Finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves

Cristian González-Avilés

Open Mathematics (2009)

  • Volume: 7, Issue: 4, page 606-616
  • ISSN: 2391-5455

Abstract

top
We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves over perfect fields. For example, if k is finitely generated over ℚ and X → C is a quadric fibration of odd relative dimension at least 11, then CH i(X) is finitely generated for i ≤ 4.

How to cite

top

Cristian González-Avilés. "Finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves." Open Mathematics 7.4 (2009): 606-616. <http://eudml.org/doc/269438>.

@article{CristianGonzález2009,
abstract = {We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves over perfect fields. For example, if k is finitely generated over ℚ and X → C is a quadric fibration of odd relative dimension at least 11, then CH i(X) is finitely generated for i ≤ 4.},
author = {Cristian González-Avilés},
journal = {Open Mathematics},
keywords = {Chow groups; Quadrics; Curves; quadrics; curves},
language = {eng},
number = {4},
pages = {606-616},
title = {Finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves},
url = {http://eudml.org/doc/269438},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Cristian González-Avilés
TI - Finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves
JO - Open Mathematics
PY - 2009
VL - 7
IS - 4
SP - 606
EP - 616
AB - We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves over perfect fields. For example, if k is finitely generated over ℚ and X → C is a quadric fibration of odd relative dimension at least 11, then CH i(X) is finitely generated for i ≤ 4.
LA - eng
KW - Chow groups; Quadrics; Curves; quadrics; curves
UR - http://eudml.org/doc/269438
ER -

References

top
  1. [1] Colliot-Thélène J.-L, Skorobogatov A., Groupe de Chow des zéro-cycles sur les fibrés en quadriques, K-Theory, 1993, 7, 477–500 http://dx.doi.org/10.1007/BF00961538[Crossref] Zbl0837.14002
  2. [2] Conrad B., Chows K/k-image and K/k-trace, and the Lang-Néron theorem, Enseign. Math. (2), 2006, 52, 37–108 
  3. [3] Fulton W., Intersection theory, Second Ed., Springer-Verlag, 1998 Zbl0885.14002
  4. [4] González-Avilés C, Algebraic cycles on Severi-Brauer schemes of prime degree over a curve, Math. Res. Lett., 2008, 15(1), 51–56 [Crossref] Zbl1146.14006
  5. [5] Gros M., 0-cycles de degré zéro sur les surfaces fibrées en coniques, J. Reine Angew. Math., 1987, 373, 166–184 Zbl0593.14005
  6. [6] Kahn B., Rost M., Sujatha R, Unramified cohomology of quadrics I, Amer. Math. J., 1998, 120(4), 841–891 http://dx.doi.org/10.1353/ajm.1998.0029[Crossref] Zbl0913.11018
  7. [7] Karpenko N., Algebro-geometric invariants of quadratic forms, Leningrad Math. J., 1991, 2(1), 119–138 
  8. [8] Karpenko N., Chow groups of quadrics and the stabilization conjecture, Adv. Soviet Math., 1991, 4, 3–8 Zbl0756.14003
  9. [9] Karpenko N., Chow groups of quadrics and index reduction formulas, Nova J. Algebra Geom., 1995, 3(4), 357–379 Zbl0902.14006
  10. [10] Karpenko N., Order of torsion in CH 4 of quadrics, Doc. Math., 1996, 1, 57–65 Zbl0867.11025
  11. [11] Karpenko N., Merkurjev A., Chow groups of projective quadrics, Leningrad Math. J., 1991, 2(3), 655–671 
  12. [12] Karpenko N., Merkurjev A., Rost projectors and Steenrod operations, Doc. Math., 2002, 7, 481–493 Zbl1030.11013
  13. [13] Lam T.Y., The algebraic theory of quadratic forms, W.A. Benjamin, Inc. Reading, Massachussettss, 1973 Zbl0259.10019
  14. [14] Milne J.S., Arithmetic Duality Theorems, Perspectives in Mathematics, vol. 1, Academic Press Inc., Orlando 1986 Zbl0613.14019
  15. [15] Parimala R., Suresh V., Zero-cycles on quadric fibrations: Finiteness theorems and the cycle map, Invent. Math., 1995, 122, 83–117 http://dx.doi.org/10.1007/BF01231440[Crossref] Zbl0865.14002
  16. [16] Rost M., Chow groups with coefficients, Doc. Math., 1996, 1, 319–393 Zbl0864.14002
  17. [17] Sherman C., Some theorems on the K-Theory of coherent sheaves, Comm. Algebra, 1979, 7(14), 1489–1508 http://dx.doi.org/10.1080/00927877908822414[Crossref] Zbl0429.18017
  18. [18] Swan R., Zero-cycles on quadric hypersurfaces, Proc. Amer. Math. Soc., 1989, 107, 43–46 http://dx.doi.org/10.2307/2048032[Crossref] 
  19. [19] Weibel C., An introduction to homological algebra, Cambridge Stud. Adv. Math., Cambridge Univ. Press, 1994, 38 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.