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Toric Fano varieties and birational morphisms
 International Mathematics Research Notices
, 2003
"... Smooth toric Fano varieties are classified up to dimension 4. In dimension 2 there are five toric Del Pezzo surfaces: P 2, P 1 × P 1 and Si, the blowup of P 2 in i points, for i = 1,2,3. There are 18 toric Fano 3folds [2, 20] and 124 toric Fano 4folds [4, 17]. In higher dimensions, little is know ..."
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Smooth toric Fano varieties are classified up to dimension 4. In dimension 2 there are five toric Del Pezzo surfaces: P 2, P 1 × P 1 and Si, the blowup of P 2 in i points, for i = 1,2,3. There are 18 toric Fano 3folds [2, 20] and 124 toric Fano 4folds [4, 17]. In higher dimensions, little
Jumping of the nef cone for Fano varieties
, 2009
"... Among all projective algebraic varieties, Fano varieties (those with ample anticanonical bundle) can be considered the simplest. Birkar, Cascini, Hacon and M c Kernan showed that the Cox ring of a Fano variety, the ring of all sections of all line bundles, is finitely generated [4]. This implies a f ..."
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Among all projective algebraic varieties, Fano varieties (those with ample anticanonical bundle) can be considered the simplest. Birkar, Cascini, Hacon and M c Kernan showed that the Cox ring of a Fano variety, the ring of all sections of all line bundles, is finitely generated [4]. This implies a
FANO VARIETIES WITH LARGE DEGREE ENDOMORPHISMS
, 901
"... We work over the complex number field C. An endomorphism of a projective variety X is a morphism f: X → X. If f is surjective, then it is automatically finite. Examples of projective varieties with an endomorphism of degree> 1 are P n and abelian varieties. In fact, these are essentially all know ..."
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known smooth examples. When X has nonnegative Kodaira dimension, this is discussed in [NZ07]. It is also conjectured that if X is a smooth Fano variety of Picard number 1 that admits a degree> 1 endomorphism, then X ∼ = P n [Am97]. This is confirmed in some cases, including when the dimension
On Landau–Ginzburg models for Fano varieties
 2008) (preprint 0707.3758). Steklov Mathematical Institute, 8 Gubkina
"... Abstract. We observe a method for finding weak LandauGinzburg models for Fano varieties and find them for smooth Fano threefolds of genera 9, 10, and 12. In the late 1980’s physicists discovered a phenomenon of Mirror Symmetry. They found that given a Calabi–Yau variety one can construct the so cal ..."
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Abstract. We observe a method for finding weak LandauGinzburg models for Fano varieties and find them for smooth Fano threefolds of genera 9, 10, and 12. In the late 1980’s physicists discovered a phenomenon of Mirror Symmetry. They found that given a Calabi–Yau variety one can construct the so
Toward the classification of higherdimensional toric Fano varieties
 Tohoku Math. J
"... The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano varieties by means of the notions of primitive collections and primitive relations due to Batyrev. By using them we can easily deal with equivariant blowups and blowdowns, and get an easy criterion to ..."
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Cited by 45 (8 self)
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The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano varieties by means of the notions of primitive collections and primitive relations due to Batyrev. By using them we can easily deal with equivariant blowups and blowdowns, and get an easy criterion
Boundedness of QFano varieties with Picard number one
, 1999
"... We prove birational boundedness of QFano varieties with Picard number one in arbitrary dimension. ..."
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We prove birational boundedness of QFano varieties with Picard number one in arbitrary dimension.
Centrally symmetric generators in toric Fano varieties
 Manuscr. Math
"... Abstract. We give a structure theorem for ndimensional smooth toric Fano varieties whose associated polytope has “many ” pairs of centrally symmetric vertices. Introduction. Smooth toric Fano varieties, together with their equivariant birational contractions, have been intensively studied in recent ..."
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Cited by 10 (0 self)
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Abstract. We give a structure theorem for ndimensional smooth toric Fano varieties whose associated polytope has “many ” pairs of centrally symmetric vertices. Introduction. Smooth toric Fano varieties, together with their equivariant birational contractions, have been intensively studied
The cone of curves of Fano varieties of coindex four
 International Journal of Math
"... Abstract. We classify the cones of curves of Fano varieties of dimension greater or equal than five and (pseudo)index dim X − 3, describing the number and type of their extremal rays. 1. ..."
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Cited by 3 (1 self)
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Abstract. We classify the cones of curves of Fano varieties of dimension greater or equal than five and (pseudo)index dim X − 3, describing the number and type of their extremal rays. 1.
FANO VARIETIES OF CUBIC FOURFOLDS CONTAINING A PLANE
, 2009
"... We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a K3 su ..."
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We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a K3
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