Rational curves on Fermat hypersurfaces

Shen, Mingmin

doi:10.1016/j.crma.2012.09.015

Cited by 7 publications

(4 citation statements)

References 6 publications

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“…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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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%

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%

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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“…See [Kollár 1996], as well as [Debarre 2001]. Following [Shen 2012], we consider the degree 5 Fermat hypersurface X : X 5 0 + X 5 1 + X 5 2 + X 5 3 + X 5 4 + X 5 5 = 0 in ‫ސ‬ 5 over an algebraically closed field k of characteristic 2. This is a nonsingular projective Fano variety.…”

Section: Introductionmentioning

confidence: 99%

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2013

Involve

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The singly periodic Scherk surfaces with higher dihedral symmetry have 2n-ends that come together based upon the value of '. These surfaces are embedded provided that 2 n < n 1 n ' < 2. Previously, this inequality has been proved by turning the problem into a Plateau problem and solving, and by using the Jenkins-Serrin solution and Krust's theorem. In this paper we provide a proof of the embeddedness of these surfaces by using some results about univalent planar harmonic mappings from geometric function theory. This approach is more direct and explicit, and it may provide an alternate way to prove embeddedness for some complicated minimal surfaces.

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Rational curves on Fermat hypersurfaces

Cited by 7 publications

References 6 publications

Very free rational curves in Fano varieties

Very free rational curves in Fano varieties

Normal bundles of rational curves on complete intersections

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