×

Rational points on log Fano threefolds over a finite field. (English) Zbl 1462.14045

Summary: We prove the \(W\mathcal{O} \)-rationality of klt threefolds and the rational chain connectedness of klt Fano threefolds over a perfect field of characteristic \(p > 5\). As a consequence, any klt Fano threefold over a finite field has a rational point.

MSC:

14J45 Fano varieties
14G05 Rational points
14E30 Minimal model program (Mori theory, extremal rays)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abhyankar, S. S.: Resolution of Singularities of Embedded Algebraic Surfaces. Pure Appl. Math. 24, Academic Press, New York (1966)Zbl 0147.20504 MR 0217069 · Zbl 0147.20504
[2] Berthelot, P., Bloch, S., Esnault, H.: On Witt vector cohomology for singular varieties. Compos. Math. 143, 363-392 (2007)Zbl 1213.14040 MR 2309991 · Zbl 1213.14040
[3] Birkar, C.: Existence of log canonical flips and a special LMMP. Publ. Math. Inst. Hautes ´Etudes Sci. 115, 325-368 (2012)Zbl 1256.14012 MR 2929730 · Zbl 1256.14012
[4] Birkar, C.: Existence of flips and minimal models for 3-folds in char p. Ann. Sci. ´Ecole Norm. Sup. (4) 49, 169-212 (2016)Zbl 1346.14040 MR 3465979 · Zbl 1346.14040
[5] Birkar, C., Waldron, J.: Existence of Mori fibre spaces for 3-folds in char p. Adv. Math. 313, 62-101 (2017)Zbl 1373.14019 MR 3649221 · Zbl 1373.14019
[6] Blickle, M., Esnault, H.: Rational singularities and rational points. Pure Appl. Math. Quart. 4, 729-741 (2008)Zbl 1162.14015 MR 2435842 · Zbl 1162.14015
[7] Campana, F.: Connexit´e rationnelle des vari´et´es de Fano. Ann. Sci. ´Ecole Norm. Sup. (4) 25, 539-545 (1992)Zbl 0783.14022 MR 1191735 · Zbl 0783.14022
[8] Cascini, P., McKernan, J., Mustat¸˘a, M.: The augmented base locus in positive characteristic. Proc. Edinburgh Math. Soc. (2) 57, 79-87 (2014)Zbl 1290.14006 MR 3165013 · Zbl 1290.14006
[9] Chatzistamatiou, A., R¨ulling, K.: Higher direct images of the structure sheaf in positive characteristic. Algebra Number Theory 5, 693-775 (2011)Zbl 1253.14013 MR 2923726 · Zbl 1253.14013
[10] Chatzistamatiou, A., R¨ulling, K.: Hodge-Witt cohomology and Witt-rational singularities. Doc. Math. 17, 663-781 (2012)Zbl 1317.14081 MR 3001634 · Zbl 1317.14081
[11] Cossart, V., Piltant, O.: Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings. J. Algebra 320, 1051-1082 (2008)Zbl 1159.14009 MR 2427629 · Zbl 1159.14009
[12] de Jong, A. J.: Families of curves and alterations. Ann. Inst. Fourier (Grenoble) 47, 599-621 (1997)Zbl 0868.14012 MR 1450427 · Zbl 0868.14012
[13] Esnault, H.: Varieties over a finite field with trivial Chow group of 0-cycles have a rational point. Invent. Math. 151, 187-191 (2003)Zbl 1092.14010 MR 1943746 · Zbl 1092.14010
[14] Fujino, O.: Special termination and reduction to pl flips. In: Flips for 3-folds and 4folds, Oxford Lecture Ser. Math. Appl. 35, Oxford Univ. Press, Oxford, 63-75 (2007) Zbl 1286.14025 MR 2359342 · Zbl 1286.14025
[15] Fujino, O., Tanaka, H.: On log surfaces. Proc. Japan Acad. Ser. A Math. Sci. 88, 109- 114 (2012)Zbl 1268.14012 MR 2989060 · Zbl 1268.14012
[16] Grothendieck, A.: ´El´ements de g´eom´etrie alg´ebrique. IV. ´Etude locale des sch´emas et des morphismes de sch´emas. II. Publ. Math. Inst. Hautes ´Etudes Sci. 24, 231 pp. (1965) Zbl 0135.39701 MR 0199181 · Zbl 0135.39701
[17] Hacon, C. D., McKernan, J.: On Shokurov’s rational connectedness conjecture. Duke Math. J. 138, 119-136 (2007)Zbl 1128.14028 MR 2309156 · Zbl 1128.14028
[18] Hacon, C. D., Xu, C.: On the three dimensional minimal model program in positive characteristic. J. Amer. Math. Soc. 28, 711-744 (2015)Zbl 1326.14032 MR 3327534 · Zbl 1326.14032
[19] Illusie, L.: Complexe de de Rham-Witt et cohomologie cristalline. Ann. Sci. ´Ecole Norm. Sup. (4) 12, 501-661 (1979)Zbl 0436.14007 MR 0565469 · Zbl 0436.14007
