Kollár, János Specialization of zero cycles. (English) Zbl 1081.14008 Publ. Res. Inst. Math. Sci. 40, No. 3, 689-708 (2004). Let \(X\) be a proper scheme over a field \(k\). Let \(\text{CH}_0(X)\) denote the Chow group of the rational equivalence classes of zero cycles on \(X\). On the other hand, there is another equivalence relation closely related to it. Two points \(x_1,x_2\in X(k)\) are called directly \(R\)-equivalent if there is a morphism \(p:\mathbb{P}^1\to X\) such that \(p(0,1)= x_1\), \(p(1,0)= x_2\). This generates an equivalence relation called \(R\)-equivalence. Let \(X(k)/R\) denote the set of \(R\)-equivalence classes of \(X(k)\). The main result in this paper says that the specialization maps on both sets of equivalence classes are isomorphisms. More precisely, let \(S\) be a local, Henselian, Dedekind ring with residue field \(k\) and quotient field \(K\). Let \(X_S\to\text{Spec\,}S\) be a smooth proper morphism. Assume that \(X_k\) is separably rationally connected. Then the specialization maps \(X_K(K)/R\to X_k(k)/R\) and \(\text{CH}_0(X_K)\to\text{CH}_0(X_k)\) are isomorphisms, for the latter of which \(k\) is assumed to be perfect. A key ingredient of the proof is a construction of good combs and their deformation theory based on the results in [T. Graber, J. Harris and J. Starr, J. Am. Math. Soc. 16, No. 1, 57–67 (2003; Zbl 1092.14063)]. Here a comb over a geometrically reduced projective curve C is defined to be a reduced projective curve \(C\cup A_1\cup\cdots\cup A_n\) where \(A_1,\dots, A_n\) are smooth rational curves, disjoint from each other and intersect \(C\) transversally in \(n\) distinct smooth points. Reviewer: Fumio Hazama (Hatoyama) Cited in 2 ReviewsCited in 14 Documents MSC: 14C15 (Equivariant) Chow groups and rings; motives 14G27 Other nonalgebraically closed ground fields in algebraic geometry 14M20 Rational and unirational varieties 14D15 Formal methods and deformations in algebraic geometry Keywords:Chow group; rationally connected variety Citations:Zbl 1092.14063 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Araujo, C. and Kollár, J., Rational curves on varieties, in: Higher dimensional varieties and rational points, Bolyai Soc. Math. Stud., 12 (2003), 13-68. · Zbl 1080.14521 [2] Artin, M., Algebraization of formal moduli. I, 1969 Global Analysis (Papers in Honor of K. Kodaira) pp. 21-71 Univ. Tokyo Press, Tokyo. · Zbl 0205.50402 [3] Colliot-Thél‘ ene, J.-L., Hilbert’s Theorem 90 for K2, with application to the Chow groups of rational surfaces, Invent. Math., 71 (1983), 1-20. · Zbl 0527.14011 · doi:10.1007/BF01393336 [4] , L’arithmétique du groupe de Chow des zéro-cycles. Les Dix- huitimes Journées Arithmétiques (Bordeaux, 1993), J. Théor. Nombres Bordeaux, 7 (1995), 51-73. [5] , Rational connectedness and Galois covers of the projective line, Ann. of Math., 151 (2000), 359-373. · Zbl 0990.12003 · doi:10.2307/121121 [6] Coray, D., Algebraic points on cubic hypersurfaces, Acta Arith., 30 (1976), 267-296. · Zbl 0294.14012 [7] Fulton, W., Intersection theory, Springer Verlag, 1984. · Zbl 0541.14005 [8] Graber, T., Harris, J. and Starr, J., Families of rationally connected varieties, J. Amer. Math. Soc., 16 (2003), 57-67. · Zbl 1092.14063 · doi:10.1090/S0894-0347-02-00402-2 [9] Kollár, J., Rational Curves on Algebraic Varieties, Springer, 1996. · Zbl 0877.14012 [10] , Low degree polynomial equations: arithmetic, geometry and topology. European Congress of Mathematics, Vol. I (Budapest, 1996) 255-288, Progr. Math., 168, Birkhäuser, Basel, 1998. · Zbl 0970.14001 [11] , Rationally connected varieties over local fields, Ann. of Math., 150 (1999), 357-367. · Zbl 0976.14016 · doi:10.2307/121107 [12] , Which are the simplest Algebraic Varieties?, Bull. AMS, 38 (2001), 409-433. · Zbl 0978.14039 · doi:10.1090/S0273-0979-01-00917-X [13] Kollár, J., Miyaoka, Y. and Mori, S., Rationally Connected Varieties, J. Alg. Geom., 1 (1992), 429-448. · Zbl 0780.14026 [14] Kollár, J. and Szabó, E., Rationally connected varieties over finite fields, Duke Math. J., 120 (2003), 251-267. · Zbl 1077.14068 · doi:10.1215/S0012-7094-03-12022-0 [15] Madore, D., Équivalence rationnell sur les hypersurfaces cubiques sur les corps p-adiques, Manuscr. Math., 110 (2003), 171-185. · Zbl 1073.14509 · doi:10.1007/s00229-002-0327-3 [16] Manin, Yu. I., Cubic forms, Nauka, 1972. · Zbl 0255.14002 [17] Milne, J., Étale cohomology, Princeton Univ. Press, 1980. · Zbl 0433.14012 [18] Moret-Bailly, L., R-équivalence simultanée de torseurs: un complément á l’article de P. Gille, J. Number Theory, 91 (2001), 293-296. János Kollár · Zbl 1076.14531 · doi:10.1006/jnth.2001.2677 [19] Moret-Bailly, L., Sur la R-equivalence de torseurs sous un groupe fini, J. Number Theory, 99 (2003), 383-404. · Zbl 1079.14031 · doi:10.1016/S0022-314X(02)00066-5 [20] Mumford, D., Varieties defined by quadratic equations, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969) 29-100. Edi- zioni Cremonese, Rome, 1970. · Zbl 0198.25801 [21] Serre, J.-P., Corps locaux, Hermann, 1962. · Zbl 0137.02601 [22] Grothendieck, A., Rev\hat etements étales et groupes fondamentale, Springer Lecture Notes, 224 (1971). 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.