Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Rational surfaces in index-one Fano hypersurfaces


Authors: Roya Beheshti and Jason Michael Starr
Journal: J. Algebraic Geom. 17 (2008), 255-274
DOI: https://doi.org/10.1090/S1056-3911-07-00459-6
Published electronically: December 5, 2007
MathSciNet review: 2369086
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Abstract | References | Additional Information

Abstract: We give the first evidence for a conjecture that a general, index-one, Fano hypersurface is not unirational: (i) a general point of the hypersurface is contained in no rational surface ruled, roughly, by low-degree rational curves, and (ii) a general point is contained in no image of a Del Pezzo surface.


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

  • 1. O. Debarre, Higher-dimensional algebraic geometry, Springer-Verlag, 2001. MR 1841091 (2002g:14001)
  • 2. F. Campana, Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. 25 (1992), 539-545. MR 1191735 (93k:14050)
  • 3. A. J. de Jong, J. Starr, Cubic fourfolds and spaces of rational curves, Illinois J. Math., 48 (2004), 415-450. MR 2085418 (2006e:14007)
  • 4. W. Fulton, R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry--Santa Cruz 1995, 45-96, Proc. Sympos. Pure Math., 62, 1995. MR 1492534 (98m:14025)
  • 5. P. A. Griffiths, On the periods of certain rational integrals, I., Annals of Math., 90 (1969), 496-451. MR 0260733 (41:5357)
  • 6. R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977. MR 0463157 (57:3116)
  • 7. V. Iskovskih, J. Manin, Three-dimensional quartics and counterexamples to the Lüroth problem, Mat. Sb. (N.S.) 86 (1971), 140-166. MR 0291172 (45:266)
  • 8. J. Kollár, Which are the simplest algebraic varieties?, Bull. Amer. Math. Soc. (N.S.) 38 (2001), 409-433. MR 1848255 (2002f:14001)
  • 9. J. Kollár, Y. Miyaoka, S. Mori, Rational connectedness and boundedness of Fano manifolds, J. Diff. Geom. 36 (1992), 765-769. MR 1189503 (94g:14021)
  • 10. M. Olsson, J. Starr, Quot functors for Deligne-Mumford stacks, Comm. Algebra 31 (2003), 4069-4096. MR 2007396 (2004i:14002)


Additional Information

Roya Beheshti
Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6
Address at time of publication: Department of Mathematics, Washington University in St. Louis, St. Louis, Missouri 63130
Email: beheshti@mast.queensu.ca; beheshti@math.wustl.edu

Jason Michael Starr
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794
Email: jstarr@math.mit.edu; jstarr@math.sunysb.edu

DOI: https://doi.org/10.1090/S1056-3911-07-00459-6
Received by editor(s): February 1, 2006
Received by editor(s) in revised form: April 10, 2006, and May 30, 2006
Published electronically: December 5, 2007
Additional Notes: The second author is supported by NSF grant DMS-0353692 and a Sloan Research Fellowship

American Mathematical Society