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Degenerations of rationally connected varieties


Authors: Amit Hogadi and Chenyang Xu
Journal: Trans. Amer. Math. Soc. 361 (2009), 3931-3949
MSC (2000): Primary 14J26, 14J45; Secondary 14E30, 14G27
DOI: https://doi.org/10.1090/S0002-9947-09-04715-1
Published electronically: March 3, 2009
MathSciNet review: 2491906
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Abstract: We prove that a degeneration of rationally connected varieties over a field of characteristic zero always contains a geometrically irreducible subvariety which is rationally connected.


References [Enhancements On Off] (What's this?)

  • [Am04] Ambro, F.; Shokurov's boundary property. J. Differential Geom. 67 (2004), no. 2, 229-255. MR 2153078 (2006d:14033)
  • [BPV84] Barth, W.; Peters, C.; Van de Ven, A.; Compact complex surfaces. Ergeb. Math. Grenz. 3, 4. Springer-Verlag, Berlin, 1984. MR 749574 (86c:32026)
  • [BCHM06] Birkar, C.; Cascini, P.; Hacon, C.; McKernan, J.; Existence of minimal models for varieties of log general type, math.AG/0610203 (2006).
  • [Ca92] Campana, F.; Connexité rationelle des variétés de Fano. Ann. Sci. École Norm. Sup. 25 (1992), 539-545. MR 1191735 (93k:14050)
  • [dJS03] de Jong, A. J.; Starr, J.; 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 (2004h:14018)
  • [Es03] Esnault, H.; Varieties over a finite field with trivial Chow group of 0-cycles have a rational point. Invent. Math. 151 (2003), no. 1, 187-191. MR 1943746 (2004e:14015)
  • [EX07] Esnault, H.; Xu, C.; Congruence for rational points over finite fields and coniveau over local fields, math.NT/0706.0972 (2007). To appear in Trans. Amer. Math. Soc.
  • [FR05] Fakhruddin, N.; Rajan, C.S.; Congruences for rational points on varieties over finite fields. Math. Ann. 333 (2005), no. 4, 797-809. MR 2195144 (2006h:14028)
  • [GHS03] Graber, T.; Harris, J.; Starr, J.; Families of rationally connected varieties. J. Amer. Math. Soc. 16 (2003), no. 1, 57-67. MR 1937199 (2003m:14081)
  • [GHMS05] Graber, T.; Harris, J.; Mazur, B.; Starr, J.; Rational connectivity and sections of families over curves. Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 5, 671-692. MR 2195256 (2006j:14044)
  • [HM06] Hacon, C.; McKernan, J.; Boundedness of pluricanonical maps of varieties of general type. Invent. Math. 166 (2006) no. 1, 1-25. MR 2242631 (2007e:14022)
  • [HM07] Hacon, C.; McKernan, J.; Shokurov's rational connectedness conjecture. Duke Math. J. 138 (2007) no. 1, 119-136. MR 2309156
  • [Kol96] Kollár, J.; Rational curves on algebraic varieties. Ergeb. Math. Grenz. 3. Folge, 32. Springer-Verlag, Berlin, 1996. MR 1440180 (98c:14001)
  • [Kol05a] Kollár, J.; A conjecture of Ax and degenerations of Fano varieties, math.AG/0512375 (2005).
  • [Kol05b] Kollár, J.; Kodaira's canonical bundle formula and subadjunction, Ch. $ 8$ in Flips for $ 3$-folds and $ 4$-folds, editor A. Corti, Oxford Lecture Series in Mathematics and its applications 35, Oxford University Press, 2007. MR 2352762
  • [Kol06] Kollár, J.; Is there a topological Bogomolov-Miyaoka-Yau inequality? math.AG/0602562 (2006).
  • [KM98] Kollár, J.; Mori, S.; Birational geometry of algebraic varieties. Cambridge Tracts in Math. 134, Cambridge University Press, Cambridge, 1998. MR 1658959 (2000b:14018)
  • [KMM92a] Kollár, J.; Miyaoka, Y.; Mori, S.; Rational connectedness and boundness of Fano manifolds. J. Diff. Geom. 36 (1992), no. 3, 765-779. MR 1189503 (94g:14021)
  • [KMM92b] Kollár, J.; Miyaoka, Y.; Mori, S.; Rationally connected varieties. J. Algebraic Geom. 1 (1992), no. 3, 429-448. MR 1158625 (93i:14014)
  • [MP06] Mori, M.; Prokhorov, Y.; On $ \mathbb{Q}$-conic bundles, math.AG/0603736 (2006). To appear in Publ. Res. Inst. Math. Sci.
  • [Re] Reid, M.; Young person's guide to canonical singularities. Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 345-414, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987. MR 927963 (89b:14016)
  • [St06] Starr, J.; Degenerations of rationally connected varieties and PAC fields, math.AG/0602649, (2006).
  • [Zh06] Zhang, Q.; Rational connectedness of log $ \mathbb{Q}$-Fano varieties. J. Reine Angew. Math. 590 (2006), 131-142. MR 2208131 (2006m:14021)

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Additional Information

Amit Hogadi
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Address at time of publication: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005 India
Email: amit@math.princeton.edu

Chenyang Xu
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: chenyang@math.princeton.edu

DOI: https://doi.org/10.1090/S0002-9947-09-04715-1
Received by editor(s): June 6, 2007
Received by editor(s) in revised form: November 29, 2007
Published electronically: March 3, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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