A bound on the degree of schemes defined by quadratic equations
type: Item
instance of: scholarly article
zbMATH Open document ID: 1263.14052
author: José Carlos Sierra
author: Alberto Alzati
published in: Forum Mathematicum
publication date: 2012-08-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1006.5857
Mathematics Subject Classification ID: 14M99
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 13D02
zbMATH DE Number: 6072027
zbMATH Keywords: 2-Veronese embedding
zbMATH Keywords: varieties defined by quadrics
zbMATH Keywords: syzygies
MaRDI profile type: Publication
OpenAlex ID: W2963103030
arXiv ID: 1006.5857
cites work: Restricting linear syzygies: algebra and geometry
cites work: Symmetric Products and Jacobians
cites work: A SECANT FORMULA
cites work: Some results on secant varieties leading to a geometric flip construction.
cites work: Equal binomial coefficients: Some elementary considerations
cites work: Projective invariants of quadratic embeddings
DOI: 10.1515/FORM.2011.081
title: A bound on the degree of schemes defined by quadratic equations
review text: Projective varieties defined by quadratic equations are a fascinating subject in algebraic geometry by many point of views; here the degree of such a scheme is studied, and a bound for it (in terms of its codimension) is given under a few general condition about syzygies of its ideal.NEWLINENEWLINELet \(X \subset {\mathbb P}^r = {\mathbb P}_{\mathbb C}^r\); a projective scheme of dimension \(n\); we will say that the scheme \(X\) is \textit{reduced in codimension zero} if a general linear space \({\mathbb P}^n\) intersects \(X\) in a reduced scheme, and we will say that \(X\) is \textit{smooth and integral in codimension one} if \(n\geq 1\) and a general linear space \({\mathbb P}^{n-1}\) intersets \(X\) in a smooth integral curve. Suppose that the ideal of \(X\) is generated by quadrics; will say that \(X\) satisfies property \(K_2\) if the trivial (Koszul) relations among those quadrics are generated by linear syzygies.NEWLINENEWLINEThen a formulation of the main result (actually, this is a corollary since the main results uses a condition weaker than \(K_2\)) is the following: Let \(X\subset {\mathbb P}^r\) be an \(n\)-dimensional scheme of degree \(d\) defined by an \(\alpha\)-dimensional linear system \(\Lambda\) of quadrics. Let \(X\) be reduced in codimension 0 and satisfying \(K_2\); if \(c=r-n\geq 2\), then: \(\alpha \geq 2c-2\) and \({d \choose 2}\leq {2c-1\choose c-1}\) (and equality holds iff \(\alpha = 2c-2\)).NEWLINENEWLINEMore can be deduced when equality holds if \(X\) is also smooth and integral in codimension one.NEWLINENEWLINEOne main tool for the proof of this result is the study of apparent double points on the 2-Veronese embedding of a suitable quadric in a linear space \({\mathbb P}^{c}\).NEWLINENEWLINENote that the bound \({d \choose 2}\leq {2c-1\choose c-1}\) is better than the obvious bound \(d\leq 2^c\), and asymptotically is of the form: \(d\leq {2^c\over \sqrt{\sqrt{\pi c}}} .\)
reviewed by: Alessandro Gimigliano
description: scientific article; zbMATH DE number 6072027