Weak Approximation for Fano Complete Intersections in Positive Characteristic
Annales de l'Institut Fourier, Volume 72 (2022) no. 4, pp. 1503-1534.

For a smooth curve B over a field k=k ¯ with char(k)=p, for every complete intersection X B in B× Speck k n of type (d 1 ,,d c ), we prove weak approximation of adelic points of X B by k(B)-points at all places of (strong) potentially good reduction, if the Fano index is 2 and if p>max(d 1 ,,d c ). This also applies to specializations of complex Fano manifolds with Picard rank 1 and Fano index 1 away from “bad primes”.

Pour une courbe lisse B sur un corps k=k ¯ de caractéristique positive p, pour chaque intersection complète X B dans B× Speck k n de type (d 1 ,,d c ), nour prouvons l’approximation faible des points adeliques de X B par des k(B)-points sur toutes les places de forte réduction potentiellement bonne, si l’indice de Fano est au moins deux et si p>max(d 1 ,,d c ). Cela s’applique également aux spécialisations des variétés de Fano complexes de nombre de Picard de rang 1 et d’indice de Fano 1 en dehors de l’ensemble des mauvais nombres premiers.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3495
Classification: 14M22, 14D10, 14G12
Keywords: weak approximation, separably rationally connected, stability
Mot clés : approximation faible, séparablement rationnellement connexe, stabilité

Starr, Jason M. 1; Tian, Zhiyu 2; Zong, Runhong 3

1 Department of Mathematics Stony Brook University Stony Brook, NY 11794 (USA)
2 Beijing International Center for Mathematical Research Peking University No.5 Yiheyuan Road Beijing, 100871 (China)
3 Department of Mathematics Nanjing University 204 Meng Minwei Building No. 22 Hankou Road Nanjing, Jiangsu, 210093 (China)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2022__72_4_1503_0,
     author = {Starr, Jason M. and Tian, Zhiyu and Zong, Runhong},
     title = {Weak {Approximation} for {Fano} {Complete} {Intersections} in {Positive} {Characteristic}},
     journal = {Annales de l'Institut Fourier},
     pages = {1503--1534},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
     number = {4},
     year = {2022},
     doi = {10.5802/aif.3495},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3495/}
}
TY  - JOUR
AU  - Starr, Jason M.
AU  - Tian, Zhiyu
AU  - Zong, Runhong
TI  - Weak Approximation for Fano Complete Intersections in Positive Characteristic
JO  - Annales de l'Institut Fourier
PY  - 2022
SP  - 1503
EP  - 1534
VL  - 72
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3495/
DO  - 10.5802/aif.3495
LA  - en
ID  - AIF_2022__72_4_1503_0
ER  - 
%0 Journal Article
%A Starr, Jason M.
%A Tian, Zhiyu
%A Zong, Runhong
%T Weak Approximation for Fano Complete Intersections in Positive Characteristic
%J Annales de l'Institut Fourier
%D 2022
%P 1503-1534
%V 72
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3495/
%R 10.5802/aif.3495
%G en
%F AIF_2022__72_4_1503_0
Starr, Jason M.; Tian, Zhiyu; Zong, Runhong. Weak Approximation for Fano Complete Intersections in Positive Characteristic. Annales de l'Institut Fourier, Volume 72 (2022) no. 4, pp. 1503-1534. doi : 10.5802/aif.3495. https://aif.centre-mersenne.org/articles/10.5802/aif.3495/

[1] Abramovich, Dan; Oort, Frans Stable maps and Hurwitz schemes in mixed characteristics, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (Contemporary Mathematics), Volume 276, American Mathematical Society, 2001, pp. 89-100 | DOI | MR | Zbl

[2] Bloch, Spencer; Srinivas, Vasudevan Remarks on correspondences and algebraic cycles, Am. J. Math., Volume 105 (1983) no. 5, pp. 1235-1253 | DOI | MR | Zbl

