Rational equivalence on some families of plane curves

Abstract

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 -3d+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 -3d+1) if d is even.