Projective generation of curves in polynomial extensions of an affine domain and a question of Nori
type: Item
instance of: scholarly article
title: Projective generation of curves in polynomial extensions of an affine domain and a question of Nori
zbMATH Open document ID: 0936.13005
author: Raja Sridharan
author: S. M. Bhatwadekar
published in: Inventiones Mathematicae
publication date: 1998-11-08T00:00:00Z
review text: Let \(A\) be a smooth affine domain of dimension \(n\) over a field \(k\). Let \(X =\text{Spec } A\) and \(Z = V(I)\) be a smooth subvariety of \(X \times {\mathbb A}^1 =\text{Spec } A[T]\) such that \(Z\) intersects \(X \times 0\) transversally in \(Y \times 0\), where \(Y\) is a smooth subvariety of \(X\) of dimension \(n-r\). Let \(P\) be a projective module of rank \(r \geq (n+3)/2\). Let \({\overline \phi}: P[T] \to I/(I^2T)\) be a surjection. \textit{M. V. Nori} in the appendix (pp. 645-646) to the paper of S. Mandal cited below asked whether one can lift \(\overline \phi\) to a surjection from \(P[T]\) to \(I\). \textit{S. Mandal} [J. Algebr. Geom. 1, No. 4, 639-646 (1992; Zbl 0796.14011)] showed that this question has an affirmative answer when \(I\) contains a monic polynomial even without the smoothness assumptions. The paper under review gives an affirmative answer when \(V(I)\) is a smooth curve and \(k\) is an infinite field. This result has several interesting consequences on projective generation of curves in polynomial extensions of \(A\). For instance, it is shown that if \(A\) is a regular affine domain over an algebraically closed field \(k\) (respectively \(\mathbb R\)) with \(\dim A = n \geq 3\) (respectively \(n \geq 2\)) such that every maximal ideal of \(A\) is a complete intersection and if \(I \subset A[T]\) is a local complete intersection ideal of height \(n\) such that \(I/I^2\) is generated by \(n\) elements, then \(I\) is generated by \(n\) elements. Moreover, the paper also gives examples showing that Nori's question has negative answer if \(r \leq (n+2)/2\) or if \(A\) is not smooth.
reviewed by: Trung Ngo Viet
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 14A05
zbMATH DE Number: 1186167
zbMATH Keywords: smooth affine domain
zbMATH Keywords: projective module
zbMATH Keywords: unimodular element
zbMATH Keywords: complete intersection
zbMATH Keywords: Euler class
zbMATH Keywords: Spec
zbMATH Keywords: generation of curves
MaRDI profile type: Publication
full work available at URL: s002220050243
OpenAlex ID: W2042165686
DOI: 10.1007/S002220050243
description: scientific article; zbMATH DE number 1186167