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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.

MSC:
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
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References:
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