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On the universal \(\mathrm {CH}_0\) group of cubic threefolds in positive characteristic

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We adapt for algebraically closed fields k of characteristic >2 two results of Voisin (On the universal \(\text {CH} _0\) group of cubic hypersurfaces, ar**v:1407.7261), on the decomposition of the diagonal of a smooth cubic hypersurface X of dimension 3 over \({\mathbb {C}}\), namely: the equivalence between Chow-theoretic and cohomological decompositions of the diagonal of those hypersurfaces and the equivalence between the algebraicity (with \(\mathbb Z_2\)-coefficients) of the minimal class \(\theta ^4/4!\) of the intermediate Jacobian J(X) of X and the cohomological (hence Chow-theoretic) decomposition of the diagonal of X. Using the second result, the Tate conjecture for divisors on surfaces defined over finite fields predicts, via a theorem of Schoen (Math Ann 311(3), 493–500, 1998), that every smooth cubic hypersurface of dimension 3 over the algebraic closure of a finite field of characteristic >2 admits a Chow-theoretic decomposition of the diagonal.

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References

  1. Altman, A.B., Kleiman, S.L.: Foundation of the theory of the Fano schemes. Compos. Math. 34(1), 3–47 (1977)

    MathSciNet  MATH  Google Scholar 

  2. Auel, A., Colliot-Thélène, J.-L., Parimala, R.: Universal unramified cohomology of cubic fourfolds containing a plane. Preprint ar**v:1310.6705

  3. Beauville, A.: Variétés de Prym et jacobiennes intermédiaires. Ann. Sci. École Norm. Sup. 10(3), 309–391 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  4. Clemens, H., Griffiths, P.: The intermediate Jacobian of the cubic threefold. Ann. Math. 95, 281–356 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  5. Colliot-Thélène, J.-L., Coray, D.: L’équivalence rationnelle sur les points fermés des surfaces rationnelles fibrées en coniques. Compos. Math. 39(3), 301–332 (1979)

    MATH  Google Scholar 

  6. Colliot-Thélène, J.-L., Szamuely, T.: Autour de la conjecture de Tate à coefficients \({\mathbb{Z} }_{\ell }\) pour les variétés sur les corps finis, In: Akhtar, R., Brosnan, P., Joshua, R. (eds.) The Geometry of Algebraic Cycles, pp. 83–98. AMS/Clay Institute Proceedings (2010)

  7. Cossart, V., Piltant, O.: Reolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin–Schreier and purely inseparable coverings. J. Algebra 320(3), 1051–1082 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cossart, V., Piltant, O.: Resolution of singularities of threefolds in positive characteristic. II. J. Algebra 321(7), 1836–1976 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Druel, S.: Espace des modules des faisceaux de rang \(2\) semi-stables de classes de Chern \(c_1=0\), \(c_2=2\) et \(c_3=0\) sur la cubique de \({\mathbb{P}}^4\). Int. Math. Res. Not. 19, 985–1004 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fulton, W.: Intersection Theory, 2nd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol 2, Springer-Verlag, Berlin (1998)

  11. Harris, J., Morrison, I.: Moduli of Curves. Graduate Texts in Mathematics 187. Springer, Berlin (1998)

    Google Scholar 

  12. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics 52. Springer, Berlin (1977)

    Google Scholar 

  13. Hartshorne, R.: Stable reflexive sheaves. Math. Ann. 254, 121–176 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hulek, K.: Projective geometry of elliptic curves. Astérisque 137 (1986)

  15. Iliev, A., Markushevich, D.: The Abel–Jacobi map for a cubic threefold and periods of Fano threefolds of degree \(14\). Doc. Math. 5, 23–47 (2000)

    MathSciNet  MATH  Google Scholar 

  16. Kollár, J.: Rational Curves on Algebraic Vareieties. volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin (1996)

  17. Markushevich, D., Tikhomirov, A.: The Abel–Jacobi map of a moduli component of vector bundles on the cubic threefold. J. Algebraic Geom. 10, 37–62 (2001)

    MathSciNet  MATH  Google Scholar 

  18. Milne, J. S.: Lectures on étale cohomology. http://www.jmilne.org/math/

  19. Milne, J.S.: Abelian varieties. In: Cornell, G., Silverman, J. (eds.) Arithmetic Geometry, pp. 103–150. Springer, Berlin (1986)

    Chapter  Google Scholar 

  20. Murre, J.P.: Algebraic cycles and algebraic aspects of cohomology and k-theory, In: Albano, A., Bardelli, F. (eds.) Algebraic Cycles and Hodge Theory, Lecture Notes in Mathematics, vol. 1594, pp. 93–152 (1994)

  21. Murre, J.P.: Algebraic equivalence modulo rational equivalence on a cubic threefold. Compos. Math. 25, 161–206 (1972)

    MathSciNet  MATH  Google Scholar 

  22. Murre, J. P.: Application of algebraic \(K\)-theory to the theory of algebraic cycles. In: Proceedings Conference on Algebraic Geometry, Sitges 1983, Lecture Notes in Mathematics, vol. 1124, pp. 216–261. Springer, Berlin (1985)

  23. Murre, J. P.: Some results on cubic threefolds. In: Classification of Algebraic Varieties and Compact Complex Manifolds, Lecture Notes in Mathematics, vol. 412, pp. 140–164. Springer, Berlin (1974)

  24. Schnell, C.: The Serre construction in codimension two. https://www.math.stonybrook.edu/~cschnell/

  25. Schoen, C.: An integral analog of the Tate conjecture for one-dimensional cycles on varieties over finite field. Math. Ann. 311(3), 493–500 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shen, M.: On relations among \(1\)-cycles on cubic hypersurfaces. J. Algebraic Geom. 23, 539–569 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Totaro, B.: The integral cohomology of the Hilbert scheme of two points. Preprint ar**v:1506.00968

  28. Tsen, C.: Quasi-algebraische-abgeschlossene Funktionenkörper. J. Chin. Math. 1, 81–92 (1936)

    Google Scholar 

  29. Voevodsky, V.: A nilpotence theorem for cycles algebraically equivalent to zero. Int. Math. Res. Not. 4, 187–198 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Voisin, C.: Remarks on zero-cycles of self-products of varieties. In: Maruyama, M. (ed.) Moduli of Vector Bundles (Proceedings of the Taniguchi Congress on Vector Bundles), pp. 265–285. Decker (1994)

  31. Voisin, C.: Abel–Jacobi map, integral Hodge classes and decomposition of the diagonal. J. Algebraic Geom. 22(1), 141–174 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Voisin, C.: On the universal \(\text{CH} _0\) group of cubic hypersurfaces. Preprint ar**v:1407.7261

  33. Welters, G.E.: Abel–Jacobi Isogenies for Certain Types of Fano threefolds. Mathematical Centre Tracts 141, Math. Centrum, Amsterdam (1981)

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  1. CMLS, Ecole Polytechnique, CNRS, Université Paris-Saclay, 91128, Palaiseau Cédex, France

    René Mboro

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  1. René Mboro
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Mboro, R. On the universal \(\mathrm {CH}_0\) group of cubic threefolds in positive characteristic. manuscripta math. 154, 147–168 (2017). https://doi.org/10.1007/s00229-016-0912-5

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