# ANNALES DE L'INSTITUT FOURIER

Rational equivalence on some families of plane curves
Annales de l'Institut Fourier, Volume 44 (1994) no. 2, p. 323-345
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 $6\left(d-2\right){d}^{2}-3d+1\right)$, the group being cyclic if $d$ is odd and the product of ${ℤ}_{2}$ and a cyclic group of order $3\left(d-2\right)\left({d}^{2}-3d+1\right)$ if $d$ is even.
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 $D=3\left(d-2\right)\left({d}^{2}-3d+1\right)$, alors $\mathrm{Pic}\left({V}_{d,1}\right)$ est un groupe cyclique d’ordre $2D$ pour $d$ impair et le produit de ${ℤ}_{2}$ par un groupe cyclique d’ordre $D$ pour $d$ est pair.
@article{AIF_1994__44_2_323_0,
author = {Miret, Josep M. and Descamps, Sebasti\'an Xamb\'o},
title = {Rational equivalence on some families of plane curves},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {44},
number = {2},
year = {1994},
pages = {323-345},
doi = {10.5802/aif.1400},
mrnumber = {95g:14006},
zbl = {0803.14013},
language = {en},
url = {https://aif.centre-mersenne.org/item/AIF_1994__44_2_323_0}
}

Rational equivalence on some families of plane curves. Annales de l'Institut Fourier, Volume 44 (1994) no. 2, pp. 323-345. doi : 10.5802/aif.1400. https://aif.centre-mersenne.org/item/AIF_1994__44_2_323_0/

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