The degree of $ \mathbb Q$-Fano threefolds

Prokhorov, Yuri

doi:10.1070/sm2007v198n11abeh003901

Cited by 21 publications

(25 citation statements)

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“…In [Nam97], Namikawa showed that every terminal Gorenstein Fano 3-fold Y can be deformed to a smooth Fano 3-fold, in particular we have Vol(−K Y ) ≤ 64. Moreover Prokhorov proved that the degree of a Q-factorial terminal non-Gorenstein Fano 3-fold of Picard rank one is bounded by 125/2 ( [Pro07]). All together, we have Vol(−K X ′ ) ≤ 64.…”

Section: Explicit Fujita-type Statementsmentioning

confidence: 99%

On the geometry of thin exceptional sets in Manin’s conjecture

Lehmann¹,

Tanimoto²

2017

Duke Math. J.

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Section: Explicit Fujita-type Statementsmentioning

confidence: 99%

On the geometry of thin exceptional sets in Manin’s conjecture

Lehmann¹,

Tanimoto²

2017

Duke Math. J.

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“…In the third row we indicate the number of candidates which do not occur as Q-Fano threefolds. This is according to this paper and [Pro07]. The fourth row shows the number baskets for which we know examples of Q-Fano threefolds.…”

Section: Introductionmentioning

confidence: 88%

“…This work is a sequel to our previous papers [Pro07], [Pro10]. Recall that a three-dimensional projective variety X is called Q-Fano threefold if it has only terminal Q-factorial singularities, Pic(X) ≃ Z, and its anticanonical divisor −K X is ample.…”

Section: Introductionmentioning

confidence: 97%

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Fano threefolds of large Fano index and large degree

Prokhorov¹

2013

Sb. Math.

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“…In fact, Kawamata's proof implies that the possible "candidate" Q-Fano 3-folds can be listed, although the volume of computation makes computer searches inevitable. This method was used in [Suz04], [BS07a], [BS07b], [Pro07], [Pro10b], [Pro10c]. See [GRDB] for explicit lists.…”

Section: Computer Search For Q-fano 3-foldsmentioning

confidence: 99%

On $\mathbb{Q}$-Fano 3-folds of Fano index 2

Prokhorov¹,

Reid²

Advanced Studies in Pure Mathematics

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To Shigefumi Mori in friendship and admiration §1. Introduction 1.1. Q-Fano 3-folds A Q-Fano 3-fold is a projective 3-fold X with at worst terminal singularities and ample anticanonical divisor −K X . Here, bearing in mind Mori's fundamental notion of extremal ray, we assume also that X is Q-factorial and has rank 1, that is, Pic X ≃ Z or equivalently, Cl X ⊗ Q ≃ Q. We define the Fano and Q-Fano index of X by:where ∼ is linear equivalence and ∼ Q is Q-linear equivalence. Clearly, q F (X) divides q Q (X), and the two coincide unless K X + qA ∈ Cl X is a nontrivial torsion element. An important invariant of a Q-Fano 3-fold is its genus g(X) := dim |−K X | − 1. Background factsKaori Suzuki [Suz04] restricts the Q-Fano index of X to one of . . . , 11, 13, 17, 19}. See also [Pro10b, Lemma 3.3]. Moreover, the following results are due to the first author. Theorem ([Pro10b]). Let X be a Q-Fano 3-fold of Q-Fano index q := q Q (X) ≥ 9. Then Cl X ≃ Z.(i) If q = 19 then X ≃ P(3, 4, 5, 7).(ii) If q = 17 then X ≃ P(2, 3, 5, 7). (iii) If q = 13 and g(X) > 4 then X ≃ P(1, 3, 4, 5).(iv) If q = 11 and g(X) > 10 then X ≃ P(1, 2, 3, 5).(v) q = 10. Theorem ([Pro10c]). Let X be a Q-Fano 3-fold of Q-Fano index q.(vi) If q = 9 and g(X) > 4 then X ≃ X 6 ⊂ P(1, 2, 3, 4, 5).Y.P. acknowledges partial support from RFBR

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The degree of $ \mathbb Q$-Fano threefolds

Abstract: We consider B-type D-branes in the Gepner model consisting of two minimal models at k = 2. This Gepner model is mirror to a torus theory. We establish the dictionary identifying the B-type D-branes of the Gepner model with A-type Neumann and Dirichlet branes on the torus.

Cited by 21 publications

References 30 publications

On the geometry of thin exceptional sets in Manin’s conjecture

On the geometry of thin exceptional sets in Manin’s conjecture

Fano threefolds of large Fano index and large degree

On $\mathbb{Q}$-Fano 3-folds of Fano index 2

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