Subvarieties of general hypersurfaces in projective space
type: Item
instance of: scholarly article
title: Subvarieties of general hypersurfaces in projective space
zbMATH Open document ID: 0823.14030
author: Geng Xu
published in: Journal of Differential Geometry
publication date: 1995-11-05T00:00:00Z
review text: The well-known Noether-Lefschetz theorem states that any curve on a general surface \(S\) of degree \(d\geq 4\) in \(\mathbb{P}^ 3\) is the complete intersection of \(S\) with another surface of degree \(k\). The main result in this paper is that, if \(d\geq 5\), the general \(S\) does not possess curves of geometric genus \(g\leq {1\over 2} d(d-3)-3\). This improves a conjecture by Harris and a previous result of Clemens. Moreover, the bound given by the author is sharp since it is achieved by the tritangent plane sections; the author also proves that if \(d\geq 6\) then these are the only curves achieving the bound. The general idea of this proof is to study the possible singularities that could possess the complete intersection of \(S\) with a surface of degree \(k\); thus the author gets a bound \(g\geq {1\over 2} dk(d-5) +2\). Then the result follows at once by just analyzing the case \(k=1\). A similar result is obtained in higher dimension. More precisely the author gives a lower bound for the geometric genus of any reduced and irreducible divisor \(M\) on a general hypersurface of \(\mathbb{P}^{n+1}\); moreover there is a bound \(C(n)\) (which can be computed explicitly for each \(n\)) such that if \(d\geq C(n)\) the bound is sharp and achieved only for hyperplane sections. The particular result for \(n=3\) states that on a general 3-fold in \(\mathbb{P}^ 4\) of degree \(d\geq 6\) an integral divisor \(M\) has geometric genus at least two. It is an open question posed by Green whether a quintic 3-fold of degree 5 in \(\mathbb{P}^ 4\) could have surfaces with geometric genus zero (a negative answer would imply Clemens conjecture that such a 3-fold has only a finite number of rational curves of each degree).
reviewed by: Enrique Arrondo
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14J30
zbMATH DE Number: 575945
zbMATH Keywords: geometric genus of divisor
zbMATH Keywords: general hypersurface
zbMATH Keywords: Noether-Lefschetz theorem
zbMATH Keywords: complete intersection
zbMATH Keywords: tritangent plane sections
zbMATH Keywords: quintic 3- fold
zbMATH Keywords: Clemens conjecture
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214454680
OpenAlex ID: W1481538715
Wikidata QID: Q115179852
DOI: 10.4310/JDG/1214454680
description: scientific article; zbMATH DE number 575945