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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.


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

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