Rational equivalence on some families of plane curves

Miret, Josep M. ; Descamps, Sebastián Xambó

Annales de l'Institut Fourier, Tome 44 (1994), p. 323-345 / Harvested from Numdam

Résumé

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 ${\rm Pic}(V_{d,\delta })$ 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 ${\rm Pic}(V_{d,1})$ est un groupe fini, vérifiant ainsi la conjecture pour $\delta =1$. Plus précisément, si $D=3(d-2)(d^2-3d+1)$, alors ${\rm Pic}(V_{d,1})$ est un groupe cyclique d’ordre $2D$ pour $d$ impair et le produit de ${\Bbb Z}_2$ par un groupe cyclique d’ordre $D$ pour $d$ est pair.

Abstract

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 ${\rm Pic}(V_{d,\delta })$ is a torsion group. In this note we study rational equivalence on some families of singular plane curves and we prove, in particular, that ${\rm Pic}(V_{d,1})$ is a finite group, so that the conjecture holds for $\delta =1$. Actually the order of ${\rm Pic}(V_{d,1})$ is $6(d-2)d^2-3d+1)$, the group being cyclic if $d$ is odd and the product of ${\Bbb Z}_2$ and a cyclic group of order $3(d-2)(d^2-3d+1)$ if $d$ is even.

Détail

Publié le : 1994-01-01

MR 95g:14006 | Zbl 0803.14013

DOI : https://doi.org/10.5802/aif.1400

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