On the classification of toric Fano 4-folds

@article{Batyrev1998OnTC,
  title={On the classification of toric Fano 4-folds},
  author={Victor V. Batyrev},
  journal={Journal of Mathematical Sciences},
  year={1998},
  volume={94},
  pages={1021-1050}
}
  • V. Batyrev
  • Published 22 January 1998
  • Mathematics
  • Journal of Mathematical Sciences
AbstractThe biregular classification of smoothd-dimensional toric Fano varieties is equivalent to the classification of special simplicial polyhedraP in ℝd, the so-called Fano polyhedra, up to an isomorphism of the standard lattice % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr… Expand
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The d -dimensional simplicial, terminal, and reflexive polytopes with at least 3d-2 vertices are classified. In particular, it turns out that all of them are smooth Fano polytopes. This improves… Expand
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Building on the work of Casagrande-Codogni-Fanelli, we develop our study on the birational geometry of the Fano fourfold Y = MS,−KS which is the moduli space of semi-stable rank-two torsion-free… Expand
Fano 4-folds having a prime divisor of Picard number 1
Fano varieties (over C) have been thoroughly classified up to dimension 3, and are widely studied also in higher dimensions. They act an important role in the Minimal Model Program, since they appear… Expand
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We find a characterization for Fano 4-folds $X$ with Lefschetz defect $\delta _{X}=3$: besides the product of two del Pezzo surfaces, they correspond to varieties admitting a conic bundle structure… Expand
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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… Expand
Anticanonical geometry of the blow-up of P 4 in 8 points and its Fano model
  • Zhixin Xie
  • 2021
Building on the work of Casagrande-Codogni-Fanelli, we develop our study on the birational geometry of the Fano fourfold Y = MS,−KS which is the moduli space of semi-stable rank-two torsion-free… Expand
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References

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This article contains a classification of special Fano varieties; we give a description of the projective models of Fano 3-folds of index r ≥ 2 and of hyperelliptic Fano 3-folds.Bibliography: 31… Expand
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(0.1) This paper applies the ideas of Mori theory [4] to toric varieties. Let X be a projective tonic variety (over any field) constructed from a simplicial fan F. The cone of effective 1-cycles… Expand
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On the classification of toric fano varieties
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  • Mathematics, Computer Science
  • Discret. Comput. Geom.
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TLDR
Toric Fano varieties are algebraic varieties associated with a special class of convex polytopes in R′' using a purely combinatorial method of proof. Expand
The boundedness of degree of Fano varieties with Picard number one
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TOROIDAL FANO VARIETIES AND ROOT SYSTEMS
In this paper it is shown that over an algebraically closed field there exist only finitely many mutually nonisomorphic toroidal Fano varieties. We give a complete description of toroidal Fano… Expand
Classification of Fano 3-folds with B2≥2
This article contains the classification of Fano 3-folds with B2≥2. There exist exactly 87 types of such 3-folds up to deformations; a Fano 3-fold is isomorphic to a product of Pl and a del Pezzo… Expand
Cubic forms; algebra, geometry, arithmetic
CH-Quasigroups and Moufang Loops. Classes of Points on Cubic Hypersurfaces. Two-Dimensional Birational Geometry. The Twenty-Seven Lines. Minimal Cubic Surfaces. The Brauer-Grothendieck Group.… Expand
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