Free and very free morphisms into a Fermat hypersurface

Bridges, Tabes; Datta, Rankeya; Eddy, Joseph; Newman, M. F.; Yu, John‐Paul J.

doi:10.2140/involve.2013.6.437

Cited by 5 publications

(5 citation statements)

References 5 publications

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“…Given a rational curve C on X, the normal bundle N C|X controls the deformations of C in X and carries essential information about the local structure of the space of rational curves. Consequently, the normal bundles of rational curves have been studied extensively when X is P n ([AR17, Con06, CR18, EV81, EV82, GS80, Ran07, Sa82, Sa80]) and more generally (see for example [Br13,CR19,K96,LT19,Sh12b]).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Very free rational curves in Fano varieties

Coskun¹,

Smith²

2021

Preprint

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Let X be a projective variety and let C be a rational normal curve on X. We compute the normal bundle of C in a general complete intersection of hypersurfaces of sufficiently large degree in X. As a result, we establish the separable rational connectedness of a large class of varieties, including general Fano complete intersections of hypersurfaces of degree at least three in flag varieties, in arbitrary characteristic. In addition, we give a new way of computing the normal bundle of certain rational curves in products of varieties in terms of their restricted tangent bundles and normal bundles on each factor.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…The paper [CR19] gives sharp bounds on the degree of very free rational curves on general Fano complete intersections in P n . In a more negative direction, certain special Fano hypersurfaces are known not to have very free curves of low degree [Br13,Sh12a].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Very free rational curves in Fano varieties

Coskun¹,

Smith²

2021

Preprint

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“…The positivity properties of normal bundles of rational curves on Fano hypersurfaces in P n can be subtle in positive characteristic. For instance, although a general Fano hypersurface X of degree d < n contains a free line, for any degree e there exist examples of Fano hypersurfaces containing no free rational curves of degree less than e [Con06, Sh12a,Br13]. Nevertheless, it is conjectured that smooth Fano hypersurfaces always contain free rational curves [Br13], although this remains open.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, although a general Fano hypersurface X of degree d < n contains a free line, for any degree e there exist examples of Fano hypersurfaces containing no free rational curves of degree less than e [Con06, Sh12a,Br13]. Nevertheless, it is conjectured that smooth Fano hypersurfaces always contain free rational curves [Br13], although this remains open. Tian [Ti15] shows that if X contains a free rational curve, then X also contains a very free rational curve.…”

Section: Introductionmentioning

confidence: 99%

Normal bundles of rational curves on complete intersections

Coskun¹,

Riedl²

2017

Preprint

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Let X ⊂ P n be a general Fano complete intersection of type (d 1 , . . . , d k ). If at least one d i is greater than 2, we show that X contains rational curves of degree e ≤ n with balanced normal bundle. If all d i are 2 and n ≥ 2k + 1, we show that X contains rational curves of degree e ≤ n − 1 with balanced normal bundle. As an application, we prove a stronger version of the theorem of Z. Tian [Ti15], Q. Chen and Y. Zhu [CZ14] that X is separably rationally connected by exhibiting very free rational curves in X of optimal degrees.

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“…The above theorem shows that on this X very free rational curves can only exist in degrees 5, 9, 10 and all e ≥ 13. In the recent paper [1], the authors showed that there is no very free rational curve in degree 5 and they explicitly gave a very free rational curve of degree 9. Definition 1.9.…”

Section: Introductionmentioning

confidence: 99%

Rational curves on Fermat hypersurfaces

Shen

2012

Comptes Rendus. Mathématique

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In this note we study rational curves on degree p r + 1 Fermat hypersurface in P p r +1 k , where k is an algebraically closed field of characteristic p.The key point is that the presence of Frobenius morphism makes the behavior of rational curves to be very different from that of charateristic 0. We show that if there exists N 0 such that for all e ≥ N 0 there is a degree e very free rational curve on X, then N 0 > p r (p r − 1).

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Free and very free morphisms into a Fermat hypersurface

Cited by 5 publications

References 5 publications

Very free rational curves in Fano varieties

Very free rational curves in Fano varieties

Normal bundles of rational curves on complete intersections

Rational curves on Fermat hypersurfaces

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