Foliations and rational connectedness in positive characteristic
Author:
Mingmin Shen
Journal:
J. Algebraic Geom. 19 (2010), 531-553
DOI:
https://doi.org/10.1090/S1056-3911-10-00552-7
Published electronically:
March 8, 2010
MathSciNet review:
2629599
Full-text PDF
Abstract |
References |
Additional Information
Abstract: In this paper, the technique of foliations in characteristic $p$ is used to investigate the difference between rational connectedness and separable rational connectedness in positive characteristic. The notion of being freely rationally connected is defined; a variety is freely rationally connected if a general pair of points can be connected by a free rational curve. It is proved that a freely rationally connected variety admits a finite purely inseparable morphism to a separably rationally connected variety. As an application, a generalized Graber-Harris-Starr type theorem in positive characteristic is proved; namely, if a family of varieties over a smooth curve has the property that its geometric generic fiber is normal and freely rationally connected, then it has a rational section after some Frobenius twisting. We also show that a freely rationally connected variety is simply connected.
References
- M. Artin, Algebraization of formal moduli. I, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 21–71. MR 0260746
- Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822
- Brian Conrad and A. J. de Jong, Approximation of versal deformations, J. Algebra 255 (2002), no. 2, 489–515. MR 1935511, DOI https://doi.org/10.1016/S0021-8693%2802%2900144-8
- David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, 1999. MR 1677117
- Olivier Debarre, Variétés rationnellement connexes (d’après T. Graber, J. Harris, J. Starr et A. J. de Jong), Astérisque 290 (2003), Exp. No. 905, ix, 243–266 (French, with French summary). Séminaire Bourbaki. Vol. 2001/2002. MR 2074059
- P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240
- J. Dieudonné and A. Grothendieck, Eléments de géométrie algébrique I, II, III, IV, Inst. Hautes Études Sci. Publi. Math. 4, 8, 11, 17, 20, 24, 28, 32 (1961–1967).
- Torsten Ekedahl, Foliations and inseparable morphisms, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 139–149. MR 927978, DOI https://doi.org/10.1090/pspum/046.2/927978
- Tom Graber, Joe Harris, and Jason Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), no. 1, 57–67. MR 1937199, DOI https://doi.org/10.1090/S0894-0347-02-00402-2
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- A. J. de Jong and J. Starr, Every rationally connected variety over the function field of a curve has a rational point, Amer. J. Math. 125 (2003), no. 3, 567–580. MR 1981034
- Nicholas M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 175–232. MR 291177
- János Kollár, Rational curves on algebraic varieties, 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. 32, Springer-Verlag, Berlin, 1996. MR 1440180
- Jun Li and Gang Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), no. 1, 119–174. MR 1467172, DOI https://doi.org/10.1090/S0894-0347-98-00250-1
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- Yoichi Miyaoka and Thomas Peternell, Geometry of higher-dimensional algebraic varieties, DMV Seminar, vol. 26, Birkhäuser Verlag, Basel, 1997. MR 1468476
- Edoardo Sernesi, Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 334, Springer-Verlag, Berlin, 2006. MR 2247603
References
- M. Artin, Algebraization of formal moduli I, in Global analysis (papers in honor of K. Kodaira), Univ. of Tokyo Press, Tokyo (1969), pp.21–71. MR 0260746 (41:5369)
- S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models, Ergeb. der Math. Grenzgeb., vol. 21, Springer-Verlag, 1990. MR 1045822 (91i:14034)
- B. Conrad and A. J. de Jong, Approximation of versal deformations, J. Algebra 255 (2002), No.2, pp.489–515. MR 1935511 (2004a:14003)
- D. A. Cox and S. Katz, Mirror symmetry and algebraic geometry, Mathmatical surveys and monographs, vol. 68, the American Mathematical Societies, 1999. MR 1677117 (2000d:14048)
- O. Debarre, Variétés rationnellement connexes, Séminaire Bourbaki, 2001–2002, exp. n$^\circ$ 905, pp. 243–266. MR 2074059 (2005g:14096)
- P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math., 36 (1969), 75–109. MR 0262240 (41:6850)
- J. Dieudonné and A. Grothendieck, Eléments de géométrie algébrique I, II, III, IV, Inst. Hautes Études Sci. Publi. Math. 4, 8, 11, 17, 20, 24, 28, 32 (1961–1967).
- T. Ekedahl, Foliations and inseparable morphisms, in Algebraic geometry, Bowdoin 1985, 139–149, Proc. Sympos. Pure Math., vol. 46, part 2, Amer. Math. Soc., Providence, RI, 1987. MR 927978 (89d:14049)
- T. Graber, J. Harris and J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc., 16 (2003), 29–55. MR 1937199 (2003m:14081)
- R. Hartshorne, Algebraic geometry, GTM 52, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
- A. J. de Jong and J. Starr, Every rationally connected variety over the function field of a curve has a rational point, Amer. J. of Math., 125 (2003), 567–580. MR 1981034 (2004h:14018)
- N. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrinttin, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 175-232. MR 0291177 (45:271)
- J. Kollár, Rational curves on algebraic varieties, Ergeb. der Math. Grenzgeb., vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180 (98c:14001)
- J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), 119–174. MR 1467172 (99d:14011)
- H. Matsumura, Commutative ring theory, translated by M. Reid, Cambridge University Press, 1986. MR 879273 (88h:13001)
- Y. Miyaoka and T. Peternell, Geometry of higher dimensional algebraic varieties, DMV Seminar 26, Birkhäuser 1997. MR 1468476 (98g:14001)
- E. Sernesi, Deformations of algebraic schemes, Grundlehren der mathematischen Wissenschaften, vol. 334, Springer-Verlag, 2006. MR 2247603 (2008e:14011)
Additional Information
Mingmin Shen
Affiliation:
Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027
Email:
mshen@math.columbia.edu
Received by editor(s):
May 30, 2008
Received by editor(s) in revised form:
October 9, 2009
Published electronically:
March 8, 2010
