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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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