The singular points of a rational plane curve and its restricted tangent bundle
type: Item
instance of: scholarly article
title: The singular points of a rational plane curve and its restricted tangent bundle
zbMATH Open document ID: 1420.14065
author: Maria-Grazia Ascenzi
published in: Journal of Algebra
publication date: 2019-06-26T00:00:00Z
review text: Let \(\phi : \mathbb{P}^1 \rightarrow \mathbb{P}^2\) be a generically one-to-one parameterization of a rational plane curve \(D \subseteq \mathbb{P}^2\) of degree \(d_D \geq 2\). Recall that \(\phi^* T_{\mathbb{P}^2}\), the pull-back to \(\mathbb{P}^1\) of the tangent bundle of \(\mathbb{P}^2\), splits as the direct sum of two line bundles. In the article under review, the author's main result shows how this splitting type depends on the minimum degree curves that are associated to \(D\). This concept of minimum degree curves, depending on the given rational parametric plane curve \(D\), is studied in some detail. A key point is that they encode invariants which relate to the singularities of \(D\). The results of the present article complement the author's earlier work [Commun. Algebra 16, No. 11, 2193--2208 (1988; Zbl 0675.14010)] as well as subsequent work of \textit{A. Gimigliano} et al. [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 12, No. 3, 587--621 (2013; Zbl 1434.14004)].
reviewed by: Nathan Grieve
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 14H20
zbMATH DE Number: 7073737
zbMATH Keywords: minimum degree relation
zbMATH Keywords: rational curve
zbMATH Keywords: restricted tangent bundle
zbMATH Keywords: singular point
zbMATH Keywords: subline bundle
zbMATH Keywords: syzygy
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2019.04.027
OpenAlex ID: W2945726360
cites work: Q4788728
cites work: The restricted tangent bundle of a rational curve in P2
cites work: The Restricted Tangent Bundle of a Rational Curve on a Quadric in P 3
cites work: Q4352969
cites work: Normal bundle of rational curves and Waring decomposition
cites work: On parameterizations of plane rational curves and their syzygies
cites work: Strata of rational space curves
cites work: The moving line ideal basis of planar rational curves
cites work: Q4317713
cites work: On the normal bundles of smooth rational space curves
cites work: On the variety of smooth rational space curves with given degree and normal bundle
cites work: Normal bundles of rational curves in \(P^ 3\).
cites work: Betti numbers for fat point ideals in the plane: A geometric approach
cites work: Stable postulation and stable ideal generation: conjectures for fat points in the plane
cites work: The role of the cotangent bundle in resolving ideals of fat points in the plane
cites work: Sur La Classification Des Fibres Holomorphes Sur La Sphere de Riemann
cites work: Q4143433
cites work: Q4352978
cites work: Q4860506
cites work: The normal bundle of a curve on a quadric
cites work: Strata of vector spaces of forms in \({R = \mathsf k[x, y]}\), and of rational curves in \({\mathbb{P}^k}\)
cites work: Q3950667
cites work: Q5752674
DOI: 10.1016/J.JALGEBRA.2019.04.027
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description: scientific article; zbMATH DE number 7073737