[20] Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem. In: Algebraic Geometry (Sendai, 1985), Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 283-360 (1987)Zbl 0672.14006 MR 0946243 · Zbl 0672.14006
[21] Keel, S.: Basepoint freeness for nef and big line bundles in positive characteristic. Ann. of Math. (2) 149, 253-286 (1999)Zbl 0954.14004 MR 1680559 · Zbl 0954.14004
[22] Kerz, M., Esnault, H., Wittenberg, O.: A restriction isomorphism for cycles of relative dimension zero. Cambridge J. Math. 4, 163-196 (2016)Zbl 1376.14008 MR 3529393 · Zbl 1376.14008
[23] Koll´ar, J.: Rational Curves on Algebraic Varieties. Ergeb. Math. Grenzgeb. 32, Springer, Berlin (1996)Zbl 0877.14012 MR 1440180
[24] Koll´ar, J.: Quotient spaces modulo algebraic groups. Ann. of Math. (2) 145, 33-79 (1997)Zbl 0881.14017 MR 1432036 · Zbl 0881.14017
[25] Koll´ar, J.: Singularities of the Minimal Model Program. Cambridge Tracts in Math. 200, Cambridge Univ. Press, Cambridge (2013)Zbl 1342.32020 MR 3057950
[26] Koll´ar, J., Miyaoka, Y., Mori, S.: Rational connectedness and boundedness of Fano manifolds. J. Differential Geom. 36, 765-779 (1992)Zbl 0759.14032 MR 1189503 · Zbl 0759.14032
[27] Koll´ar, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge (1998)Zbl 1143.14014 MR 1658959 · Zbl 0926.14003
[28] Maddock, Z.: Regular del Pezzo surfaces with irregularity. J. Algebraic Geom. 25, 401- 429 (2016)Zbl 1336.14026 MR 3493588 · Zbl 1336.14026
[29] Poonen, B.: Bertini theorems over finite fields. Ann. of Math. 160, 1099-1127 (2004) Zbl 1084.14026 MR 2144974 · Zbl 1084.14026
[30] Prokhorov, Yu. G., Shokurov, V. V.: Towards the second main theorem on complements. J. Algebraic Geom. 18, 151-199 (2009)Zbl 1159.14020 MR 2448282 · Zbl 1159.14020
[31] Schr¨oer, S.: Weak del Pezzo surfaces with irregularity. Tohoku Math. J. (2) 59, 293-322 (2007)Zbl 1135.14033 MR 2347424 · Zbl 1135.14033
[32] Schwede, K., Smith, K. E.: Globally F -regular and log Fano varieties. Adv. Math. 224, 863-894 (2010)Zbl 1193.13004 MR 2628797 · Zbl 1193.13004
[33] Serre, J.-P.: Local Fields. Grad. Texts in Math. 67, Springer, New York (1979) Zbl 0423.12016 MR 0554237 · Zbl 0423.12016
[34] Shokurov, V. V.: On rational connectedness. Mat. Zametki 68, 771-782 (2000) (in Russian); English transl.: Math. Notes 68, 652-660 (2000)Zbl 1047.14006 MR 1835458 · Zbl 1047.14006
[35] Tanaka, H.: Minimal models and abundance for positive characteristic log surfaces. Nagoya Math. J. 216, 1-70 (2014)Zbl 1311.14020 MR 3319838 · Zbl 1311.14020
[36] Tanaka, H.: The X-method for klt surfaces in positive characteristic. J. Algebraic Geom. 24, 605-628 (2015)Zbl 1338.14017 MR 3383599 · Zbl 1338.14017
[37] Tanaka, H.: Abundance theorem for semi log canonical surfaces in positive characteristic. Osaka J. Math. 53, 535-566 (2016)Zbl 1353.14020 MR 3492812 · Zbl 1353.14020
[38] Tanaka, H.: Semiample perturbations for log canonical varieties over an F -finite field containing an infinite perfect field. Int. J. Math. 28, art. 1750030, 13 pp. (2017) Zbl 1387.14058 MR 3655076 · Zbl 1387.14058
[39] Tanaka, H.: Behavior of canonical divisors under purely inseparable base changes. J. Reine Angew. Math. 744, 237-264 (2018)Zbl 06971602 MR 3871445 · Zbl 1468.14031
[40] Wang, Y.: On relative rational chain connectedness of threefolds with anti-big canonical divisors in positive characteristics. Pacific J. Math. 290, 231-245 (2017) Zbl 1390.14153MR3673085 · Zbl 1390.14153
[41] Xu, C.: Finiteness of algebraic fundamental groups. Compos. Math. 150, 409-414 (2014)Zbl 1291.14057 MR 3187625 · Zbl 1291.14057
[42] Xu, C.: On the base-point-free theorem of 3-folds in positive characteristic. J. Inst. Math. Jussieu 14, 577-588 (2015)Zbl 1346.14020 MR 3352529 · Zbl 1346.14020
[43] Zhang, Q.: Rational connectedness of log Q-Fano varieties. J. Reine Angew. Math. 590, 131-142 (2006)Zbl 1093.14059 MR 2208131 · Zbl 1093.14059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.