Counting rational points on the Cayley ruled cubic

Bretèche, Régis de la; Browning, Tim; Salberger, Per

doi:10.1007/s40879-015-0049-1

Cited by 5 publications

(13 citation statements)

References 17 publications

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“…All these varieties are spherical varieties; more precisely, flag varieties and toric varieties are special cases of horospherical varieties (which are toric bundles over flag varieties, at least after blow-ups); wonderful compactifications of semi-simple groups are special cases of wonderful varieties. This approach has also been applied to some nonspherical varieties, namely equivariant compactifications of vector groups [CLT02] and Cayley's singular ruled cubic surface [BBS16].…”

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confidence: 99%

Manin's conjecture for certain spherical threefolds

Derenthal

Gagliardi

2018

Advances in Mathematics

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We prove Manin's conjecture on the asymptotic behavior of the number of rational points of bounded anticanonical height for a spherical threefold with canonical singularities and two infinite families of spherical threefolds with log terminal singularities. Moreover, we show that one of these families does not satisfy a conjecture of Batyrev and Tschinkel on the leading constant in the asymptotic formula. Our proofs are based on the universal torsor method, using Brion's description of Cox rings of spherical varieties.

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confidence: 99%

Manin's conjecture for certain spherical threefolds

Derenthal

Gagliardi

2018

Advances in Mathematics

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“…This conjecture was proven to be correct for the non-normal Cayley ruled surface t 0 t 1 t 2 = t 2 0 t 3 + t 3 1 by de la Bretèche, Browning and Salberger [5]. In this case the infinitely many lines all contribute to the main term precisely in the way as predicted by the conjecture.…”

Section: Introductionmentioning

confidence: 73%

The Batyrev-Tschinkel conjecture for a non-normal cubic surface and its symmetric square

Gubela,

Lyczak

2020

Preprint

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We complete the study of points of bounded height on irreducible non-normal cubic surfaces by doing the point count on the cubic surface W given by t 2 0 t 2 = t 2 1 t 3 over any number field. We show that the order of growth agrees with a conjecture by Batyrev and Manin and that the constant reflects the geometry of the variety as predicted by a conjecture of Batyrev and Tschinkel. We then provide the point count for its symmetric square Sym 2 W . Although we can explain the main term of the counting function, the Batyrev-Manin conjecture is only satisfied after removing a thin set. Finally we interpret the main term of the count on Sym 2 (P 2 × P 1 ) done by Le Rudulier using these conjecture.

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“…and {Q [a:b] } covers Q as long as [a : b] goes thorough P 1 (Q). From this, it is possible to interpret Theorem 1.1 in the framework of the generalized Manin's conjecture by Batyrev and Tschinkel [1], as was done in the work of de la Bretèche, Browning, and Salberger [3]. However, we will not pursue such an explanation here.…”

Section: Introductionmentioning

confidence: 97%

“…). Then Lemma 2.1(i) gives (DM)(X, X + H; Y, Y + J) = XY 4P (ψ) +3 2 P ′ (ψ) + O(XJL 2 + Y HL 2 ) HJ. Since D is a linear operator, this together with Proposition 4.1 implies that…”

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confidence: 99%