Rational equivalence on some families of plane curves
Annales de l'Institut Fourier, Volume 44 (1994) no. 2, p. 323-345

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 2 and a cyclic group of order 3(d-2)(d 2 -3d+1) if d is even.

Si V d,δ est la variété des courbes planes irréductibles de degré d avec exactement δ nœuds comme singularités, Diaz-Harris (1986) ont conjecturé que Pic (V d,δ ) 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 Pic (V d,1 ) est un groupe fini, vérifiant ainsi la conjecture pour δ=1. Plus précisément, si D=3(d-2)(d 2 -3d+1), alors Pic (V d,1 ) est un groupe cyclique d’ordre 2D pour d impair et le produit de 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},
     zbl = {0803.14013},
     mrnumber = {95g:14006},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1994__44_2_323_0}
}
Miret, Josep M.; Descamps, Sebastián Xambó. 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. http://www.numdam.org/item/AIF_1994__44_2_323_0/

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