Remarks on the defining equations of smooth threefolds in \({\mathbb P}^5\)
type: Item
instance of: scholarly article
title: Remarks on the defining equations of smooth threefolds in \({\mathbb P}^5\)
zbMATH Open document ID: 1066.14056
DOI: 10.1023/A:1022133506844
author: Youngook Choi
author: Sijong Kwak
published in: Geometriae Dedicata
publication date: 2003-04-27T00:00:00Z
review text: The authors are interested in the vanishing of \(H^1(\mathbb P^n, {\mathcal I}_X (t))\), for \(t\) small, when \(X\) is a smooth threefold in \(\mathbb P^5\) over the complex numbers. (When such vanishing occurs, \(X\) is said to be \(t\)-normal.) In particular, the authors are motivated by the following conjecture of Peskine and Zak: Let \(X\) be a nondegenerate, not necessarily smooth, projective variety of codimension \(e\) in \(\mathbb P^n\). Then (1) \(H^i(\mathbb P^n, {\mathcal I}_X(j)) = 0\) for \(i \geq 1, j \geq 0, i+j < \dim(X)/(e-1)\); (2) For \(i \geq 1\), \(j \geq 0\), \(i+j = \dim(X)/(e-1)\), it is possible to describe all varieties for which \(H^i(\mathbb P^n, {\mathcal I}_X(j)) \neq 0\). It has also been conjectured by Peskine and Van de Ven that all smooth threefolds in \(\mathbb P^5\) are 2-normal except the Palatini scroll of degree 7. On the other hand, \textit{N. Yu. Netsvetaev} [in: Topology and geometry, Rohlin Semin. 1984-1986, Lect. Notes Math. 1346, 433--453 (1988; Zbl 0664.14028)] and \textit{A. Arsie} [\texttt{http://arxiv.org/abs/math.AG/0005142}] have results of the type that if \(X\) has low codimension and can be defined by a small number of equations then in fact it is a complete intersection (and so in particular those cohomology modules vanish). In this paper, the authors first give some examples that show the sharpness of Arsie's result. Their main result, however, is that if \(X\) is a smooth threefold in \(\mathbb P^5\) cut out scheme theoretically by at most four equations then \(X\) is \(t\)-normal for \(t \leq 3\). Furthermore, if a technical (but natural) surjectivity condition holds then they also conclude that \(X\) is 4-normal. They do this by studying the cohomology of a certain rank three vector bundle on \(\mathbb P^5\) associated to \(X\).
reviewed by: Juan C. Migliore
Mathematics Subject Classification ID: 14M07
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14F17
zbMATH DE Number: 1901491
zbMATH Keywords: 2-normality
zbMATH Keywords: complete intersection
zbMATH Keywords: vanishing theorem
zbMATH Keywords: regularity matrix
MaRDI profile type: Publication
full work available at URL: a:1022133506844
OpenAlex ID: W109058852
description: scientific article; zbMATH DE number 1901491