Rational equivalence on some families of plane curves (Q1337026)

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.