Birational geometry of rationally connected
      manifolds via quasi-lines

Ciliberto, Ciro; Geramita, Antony V.; Harbourne, Brian; Miró‐Roig, Rosa M.; Ranestad, Kristian

doi:10.1515/9783110199703.317

Cited by 13 publications

(12 citation statements)

References 15 publications

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“…The following simple remark, which is surely well known to the experts, will play a central role several times in our analysis, see also [32,Lemma 2.2] and [52] for related statements. Since we were unable to find a precise reference for the generality needed, we also include a proof.…”

Section: Rationality Of X(r + 1 N δ) and Of The General Curve Of The ...mentioning

confidence: 99%

“…The classical roots of this type of results go back to C. Segre, [59], who proved that dim( X(2, 2, 2) ) ≤ 5 and that X(2, 2, 2) ⊂ P 5 is projectively equivalent to the Veronese surface. Bompiani generalized this result in [6] to dim( X(r + 1, 2, δ) ) ≤ r+1+δ r+1 − 1 with equality holding if and only if X(r + 1, 2, δ) is projectively equivalent to the δ-Veronese embedding of P r+1 , see Theorem 2.2 here and also [32] and [64].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Varieties $n$-covered by curves of degree $\delta$

Pirio¹,

Russo²

2013

Comment. Math. Helv.

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Consequently, the maps I, G and T ω are projective automorphisms of the cubic X 3 J . Let Conf(J) be the conformal group of J, that is the subgroup of P GL(Z 2 (J)) generated by I and the maps G g and T ω for all ω ∈ J and all g ∈ Str(J). From Lemma 4.6, it follows that Conf(J) is a subgroup of the group of projective automorphisms of X 3 J .Proposition 4.7. The group of projective automorphisms of X 3 J acts transitively on 3uple's of general points of X 3 J . In fact, if x 1 , x 2 , x 3 ∈ J are sufficiently general, it exists µ ∈ Conf(J) such that µ(ν 3 (x 1 )) = 0 J , µ(ν 3 (x 2 )) = 1 J and µ(ν 3 (x 3 )) = ∞ J .Proof. First let G = Str(J) · e be the orbit in J of the unit e under the linear action of the structural group. By [61, Theorem 6.5], one knows that G is a Zariski-open subset of J.Assume that x, y, z ∈ J are such that (1) y 1 = y − x and z 1 = z − x are invertible; (2) z 2 = z −1 1 − y −1 1 ∈ G , i.e z 2 = g(e) for a certain g ∈ Str(J). Then setWe leave to the reader to verify that µ(ν 3 (x)) = ∞ J , µ(ν 3 (y)) = 0 J and µ(ν 3 (z)) = 1 J . Since ν 3 (J) is dense in X 3 J , the conclusion follows.Let x 1 , x 2 , x 3 ∈ J and let µ ∈ Conf(J) be as in the statement of Proposition 4.7. Since µ ∈ P GL(Z 2 (J)), the curve µ −1 (C J ) is a rational normal curve of degree 3 passing through the points ν 3 (x i ) for i = 1, 2, 3. Since µ is a projective automorphism of X 3 J (by Lemma 4.6 above), this twisted cubic curve is also contained in X 3 J . Thus we have proved the following result.Corollary 4.8. The twisted cubic X 3 J associated to a Jordan algebra J of rank three is 3-covered by rational normal curves of degree 3.Let k be the dimension of a Jordan algebra J of rank 3. Then X 3 J is a non-degenerate algebraic subvariety of the projective space PZ 2 (J), whose dimension is 2k + 1. Since π(k, 3, 3) = 2k + 2, one obtains that X 3 J ⊂ PZ 2 (J) = P 2k+1 is an example of X(k, 3, 3). Thus the cubics X 3 J associated to Jordan algebras of rank 3 are examples of varieties of

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Section: Rationality Of X(r + 1 N δ) and Of The General Curve Of The ...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Varieties $n$-covered by curves of degree $\delta$

Pirio¹,

Russo²

2013

Comment. Math. Helv.

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“…On the other hand, the algebraic geometers are interested in complex projective manifolds which admit an embedding of the Riemann sphere with positive normal bundle, as these are the 'rationally connected manifolds' -see the paper [6] of Paltin Ionescu (and the references therein) to whom I am grateful for kindly drawing my attention to this important fact.…”

Section: Introductionmentioning

confidence: 99%

On the embeddings of the Riemann sphere with nonnegative normal bundles

Pantilie¹

2018

Electronic Research Announcements

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“…Finally, as any Fano manifold is compact simply-connected and admits rational curves as in Corollary 3.2 (see [10] and the references therein) from Corrolary 3.1 we obtain the following fact [3] : the projective space is the only Fano manifold which admits a projective structure (compare [7, (5.3)] , [6] , [10] ).…”

Section: Applicationsmentioning

confidence: 99%

Projective structures and $ρ$-connections

Pantilie

2016

Preprint

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We extend T. Y. Thomas's approach to the projective structures, over the complex analytic category, by involving the ρ-connections. This way, a better control of the projective flatness is obtained and, consequently, we have, for example, the following application: if the twistor space of a quaternionic manifold P is endowed with a complex projective structure then P can be locally identified, through quaternionic diffeomorphisms, with the quaternionic projective space.

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Birational geometry of rationally connected manifolds via quasi-lines

Cited by 13 publications

References 15 publications

Varieties $n$-covered by curves of degree $\delta$

Varieties $n$-covered by curves of degree $\delta$

On the embeddings of the Riemann sphere with nonnegative normal bundles

Projective structures and $ρ$-connections

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