[3] Bonavero, Laurent; Höring, Andreas Counting conics in complete intersections, Acta Math. Vietnam., Volume 35 (2010) no. 1, pp. 23-30 | MR | Zbl

[4] Chatzistamatiou, Andre; Levine, Marc Torsion orders of complete intersections (2016) (https://arxiv.org/abs/1605.01913)

[5] Deligne, Pierre; Illusie, Luc Relèvements modulo p 2 et décomposition du complexe de de Rham, Invent. Math., Volume 89 (1987) no. 2, pp. 247-270 | DOI | MR | Zbl

[6] Gounelas, Frank; Javanpeykar, Ariyan Invariants of Fano varieties in families, Mosc. Math. J., Volume 18 (2018) no. 2, pp. 305-319 | DOI | MR | Zbl

[7] Graber, Tom; Harris, Joe; Starr, Jason Michael Families of rationally connected varieties, J. Am. Math. Soc., Volume 16 (2003) no. 1, pp. 57-67 | DOI | MR | Zbl

[8] Grothendieck, Alexander Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas., Publ. Math., Inst. Hautes Étud. Sci., Volume 20 (1964), pp. 101-355 ibid. 24 (1965), p. 5-231; ibid. 28 (1966), p. 5-255; ibid. 32 (1967), p. 5-361 | Numdam | MR | Zbl

[9] Gutt, Jan Aleksander Hwang–Mok rigidity of cominuscule homogeneous varieties in positive characteristic, Ph. D. Thesis, State University of New York at Stony Brook (2013) | MR

[10] Hassett, Brendan; Tschinkel, Yuri Weak approximation over function fields, Invent. Math., Volume 163 (2006) no. 1, pp. 171-190 | DOI | MR | Zbl

[11] de Jong, Aise J.; Starr, Jason Michael Every rationally connected variety over the function field of a curve has a rational point, Am. J. Math., Volume 125 (2003) no. 3, pp. 567-580 | MR | Zbl

[12] Kollár, János Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 32, Springer, 1996, viii+320 pages | DOI | MR

[13] Kollár, János; Smith, Karen E.; Corti, Alessio Rational and nearly rational varieties, Cambridge Studies in Advanced Mathematics, 92, Cambridge University Press, 2004, vi+235 pages | DOI | MR

[14] Reid, Miles Bogomolov’s theorem c 1 2 4c 2 , Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) (1978), pp. 623-642 | MR

[15] Rojtman, A. A. The torsion of the group of 0-cycles modulo rational equivalence, Ann. Math., Volume 111 (1980) no. 3, pp. 553-569 | DOI | MR | Zbl

[16] Starr, Jason Michael; Tian, Zhiyu Separable rational connectedness and weak approximation in positive characteristic (https://arxiv.org/abs/1907.07041)

[17] Starr, Jason Michael; Tian, Zhiyu; Zong, Runhong Weak approximation for Fano complete intersections in positive characteristic (https://arxiv.org/abs/1811.02466)

[18] Tian, Zhiyu Separable rational connectedness and stability, Rational points, rational curves, and entire holomorphic curves on projective varieties (Contemporary Mathematics), Volume 654, American Mathematical Society, 2015, pp. 155-159 | DOI | MR | Zbl

[19] Tian, Zhiyu; Zong, Runhong Weak approximation for isotrivial families, J. Reine Angew. Math., Volume 752 (2019), pp. 1-23 | DOI | MR | Zbl

[20] Totaro, Burt Hypersurfaces that are not stably rational, J. Am. Math. Soc., Volume 29 (2016) no. 3, pp. 883-891 | DOI | MR | Zbl

[21] Tsen, Chiungtze Zur Stufentheorie der Quasialgebraisch-Abgeschlossenheit kommutativer Körper, J. Chin. Math. Soc., Volume 1 (1936), pp. 81-92 | Zbl

[22] Voisin, Claire Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal, J. Algebr. Geom., Volume 22 (2013) no. 1, pp. 141-174 | DOI | MR | Zbl

Cited by Sources: