Quadrics through a set of points and their syzygies
type: Item
instance of: scholarly article
title: Quadrics through a set of points and their syzygies
zbMATH Open document ID: 0815.14034
DOI: 10.1007/BF02571888
published in: Mathematische Zeitschrift
publication date: 1995-07-02T00:00:00Z
full work available at URL: https://eudml.org/doc/174713
review text: Let \(X\) be a nondegenerate projective variety in \(\mathbb{P}^ n = P^ n_ k\), where \(k\) is an algebraically closed field. Let \(I = I(X)\) denote the defining ideal of \(X\) in the polynomial ring \(R : = k[X_ 0, \dots, X_ n]\) and \(A : = R/I\) its homogeneous coordinate ring. The graded \(R\)- module \(A\) has a minimal free resolution \(\mathbb{E} : 0 \to E_ h \to \cdots \to E_ 1 \to R \to A \to 0\) where \(h = hd_ R (A)\) and \(E_ i = \oplus^{\beta_ i}_{j=1} R(-d_{ij})\). Several recent investigations and conjectures relate the numerical invariants of the resolution with the geometrical properties of \(X\). In this paper we are mainly concerned with the ``linear part'' of the resolution. Hence for every \(i = 1, \dots, h\), we let \(a_ i (X) = \dim_ k [\text{Tor}^ R_ i (A, k)]_{i + 1}\) to be the multiplicity of the shift \(i + 1\) in \(E_ i\). Hence we are interested in projective varieties \(X\) which lie on some quadric and we want to study the syzygies of the quadrics passing through \(X\). The main idea coming from M. Green is that a long linear strand in the resolution has a uniform and simple motivation. Following this approach we start by proving that \(a_ i \neq 0\) in the following geometric situations. Either \(X\) is contained on a variety of minimal degree and dimension \(n - i\), or \(X\) is contained in the union of two linear subspaces of \(\mathbb{P}^ n\) of dimension \(k\) and \(r\) where \(k, r < n\) and \(r + k = 2n - i - 1\). A natural question is to what extent the converse of the above result is true. In section 1 we prove that this is the case if \(i = 1\) (trivial) or \(i = n\). -- In section 2 we consider the case \(i = n - 1\) and we restrict ourselves to reduced schemes which are zero dimensional. -- A basic result by M. Green, the so-called strong Castelnuovo lemma, says that for a set \(X\) of distinct points in linearly general position in \(\mathbb{P}^ n\), we have \(a_{n-1} \neq 0\) if and only if the points are on a rational…
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 14P05
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 13D02
zbMATH DE Number: 737160
zbMATH Keywords: set of points in projective space
zbMATH Keywords: syzygies of quadrics
zbMATH Keywords: rational normal scroll
zbMATH Keywords: strong Castelnuovo lemma
zbMATH Keywords: minimal resolution conjecture
author: Maria Pia Cavaliere
author: M. E. Rossi
author: G. Valla
describes a project that uses: CoCoA
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 737160