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On the degree of algebraic cycles on hypersurfaces

Matthias Paulsen EMAIL logo


Let X 4 be a very general hypersurface of degree d 6 . Griffiths and Harris conjectured in 1985 that the degree of every curve C X is divisible by d. Despite substantial progress by Kollár in 1991, this conjecture is not known for a single value of d. Building on Kollár’s method, we prove this conjecture for infinitely many d, the smallest one being d = 5005 . The set of these degrees d has positive density. We also prove a higher-dimensional analogue of this result and construct smooth hypersurfaces defined over that satisfy the conjecture.

Funding statement: This project received partial funding by the DFG project “Topological properties of algebraic varieties” (grant no. 416054549) and by the ERC grant “RationAlgic” (grant no. 948066). The final part of this project was carried out while the author was in residence at Institut Mittag-Leffler, supported by the Swedish Research Council under grant no. 2016-06596.


I am grateful to my supervisor Stefan Schreieder for many helpful suggestions and comments concerning this work. Further, I would like to thank the anonymous referee for useful remarks that improved the exposition.


[1] E. Ballico, F. Catanese and C. Ciliberto (eds.), Classification of irregular varieties. Minimal models and abelian varieties, Lecture Notes in Math. 1515, Springer, Berlin (1992). 10.1007/BFb0098332Search in Google Scholar

[2] R. Beheshti and D. Eisenbud, Fibers of generic projections, Compos. Math. 146 (2010), no. 2, 435–456. 10.1112/S0010437X09004503Search in Google Scholar

[3] A. A. Buhštab, On those numbers in an arithmetic progression all prime factors of which are small in order of magnitude, Dokl. Akad. Nauk SSSR (N.S.) 67 (1949), 5–8. Search in Google Scholar

[4] O. Debarre, K. Hulek and J. Spandaw, Very ample linear systems on abelian varieties, Math. Ann. 300 (1994), no. 2, 181–202. 10.1007/BF01450483Search in Google Scholar

[5] K. Dickman, On the frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astron. Fys. 22A (1930), no. 10, 1–14. Search in Google Scholar

[6] P. Griffiths and J. Harris, On the Noether–Lefschetz theorem and some remarks on codimension-two cycles, Math. Ann. 271 (1985), no. 1, 31–51. 10.1007/BF01455794Search in Google Scholar

[7] S. Lefschetz, L’analysis situs et la géométrie algébrique, Gauthier-Villars, Paris 1924. Search in Google Scholar

[8] J. N. Mather, Generic projections, Ann. of Math. (2) 98 (1973), 226–245. 10.2307/1970783Search in Google Scholar

[9] F. Mertens, Ein Beitrag zur analytischen Zahlentheorie, J. reine angew. Math. 78 (1874), 46–62. 10.1515/9783112389843-002Search in Google Scholar

[10] C. Soulé and C. Voisin, Torsion cohomology classes and algebraic cycles on complex projective manifolds, Adv. Math. 198 (2005), no. 1, 107–127. 10.1016/j.aim.2004.10.022Search in Google Scholar

[11] B. Totaro, On the integral Hodge and Tate conjectures over a number field, Forum Math. Sigma 1 (2013), Paper No. e4. 10.1017/fms.2013.3Search in Google Scholar

[12] C. Voisin, Sur une conjecture de Griffiths et Harris, Algebraic curves and projective geometry (Trento 1988), Lecture Notes in Math. 1389, Springer, Berlin (1989), 270–275. 10.1007/BFb0085939Search in Google Scholar

[13] X. Wu, On a conjecture of Griffiths–Harris generalizing the Noether–Lefschetz theorem, Duke Math. J. 60 (1990), no. 2, 465–472. 10.1215/S0012-7094-90-06018-1Search in Google Scholar

Received: 2021-10-04
Revised: 2022-04-30
Published Online: 2022-06-25
Published in Print: 2022-09-01

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