Rational equivalence on some families of plane curves
type: Item
instance of: scholarly article
title: Rational equivalence on some families of plane curves
zbMATH Open document ID: 0803.14013
publication date: 1994-12-15T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75064
review text: If \(V_{d, \delta}\) denotes the variety of irreducible plane curves of degree \(d\) with exactly \(\delta\) nodes as singularities, \textit{S. Diaz} and \textit{J. Harris} [Trans. Am. Math. Soc. 309, No. 1, 1--34 (1988; Zbl 0677.14003) and in Algebraic Geometry, Proc. Conf., Sundance 1986, Lect. Notes Math. 1311, 23--50 (1988; Zbl 0677.14004)] have conjectured that \(\text{Pic}(V_{d, \delta})\) is a torsion group. In this note we study rational equivalence on some families of singular plane curves and we prove, in particular, that \(\text{Pic}(V_{d,1})\) is a finite group, so that the conjecture holds for \(\delta=1\). Actually the order of \(\text{Pic}(V_{d,1})\) is \(6(d-2) (d^ 2 - 3d + 1)\), the group being cyclic if \(d\) is odd and the product of \(\mathbb Z_ 2\) and a cyclic group of order \(3(d-2) (d^ 2 - 3d + 1)\) if \(d\) is even.
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14C22
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14N10
Mathematics Subject Classification ID: 14H20
zbMATH DE Number: 672143
zbMATH Keywords: Picard groups
zbMATH Keywords: intersection rings
zbMATH Keywords: rational equivalence on families of singular plane curves
author: Josep M. Miret Biosca
author: Sebastian Xambó
reviewed by: Sebastian Xambó
MaRDI profile type: Publication
OpenAlex ID: W2331077764
cites work: Geometry of severi varieties, II: Independence of divisor classes and examples
cites work: Q3328651
cites work: On the Severi problem
cites work: Q4206875
cites work: Q4011037
cites work: On nodal plane curves
DOI: 10.5802/AIF.1400
published in: Annales de l’institut Fourier
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description: scientific article; zbMATH DE number 672143