Toric Fano varieties and birational morphisms

@article{Casagrande2001ToricFV,
  title={Toric Fano varieties and birational morphisms},
  author={Cinzia Casagrande},
  journal={International Mathematics Research Notices},
  year={2001},
  volume={2003},
  pages={1473-1505}
}
  • C. Casagrande
  • Published 1 December 2001
  • Mathematics
  • International Mathematics Research Notices
In this paper we study smooth toric Fano varieties using primitive relations and toric Mori theory. We show that for any irreducible invariant divisor D in a toric Fano variety X, we have $0\leq\rho_X-\rho_D\leq 3$, for the difference of the Picard numbers of X and D. Moreover, if $\rho_X-\rho_D>0$ (with some additional hypotheses if $\rho_X-\rho_D=1$), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have $\rho_X\leq 9$. In the second part of… 

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