Biregular theory of fano 3-folds

Iskovskih, V A; Šokurov, V V

doi:10.1007/bfb0066644

Cited by 18 publications

(11 citation statements)

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“…Now by (1.5) it is sufficient to show that p3 is the only object among those listed in i... v) for which the ample generator H of Pic (X) satisfies n 3 = 1. In cases i)--iv) this is immediate in view of table 21 of [6]; on the other hand, X cannot be as in v) since the Fano genus of X 1 3 ~ H 3…”

Section: ) []mentioning

confidence: 92%

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Projective manifolds with the same homology as ? k

Lanteri

Struppa

1986

Monatshefte f�r Mathematik

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Abstract.Complex projective threefolds having the same integral homology groups as P 3 are classified.'This classification is used to improve a result of Fujita on manifolds whose integral cohomology ring is the same as that of pk and applies to the problem of classifying polarized manifolds (X, A), A being a nonsingular hypersurface such that H,. (A, Z) -H,. (X, 7/) for i ~< 2 dim A.The main result of this paper is the following (Thm. 2.1): the only compact K/ihler threefold having the same integral cohomology ring as P 3 is P 3 itself. This extends results of FUJITA ([3], [4] App. 2) and uses different arguments. Actually our proof relies on a partial classification of projective threefolds having the same integral homology groups as p3 (Thm. 1.5). We obtain this classification combining Yau's inequality for manifolds with ample canonical bundle with Iskowskih's classification of Fano threefolds. However, due to a gap in Iskowskih's proof, the "pathological Fano threefolds" still constitute an undecided case (case v) in Thm. 1.5. Theorem 1.5 also applies to the study of the pairs (X, A), where A is a smooth surface contained as an ample divisor in a projective threefold X and satisfying Hq (A, Z) '~ Hq ()if, 7/) for q ~< 4 (Cor. 2.3).We are indebted to the referee for calling our attention to [7].

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Section: ) []mentioning

confidence: 92%

“…IfKx is not ample, then Xis a Fano threefold of first kind, by (1.1). Then cases i)--iv) follow from [6] recalling that the intermediate jacobian of X is trivial. Case v) follows from ( [7], pp.…”

Section: ) []mentioning

confidence: 94%

Projective manifolds with the same homology as ? k

Lanteri

Struppa

1986

Monatshefte f�r Mathematik

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“…It is well known that if (X, H ) is a del Pezzo threefold of degree one then (X, H ) is isomorphic to (V 1 , − 1 2 K V 1 ), and if | − K X | is not base point free then X is isomorphic to V 1 or P 1 × S 1 ([13], [14]). …”

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…By Theorem 5.1 in [19], we have Pic(X) ∼ = π * 1 Pic(V 1 ) ⊕ π * 2 Pic(P 1 ). Since Pic(V 1 ) is generated by − 1 2 K V 1 and | − t 2 K V 1 | is free for t ≥ 2 ([13], [14]), |H | is free unless (X, H ) ∼ = (V 1 , π * 1 (− 1 2 K V 1 ) + π * 2 A) as in the corollary. However for a given point p of P 1 , S := π * 2 (p) is a del Pezzo surface of degree one and H | S is the anticanonical divisor of S. Thus (H ) = 1 2 and it completes the proof.…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

Seshadri constants and Fano manifolds

Lee

2003

Mathematische Zeitschrift

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In this paper, we give a lower bound of Seshadri constants on smooth Fano varieties. More precisely, we show that on a smooth Fano manifold of dimension n whose anticanonical system is base point free, Seshadri constants of ample divisors are bounded from below by one over n − 2. As a corollary we recover the earlier result on Fano threefolds. (2000): 14J45, 14N30. Mathematics Subject Classification

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“…Many interesting cases from that classification, and especially among the prime Fano manifolds of index one, are obtained by taking suitable sections (mostly linear sections) of certain rational homogeneous spaces, and Mukai wrote a wonderful series of papers about their astonishing geometry (see for example [5], and the more general reference [2]). …”

Section: Introductionmentioning

confidence: 99%

On Fano manifolds of Picard number one

Manivel

2015

Math. Z.

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Abstract. Küchle classified the Fano fourfolds that can be obtained as zero loci of global sections of homogeneous vector bundles on Grassmannians. Surprisingly, his classification exhibits two families of fourfolds with the same discrete invariants. Kuznetsov asked whether these two types of fourfolds are deformation equivalent. We show that the answer is positive in a very strong sense, since the two families are in fact the same! This phenomenon happens in higher dimension as well.

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Biregular theory of fano 3-folds

Cited by 18 publications

References 3 publications

Projective manifolds with the same homology as ? k

Projective manifolds with the same homology as ? k

Seshadri constants and Fano manifolds

On Fano manifolds of Picard number one

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