Skip to main content
Log in

Classification of terminal simplicial reflexive d-polytopes with 3d − 1 vertices

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We classify terminal simplicial reflexive d-polytopes with 3d − 1 vertices. They turn out to be smooth Fano d-polytopes. When d is even there is one such polytope up to isomorphism, while there are two when d is uneven.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Batyrev V.V. (1982). Toroidal Fano 3-folds. Math. USSR-Izv. 19: 13–25

    Article  MATH  Google Scholar 

  2. Batyrev V.V. (1991). On the classification of smooth projective toric varieties. Tohoku Math. J. 43: 569–585

    MATH  MathSciNet  Google Scholar 

  3. Batyrev V.V. (1994). Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebraic Geom. 3(3): 493–535

    MATH  MathSciNet  Google Scholar 

  4. Batyrev V.V. (1999). On the classification of toric Fano 4-folds. J. Math. Sci. (New York) 94: 1021–1050

    Article  MATH  MathSciNet  Google Scholar 

  5. Casagrande C. (2003). Centrally symmetric generators in toric Fano varieties. Manuscr. Math. 111: 471–485

    Article  MATH  MathSciNet  Google Scholar 

  6. Casagrande C. (2006). The number of vertices of a Fano polytope. Ann. Inst. Fourier 56: 121–130

    MATH  MathSciNet  Google Scholar 

  7. Debarre, O.: Fano varieties. In: Higher dimensional varieties and rational points (Budapest 2001), Bolyai Soc. Math. Stud., vol. 12, pp. 93–132. Springer, Berlin (2003)

  8. Ewald G. (1988). On the classification of toric Fano varieties. Discrete Comput. Geom. 3: 49–54

    Article  MATH  MathSciNet  Google Scholar 

  9. Kleinschmidt P. (1988). A classification of toric varieties with few generators. Aequationes Math 35(2–3): 254–266

    Article  MathSciNet  MATH  Google Scholar 

  10. Kreuzer, M., Nill, B.: Classification of toric Fano 5-folds. Preprint, math.AG/0702890 (2007)

  11. Kreuzer M. and Skarke H. (1998). Classification of reflexive polyhedra in three dimensions. Adv. Theor. Math. Phys. 2: 853–871

    MATH  MathSciNet  Google Scholar 

  12. Kreuzer M. and Skarke H. (2000). Complete classification of reflexive polyhedra in four dimensions. Adv. Theor. Math. Phys. 4: 1209–1230

    MATH  MathSciNet  Google Scholar 

  13. Nill B. (2005). Gorenstein toric Fano varieties. Manuscr. Math. 116: 183–210

    Article  MATH  MathSciNet  Google Scholar 

  14. Nill, B.: Classification of pseudo-symmetric simplicial reflexive polytopes. In: Algebraic and geometric combinatorics, Contemp. Math., vol. 423, pp. 269–282. Am. Math. Soc., Providence (2006)

  15. Sato H. (2000). Toward the classification of higher-dimensional Toric Fano varieties. Tohoku Math. J. 52: 383–413

    MATH  MathSciNet  Google Scholar 

  16. Voskresenskij V.E. and Klyachko A. (1985). Toric Fano varieties and systems of roots. Math. USSR-Izv. 24: 221–244

    Article  MATH  Google Scholar 

  17. Watanabe K. and Watanabe M. (1982). The classification of Fano 3-folds with torus embeddings. Tokyo Math. J. 5: 37–48

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mikkel Øbro.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Øbro, M. Classification of terminal simplicial reflexive d-polytopes with 3d − 1 vertices. manuscripta math. 125, 69–79 (2008). https://doi.org/10.1007/s00229-007-0133-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-007-0133-z

Mathematics Subject Classification (2000)

Navigation