Inductively computable unions of fat linear subspaces
type: Item
instance of: scholarly article
title: Inductively computable unions of fat linear subspaces
zbMATH Open document ID: 1348.13029
author: Giuliana Fatabbi
author: Brian Harbourne
author: Anna Lorenzini
published in: Journal of Pure and Applied Algebra
publication date: 2015-08-31T00:00:00Z
review text: The paper introduces complexes of linear varieties, called inclics (for INductively Constructible LInear ComplexeS). An inclic \[ \mathcal{C}=\mathcal{C}(n,r,s;L_0,L_1,\dots, L_r,H_0,H_1,\dots, H_s) \] is a collection of linear subvarieties \(L_0,L_1,\dots, L_r,H_0,H_1,\dots, H_s\subset \mathbb{P}^n\) such that the following conditions hold: (C1) \(H_0,H_1,\dots, H_s\) are distinct hyperplanes; (C2) \(L_i\subseteq H_0\) for \(i>0\) but \(L_0\not\subseteq H_0\); (C3) if \(L_i\subseteq L_j\), then \(i=j\); (C4) for all \(i\geq 0\) and \(j>0\) we have \(L_i\not\subseteq H_i\). By assigning an order of vanishing (i.e., a multiplicity) to each member of the complex, the authors obtain fat linear varieties (fat points if all of the linear varieties are points). The scheme theoretic union of these fat linear varieties gives an inclic scheme \[ X=\sum_{i\geq 0}l_iL_i+\sum_{j>0}h_jH_j \] by which we mean the scheme defined by the ideal \(I_X=(\cap_{i\geq 0}I_{L_i}^{l_i})\cap(\cap_{j>0}I_{H_j}^{h_j})\), where \(l_i\) and \(h_j\) are non-negative integers and \(I_X\subset R=K[\mathbb{P}^n]\). Clearly it is possible to choose coordinates such that \(I_{H_0}=(x_0)\) so \(R'=K[H_0]=K[x_1,\dots, x_n]\). Since \(L_0\not\subseteq H_0\), we may also assume that \(I_{L_0}=(x_{k+1}, \dots, x_n)\) for some \(0\leq k<n\) and denote the linear forms defining \(H_j\) for \(j>0\) by \(\eta_j\). Then define \(Y=l_1L_1+\cdots l_rL_r\) of \(\mathbb{P}^n\), \(Y'=Y\cap H_0\) and \(Y_i'=Y_i\cap H_0\) for \(Y_i=l_1(i)L_1+\cdots +l_r(i)L_r\), where \(l_j(i)=\max(0, l_j-i)\). Thus \(Y_0'=Y'\) and \(I_{Y_i'}=I_{Y_i}\cap K[x_1, \dots, x_n]\). Moreover, \(I_{Y_i}=I_Y:(x_0^i)\). Then finally set \(Z=l_0L_0\), \(W=Y\cup Z\), \(X=W\cup (\bigcup_{j>0}h_jH_j)\), \(L_0'=L_0\cap H_0\), \(Z'=Z\cap H_0\), \(W'=W\cap H_0\) and \(X'=X\cap H_0\). For such a inclic scheme \(X\), the author show there is an inductive procedure for computing the Hilbert function and a resolution of its defining ideal \(I_X\), regardl…
reviewed by: Cristiano Bocci
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 13A02
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 6476974
zbMATH Keywords: inclics
zbMATH Keywords: symbolic powers
zbMATH Keywords: Waldschmidt constant
MaRDI profile type: Publication
full work available at URL: j.jpaa.2015.05.023
OpenAlex ID: W592838380
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DOI: 10.1016/J.JPAA.2015.05.023
description: scientific article; zbMATH DE number 6476974