Rational Curves on Smooth Cubic Hypersurfaces

Coskun, Izzet; Starr, Jason Michael

doi:10.1093/imrn/rnp102

Cited by 31 publications

(55 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the setting d = 3 of cubic hypersurfaces it is possible to obtain results for all smooth hypersurfaces in the family. Thus Coskun and Starr [7] have shown that M 0,0 (X, e) is irreducible and of dimension µ for any smooth cubic hypersurface X ⊂ P n over C, provided that n > 4. (If n = 4 then M 0,0 (X, e) has two irreducible components of the expected dimension µ = 2e.…”

Section: Introductionmentioning

confidence: 99%

Rational curves on smooth hypersurfaces of low degree

Browning

Vishe

2017

Alg. Number Th.

View full text Add to dashboard Cite

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Rational curves on smooth hypersurfaces of low degree Tim Browning and Pankaj VisheWe establish the dimension and irreducibility of the moduli space of rational curves (of fixed degree) on arbitrary smooth hypersurfaces of sufficiently low degree. A spreading out argument reduces the problem to hypersurfaces defined over finite fields of large cardinality, which can then be tackled using a function field version of the Hardy-Littlewood circle method, in which particular care is taken to ensure uniformity in the size of the underlying finite field.

show abstract

Section: Introductionmentioning

confidence: 99%

Rational curves on smooth hypersurfaces of low degree

Browning

Vishe

2017

Alg. Number Th.

View full text Add to dashboard Cite

show abstract

“…To derive Theorem 5.1 from Proposition 5.5, we require facts about hypersurfaces that are tedious but easy to check. In characteristic 0, many of these statements are immediate, as smooth hypersurfaces all have finitely many Eckardt points (see the discussion under Corollary 2.2 in [7]) and the Fermat hypersurface {X d 0 + · · · + X d r = 0} ⊂ P r contains Eckardt points and planes, and is smooth when the characteristic does not divide d.…”

Section: 2mentioning

confidence: 98%

Collections of Hypersurfaces Containing a Curve

Tseng

2018

International Mathematics Research Notices

View full text Add to dashboard Cite

We consider the closed locus parameterizing k-tuples of hypersurfaces that have positive dimensional intersection and fail to intersect properly, and show in a large range of degrees that its unique irreducible component of maximal dimension consists of tuples of hypersurfaces whose intersection contains a line. We then apply our methods in conjunction with a known reduction to positive characteristic argument to find the unique component of maximal dimension of the locus of hypersurfaces with positive dimensional singular loci. We will also find the components of maximal dimension of the locus of smooth hypersurfaces with a higher dimensional family of lines through a point than expected. Problem 1.2. Does Z have a unique component of maximal dimension, consisting of tuples (F 1 , . . . , F k ) of hypersurfaces all containing the same r − k + 1 dimensional linear space?The answer to Problem 1.2 is negative as it stands. For example, if r = 3 and the degrees are d 1 = 2, d 2 = 2, and d 3 = 100, then the locus of 3-tuples of hypersurfaces all containing the same line is codimension 103, while the second quadric will be equal to the first quadric in codimension 9. Even if the degrees are all equal, we can let r = 4, k = 2, and the degrees be d 1 = 2, d 2 = 2, where the two quadrics will contain a plane in codimension 16, but are equal in codimension 14.

show abstract

“…Building on pioneering work of Harris, Roth and Starr [12], Riedl and Yang [22] have proved that M 0,0 (Z, d) is an irreducible, local complete intersection scheme of the expected dimension, provided that Z is general and n k + 3. This can be extended to all smooth hypersurfaces when k = 3, thanks to work of Coskun and Starr [7]. Finally, for n > 2 k−1 (5k − 4), Browning and Vishe [5] have adapted the Hardy-Littlewood circle method to handle the space of degree d rational curves on arbitrary smooth hypersurfaces of degree k in P n−1 .…”

Section: Introductionmentioning

confidence: 99%