Rational equivalence on some families of plane curves

Miret, Josep M.; Descamps, Sebastián Xambó

Annales de l'Institut Fourier, Tome 44 (1994) no. 2, pp. 323-345.

Si $V_{d,\delta }$ est la variété des courbes planes irréductibles de degré $d$ avec exactement $\delta$ nœuds comme singularités, Diaz-Harris (1986) ont conjecturé que $\mathrm{Pic}\left(V_{d,\delta }\right)$ est un groupe de torsion. Ici nous étudions l’équivalence rationnelle de certaines familles de courbes planes singulières et cela nous permet, en particulier, de montrer que $\mathrm{Pic}\left(V_{d,1}\right)$ est un groupe fini, vérifiant ainsi la conjecture pour $\delta =1$. Plus précisément, si $D=3(d-2)(d^2-3d+1)$, alors $\mathrm{Pic}\left(V_{d,1}\right)$ est un groupe cyclique d’ordre $2D$ pour $d$ impair et le produit de ${\mathbb{Z}}_2$ par un groupe cyclique d’ordre $D$ pour $d$ est pair.

If $V_{d,\delta }$ denotes the variety of irreducible plane curves of degree $d$ with exactly $\delta$ nodes as singularities, Diaz and Harris (1986) have conjectured that $\mathrm{Pic}\left(V_{d,\delta }\right)$ is a torsion group. In this note we study rational equivalence on some families of singular plane curves and we prove, in particular, that $\mathrm{Pic}\left(V_{d,1}\right)$ is a finite group, so that the conjecture holds for $\delta =1$. Actually the order of $\mathrm{Pic}\left(V_{d,1}\right)$ 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.

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