE number 7628606