Smoothness of the General Anticanonical Divisor on a Fano 3-Fold

Šokurov, V V

doi:10.1070/im1980v014n02abeh001123

Cited by 55 publications

(46 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This assumption is somewhat weaker than the original assumption of Iskovskih and true for dimension two, three ( [2,3]) and four ( [4,5]). Actually the fundamental divisors of Fano n-folds of index greater than one are very ample at least for dimension n ≤ 4.…”

Section: The Observationmentioning

confidence: 92%

A Remark on Fano Manifolds of Picard Number One and Index Greater Than One

Lee¹

2016

Journal of the Chungcheong Mathematical Society

View full text Add to dashboard Cite

show abstract

Section: The Observationmentioning

confidence: 92%

A Remark on Fano Manifolds of Picard Number One and Index Greater Than One

Lee¹

2016

Journal of the Chungcheong Mathematical Society

View full text Add to dashboard Cite

show abstract

“…The conjecture is affirmative in case of Gorenstein canonical Fano 3-folds [Sho79b] and [Reid83]. In §1, we prove the following:…”

Section: No H (−Kmentioning

confidence: 95%

On classification of ℚ-fano 3-folds of Gorenstein index 2. II

Takagi¹

2002

Nagoya Mathematical Journal

View full text Add to dashboard Cite

Abstract. In the previous paper, we obtained a list of numerical possibilities of -Fano 3-folds X with Pic X = ¡ (−2KX) and h 0 (−KX) ≥ 4 containing index 2 points P such that (X, P ) ({xy + z 2 + u a = 0}/¡ 2(1, 1, 1, 0), o) for some a ∈ ¢ . Moreover we showed that such an X is birational to a simpler Mori fiber space. In this paper, we prove their existence except for a few cases by constructing a Mori fiber space with desired properties and reconstructing X from it. Notation and Conventions N:The set of positive integers. ∼: Linear equivalence. ≡: Numerical equivalence. F n : Segre-del Pezzo scroll of degree n. F n,0 : Surface obtained by contracting the negative section of F n . Q 3 : Smooth quadric 3-fold. ODP: Ordinary double point, i.e., singularity analytically isomorphic to {xy + z 2 + u 2 = 0 ⊂ C 4 }. QODP: Singularity analytically isomorphic to {xy + z 2 + u 2 = 0 ⊂ C 4 /Z 2 (1, 1, 1, 0)}. B i (1 ≤ i ≤ 5): Factorial Gorenstein terminal Fano 3-fold of Fano index 2, and with Picard number 1 and (−K) 3 = 8i, where K is the canonical divisor. A 2g−2 (1 ≤ g ≤ 12 and g = 11): Factorial Gorenstein terminal Fano 3-fold of Fano index 1, and with Picard number 1 and genus g. Abuse of notation: We use the same notation for transforms of curves by birational maps as original ones. In this paper we work over C, the complex number field.Definition 0.0. (Q-Fano variety) Let X be a normal projective variety. X is said to be a terminal (resp. canonical, klt, etc.) Q-Fano variety if X has only terminal (resp. canonical, Kawamata log terminal, etc.) singularities and −K X is ample. By replacing 'ample' with 'nef and big', terminal (resp. canonical, klt, etc.) weak Q-Fano varieties are similarly defined. If X has only terminal singularities, then we say that X is a Q-Fano variety for short and if X has only Gorenstein terminal (resp. canonical, klt, etc.) singularities, we say that X is a Gorenstein terminal (resp. canonical, klt, etc.) Fano variety.Let I(X) := min{I | IK X is a Cartier divisor} and we call I(X) the Gorenstein index of X.Write I(X)(−K X ) ≡ r(X)H(X), where H(X) is a primitive Cartier divisor and r(X) ∈ N. (Note that H(X) is unique since Pic X is torsion free.) Then we call r(X)/I(X) the Fano index of X and denote it by F (X).In the previous paper [Taka02], we formulate a generalization of Takeuchi's method [Take89] for the classification of smooth Fano 3-folds and use it for a partial classification of Q-Fano 3-folds X with the following properties.Main Assumption 0.1.(1) The Picard number of X is 1, (2) the Gorenstein index of X is 2, (3) the Fano index of X is 1/2, (4) h 0 (−K X ) ≥ 4, and (5) there exists an index 2 point P such thatLet f : Y → X be the weighted blow-up at P with weight 1 2 (1, 1, 1, 2). In the previous paper [Taka02], we proved that Y is a weak Q-Fano 3-fold and obtained the following diagram. We use the following notation.Notation 0.2.• E := the strict transform of E on Y ,• Rational numbers z and u are defined as follows. In case f is birational, the• N is the number of as above. T...

show abstract

“…The main difficulty in constructing an explicit example is to find a smooth divisor in the appropriate linear system on the Fano 3-fold from the list. On the other hand the existence of such a divisor follows from [13] so it is easy to make a list of numerical data of 89 Calabi-Yau manifolds. Since the examples with Picard number 1 are well known, we only include examples with Picard number ≥ 2, i.e.…”

Section: S Cynkmentioning

confidence: 99%