Minimal free resolution for points on surfaces
type: Item
instance of: scholarly article
title: Minimal free resolution for points on surfaces
zbMATH Open document ID: 1262.13020
author: Joan Pons-Llopis
author: Rosa M. Miró-Roig
published in: Journal of Algebra
publication date: 2012-11-09T00:00:00Z
review text: Let \(X \subset \mathbb{P}^n\) be a subscheme and let \[ F_\bullet: 0 \longrightarrow F_{n+1} {\longrightarrow} F_n \longrightarrow \cdots \longrightarrow F_1 {\longrightarrow} F_0 {\longrightarrow} R/I(X) \longrightarrow 0 \] be its minimal graded free resolution. The graded Betti numbers \(b_{ij}(X)\) are defined as \[ F_i=\bigoplus_{j \in \mathbb{Z}}R(-i-j)^{b_{i,j}(X)}. \] Let \(Z \subset X\) be a set of distinct points on \(X\) with homogeneous ideal \(I(X)\). A very long-standing problem in algebraic geometry and commutative algebra is to determine the Hilbert function of \(k[x_0, \dots, x_n]/I(Z)\). Then a much more subtle question is to understand all of the syzygies of \(I(Z)\), i.e. to find the minimal graded free resolution of \(k[x_0, \dots, x_n]/I(Z)\) or, at least, the graded Betti numbers \(b_{i,j}(Z)\) of \(Z\). In 1993, \textit{D. Lorenzini} [J. Algebra 156, No. 1, 5--35 (1993; Zbl 0811.13008)] stated the so-called Minimal Resolution Conjecture (MRC) that, grosso modo, says that a general set \(Z\) of \(s\) distinct points in \(\mathbb{P}^n\) has no ghost terms, i.e. \(b_{i+1,j}(Z) \cdot b_{i,j+1}(Z) =0\) for all \(i,j\). The conjecture holds for \(n\leq 4\) and for large values of \(s\) for any \(n\), but it fails for all \(n\geq 6\), \(n\not= 9\). In 1998, \textit{M. Mustaţă} [Matematiche 53, Suppl., 53--81 (1998; Zbl 0943.13010)] introduced a generalized version of MRC for distinct points in arbitrary projective varieties. Let \(\mathrm{Hilb}^s(X)\) be the Hilbert scheme parameterizing \(0-\)dimensional subschemes of \(X\) of length \(s\) and let \(H^s_0\) be the irreducible component whose general points correspond to a set \(Z\) of \(s\) distinct points on \(X\). \textit{Mustaţǎ's conjecture}. Let \(X \subset \mathbb{P}^n\) be a projective variety, let \(P_X(t)\) be its Hilbert polynomial and \(m=\mathrm{reg}(X)\). Let \(s\) be an integer such that \(P_X(r-1) \leq s < P_X(r )\) for some \(r\geq m+1\). Then, there is a non-empty open subset \(U^…
reviewed by: Cristiano Bocci
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 6104541
zbMATH Keywords: Betti numbers
zbMATH Keywords: minimal free resolution
zbMATH Keywords: \(0\)-dimensional schemes
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.01.034
OpenAlex ID: W2072963913
cites work: Thin algebras of embedding dimension three
cites work: Q3758018
cites work: The Minimal Resolution Conjecture for Points on the Cubic Surface
cites work: Exterior algebra methods for the minimal resolution conjecture.
cites work: An inequality for Hilbert series of graded algebras.
cites work: Resolutions of generic points lying on a smooth quadric
cites work: The minimal resolution of the ideal of a general arrangement of a big number of points in \(\mathbb{P}^ n\)
cites work: THE LINEAR SYZYGIES OF GENERIC FORMS
cites work: Reducibility of the families of O-dimensional schemes on a variety
cites work: Compressed Algebras: Artin Algebras Having Given Socle Degrees and Maximal Length
cites work: Power sums, Gorenstein algebras, and determinantal loci. With an appendix `The Gotzmann theorems and the Hilbert scheme' by Anthony Iarrobino and Steven L. Kleiman
cites work: The minimal resolution conjecture
cites work: Minimal resolution of relatively compressed level algebras
cites work: Minimal free resolutions of general points lying on cubic surfaces in \(\mathbb P^3\)
cites work: Reducibility of Punctual Hilbert Schemes of Cone Varieties
cites work: Q4939592
cites work: The minimal free resolution of the homogeneous ideal of \(s\) general points in \(\mathbb{P}^ 4\)
DOI: 10.1016/J.JALGEBRA.2012.01.034
description: scientific article; zbMATH DE number 6104541