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Rational equivalence on some families of plane curves

Josep M. Miret; Sebastián Xambó Descamps — 1994

Annales de l'institut Fourier

If $V_{d, δ}$ denotes the variety of irreducible plane curves of degree $d$ with exactly $δ$ nodes as singularities, Diaz and Harris (1986) have conjectured that $Pic (V_{d, δ})$ is a torsion group. In this note we study rational equivalence on some families of singular plane curves and we prove, in particular, that $Pic (V_{d, 1})$ is a finite group, so that the conjecture holds for $δ = 1$ . Actually the order of $Pic (V_{d, 1})$ is $6 (d - 2) d^{2} - 3 d + 1)$ , the group being cyclic if $d$ is odd and the product of $ℤ_{2}$ and a cyclic group of order $3 (d - 2) (d^{2} - 3 d + 1)$ if $d$ is even.

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