Triads and families of space curves
type: Item
instance of: scholarly article
title: Triads and families of space curves
zbMATH Open document ID: 0976.14019
author: Robin Hartshorne
author: Mireille Martin-Deschamps
author: Daniel Perrin
published in: Mathematische Annalen
publication date: 2000-02-07T00:00:00Z
review text: Many interesting properties of curves \(C\) in the projective space \({\mathbb P}^3\) can be read in the Hartshorne-Rao module \(\bigoplus^nH^1({\mathcal I}_C(n))\), which contains informations on the extrinsic geometry of \(C\). Unfortunately, when \(C\) is replaced by a flat family of curves \({\mathcal C}\) defined over the local ring \(A\), the naive definition of the Hartshorne-Rao module is useless because it does not commute with base change. The authors propose a different approach, namely they associate to any curve a triad, i.e. a complex of free modules \(L_\bullet= L_1\to L_0\to L_{-1}\) whose middle cohomology is the module \(\bigoplus^nH^1({\mathcal I}_C(n))\). This triad is obtained from sheaf resolutions \(0\to {\mathcal P}\to {\mathcal N}\to {\mathcal I}_C\to 0\) and \(0\to {\mathcal N}\to {\mathcal L}_1 \to {\mathcal L}_0\to {\mathcal L}_{-1}\to 0\), such that \({\mathcal P}\) and the \({\mathcal L}_i\)'s are sums of line bundles. When \(C\) is replaced by a family in \({\mathbb P}^3_A\), \(A\) a local ring, then the triad behaves properly with respect to the base change. The authors introduce a general theory of triads and triadic sheaves and they prove that triads associated to families are good surrogates for the Hartshorne-Rao modules of curves in many senses. For instance, they show that two families of curves are in the same biliaison classes if and only if the corresponding triads are pseudo-isomorphic (up to twisting). Several examples of concrete computation of triads associated to families of curves are provided.
reviewed by: Luca Chiantini
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14H10
zbMATH DE Number: 1384375
zbMATH Keywords: space curves
zbMATH Keywords: Hartshorne-Rao module
MaRDI profile type: Publication
full work available at URL: s002080050323
OpenAlex ID: W2067533760
DOI: 10.1007/S002080050323
description: scientific article; zbMATH DE number 1384375