If denotes the variety of irreducible plane curves of degree with exactly nodes as singularities, Diaz and Harris (1986) have conjectured that is a torsion group. In this note we study rational equivalence on some families of singular plane curves and we prove, in particular, that is a finite group, so that the conjecture holds for . Actually the order of is , the group being cyclic if is odd and the product of and a cyclic group of order if is even.
Si est la variété des courbes planes irréductibles de degré avec exactement nœuds comme singularités, Diaz-Harris (1986) ont conjecturé que 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 est un groupe fini, vérifiant ainsi la conjecture pour . Plus précisément, si , alors est un groupe cyclique d’ordre pour impair et le produit de par un groupe cyclique d’ordre pour 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}, pages = {323--345}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {44}, number = {2}, year = {1994}, doi = {10.5802/aif.1400}, mrnumber = {95g:14006}, zbl = {0803.14013}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1400/} }
TY - JOUR AU - Miret, Josep M. AU - Descamps, Sebastián Xambó TI - Rational equivalence on some families of plane curves JO - Annales de l'Institut Fourier PY - 1994 SP - 323 EP - 345 VL - 44 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.1400/ DO - 10.5802/aif.1400 LA - en ID - AIF_1994__44_2_323_0 ER -
%0 Journal Article %A Miret, Josep M. %A Descamps, Sebastián Xambó %T Rational equivalence on some families of plane curves %J Annales de l'Institut Fourier %D 1994 %P 323-345 %V 44 %N 2 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.1400/ %R 10.5802/aif.1400 %G en %F 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://archive.numdam.org/articles/10.5802/aif.1400/
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