Rational equivalence on some families of plane curves

Josep M. Miret; Sebastián Xambó Descamps

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 2, page 323-345
  • ISSN: 0373-0956

Abstract

top
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 - 3 d + 1 ) , the group being cyclic if d is odd and the product of ℤ 2 and a cyclic group of order 3 ( d - 2 ) ( d 2 - 3 d + 1 ) if d is even.

How to cite

top

Miret, Josep M., and Descamps, Sebastián Xambó. "Rational equivalence on some families of plane curves." Annales de l'institut Fourier 44.2 (1994): 323-345. <http://eudml.org/doc/75064>.

@article{Miret1994,
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.},
author = {Miret, Josep M., Descamps, Sebastián Xambó},
journal = {Annales de l'institut Fourier},
keywords = {Picard groups; intersection rings; rational equivalence on families of singular plane curves},
language = {eng},
number = {2},
pages = {323-345},
publisher = {Association des Annales de l'Institut Fourier},
title = {Rational equivalence on some families of plane curves},
url = {http://eudml.org/doc/75064},
volume = {44},
year = {1994},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 2
SP - 323
EP - 345
AB - 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.
LA - eng
KW - Picard groups; intersection rings; rational equivalence on families of singular plane curves
UR - http://eudml.org/doc/75064
ER -

References

top
  1. [1] S. DIAZ, J. HARRIS, Geometry of the Severi varieties I, preprint (1986). 
  2. [2] S. DIAZ, J. HARRIS, Geometry of Severi varieties, II : Independence of divisor classes and examples, in : Algebraic Geometry, LN 1311, Springer-Verlag (Proceeding Sundance 1986), edited by Holme-Speiser), 23-50. Zbl0677.14004MR89g:14014
  3. [3] W. FULTON, Intersection Theory, Ergebnisse 3.Folge, Band 2, Springer-Verlag, 1984. Zbl0541.14005MR85k:14004
  4. [4] J. HARRIS, On the Severi problem, Inventiones Math., 84 (1986), 445-461. Zbl0596.14017MR87f:14012
  5. [5] J.M. MIRET, S. XAMBÓ, Geometry of complete cuspidal cubics, in : Algebraic Curves and Projective Geometry, LN 1389, Springer-Verlag (Proceedings Trento 1988, edited by Ballico and Ciliberto), 1989, 195-234. Zbl0688.14050MR90k:14059
  6. [6] J.M. MIRET, S. XAMBÓ, On the Geometry of nodal plane cubics : the condition p, in : Enumerative Geometry : Zeuthen Symposium, Contemporary Mathematics 123 (1991), AMS (Proceedings of the Zeuthen Symposium 1989, edited by S. Kleiman and A. Thorup). Zbl0766.14041
  7. [7] Z. RAN, On nodal plane curves, Inventiones Math., 86 (1986), 529-534. Zbl0644.14009MR87j:14039
  8. [8] F. SEVERI, Anhang F in : Vorlesungen ber algebraische Geometrie, Teubner, 1921. JFM48.0687.01

NotesEmbed ?

top

You must be logged in to post comments.