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Rational equivalence on some families of plane curves. (English) Zbl 0803.14013
If \(V_{d, \delta}\) denotes the variety of irreducible plane curves of degree \(d\) with exactly \(\delta\) nodes as singularities, S. Diaz and 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.

14H10 Families, moduli of curves (algebraic)
14C22 Picard groups
14C15 (Equivariant) Chow groups and rings; motives
14N05 Projective techniques in algebraic geometry
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14H20 Singularities of curves, local rings
Full Text: DOI Numdam EuDML
[1] S. DIAZ, J. HARRIS, Geometry of the Severi varieties I, preprint (1986).
[2] S. DIAZ, J. HARRIS, Geometry of Severi varieties, II : independence of divisor classes and examples, in : Algebraic Geometry, LN 1311, Springer-Verlag (Proceeding Sundance 1986), edited by Holme-Speiser), 23-50. · Zbl 0677.14004
[3] W. FULTON, Intersection theory, Ergebnisse 3.Folge, Band 2, Springer-Verlag, 1984. · Zbl 0541.14005
[4] J. HARRIS, On the Severi problem, Inventiones Math., 84 (1986), 445-461. · Zbl 0596.14017
[5] J.M. MIRET, S. XAMBÓ, Geometry of complete cuspidal cubics, in : Algebraic Curves and Projective Geometry, LN 1389, Springer-Verlag (Proceedings Trento 1988, edited by Ballico and Ciliberto), 1989, 195-234. · Zbl 0688.14050
[6] J.M. MIRET, S. XAMBÓ, On the geometry of nodal plane cubics : the condition p, in : Enumerative Geometry : Zeuthen Symposium, Contemporary Mathematics 123 (1991), AMS (Proceedings of the Zeuthen Symposium 1989, edited by S. Kleiman and A. Thorup). · Zbl 0766.14041
[7] Z. RAN, On nodal plane curves, Inventiones Math., 86 (1986), 529-534. · Zbl 0644.14009
[8] F. SEVERI, Anhang F in : vorlesungen ber algebraische geometrie, Teubner, 1921. · JFM 48.0687.01
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