Rational curves of degree 11 on a general quintic 3-fold
type: Item
instance of: scholarly article
zbMATH Open document ID: 1260.14046
author: Ethan Cotterill
published in: The Quarterly Journal of Mathematics
publication date: 2012-09-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/0711.2758
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 6078027
zbMATH Keywords: rational curve
zbMATH Keywords: incidence scheme
zbMATH Keywords: normal bundle
zbMATH Keywords: generic initial ideal
describes a project that uses: Macaulay2
MaRDI profile type: Publication
OpenAlex ID: W2051211433
Wikidata QID: Q56568236
arXiv ID: 0711.2758
DOI: 10.1093/QMATH/HAR001
title: Rational curves of degree 11 on a general quintic 3-fold
review text: Twenty-five years ago Clemens conjectured that the number of smooth rational curves \(C\) of fixed degree \(d\) on a general quintic \(3\)-fold \(F\) in \(\mathbb P ^4\) is finite. Clemens' conjecture holds provided the incidence scheme NEWLINE\[NEWLINE\phi _d := \{ (C,F) \mid F\text{ is a quintic treefold in }\mathbb P ^4 \text{ containing a rational curve } C \text{ of degree }d \}NEWLINE\]NEWLINE is irreducible.NEWLINENEWLINEThe author proves in this paper that in degree \(d=11\) the incidence scheme \(\phi _{11}\) is irreducible, on a general quintic \(F\) in \(\mathbb P ^4\) there are only finitely many smooth rational curves \(C\) of degree \(11\) and each curve \(C\) is embedded in \(F\) with normal bundle \(\mathcal O (-1) \oplus \mathcal O (-1)\). Moreover the author proves that in degree \(11\) there are no singular, reduced and irreducible rational curves, nor any reduced, reducible and connected curves with rational components on \(F\). The author proved in [Commun. Algebra 33, No. 6, 1833--1872 (2005; Zbl 1079.14048)] the analogous result for \(d=10\). To do so he introduced a technique based on a combinatorial analysis of the Borel ideals that are generic initial ideals \(\mathrm{gin}(I_C)\) of the ideals \(I_C\) defining the rational curves \(C\), they are degenerations of the ideals \(I_C\). He used also the stratification of the mapping space \(\{ f: \mathbb P ^1 \rightarrow \mathbb P ^4 \mid f\text{ is a degree }d\) morphism
reviewed by: Luciana Ramella
description: scientific article; zbMATH DE number 6078027