Imaginary quadratic points on toric varieties via universal torsors

Pieropan, Marta

doi:10.1007/s00229-015-0817-8

Cited by 6 publications

(5 citation statements)

References 30 publications

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“…Work on points of bounded height goes back to [71], followed immediately by [16] and more recently by [17]. Work of Schindler and the first author [56] investigates the distribution of Campana points on toric varieties. Recent work of Xiao [72] extends our results to biequivariant compactifications of the Heisenberg group.…”

Section: Campana Orbifolds Campana Points and The Conjecturementioning

confidence: 99%

“…Indeed, for the line bundle

L = O (1)

, we have

a (false(X, D_{ε} false), L) = 3 / 2

and

b = b (F, false(X, D_{ε} false), L) = 1

, so Conjecture 1.1 predicts a counting formula for Campana points of bounded height that grows like

c T^{3 / 2}

T \to \infty

, which is correct. The upper bound is, in fact, sharp; see [56, Theorem 1.2]. The lower bound () shows that counting Campana points and weak Campana points of bounded height in the same setting can lead to different asymptotics.…”

Section: Campana Orbifolds Campana Points and The Conjecturementioning

confidence: 99%

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Campana points of bounded height on vector group compactifications

Pieropan

Smeets

Tanimoto

et al. 2020

Proc. London Math. Soc.

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We initiate a systematic quantitative study of subsets of rational points that are integral with respect to a weighted boundary divisor on Fano orbifolds. We call the points in these sets Campana points. Earlier work of Campana and subsequently Abramovich shows that there are several reasonable competing definitions for Campana points. We use a version that delineates well different types of behavior of points as the weights on the boundary divisor vary. This prompts a Manin-type conjecture on Fano orbifolds for sets of Campana points that satisfy a klt (Kawamata log terminal) condition. By importing work of Chambert-Loir and Tschinkel to our setup, we prove a log version of Manin's conjecture for klt Campana points on equivariant compactifications of vector groups. Contents

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Section: Campana Orbifolds Campana Points and The Conjecturementioning

confidence: 99%

“…Indeed, for the line bundle

L = O (1)

, we have

a (false(X, D_{ε} false), L) = 3 / 2

and

b = b (F, false(X, D_{ε} false), L) = 1

, so Conjecture 1.1 predicts a counting formula for Campana points of bounded height that grows like

c T^{3 / 2}

T \to \infty

Section: Campana Orbifolds Campana Points and The Conjecturementioning

confidence: 99%

Campana points of bounded height on vector group compactifications

Pieropan

Smeets

Tanimoto

et al. 2020

Proc. London Math. Soc.

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“…The only cases where the role of Cox rings is made rigorous seem to be split toric varieties [Sal98] and split weak del Pezzo surfaces [FP16]. For these classes of varieties, the explicit description of Cox rings from [Cox95,HT04,Der14] is successfully applied to prove many cases of Manin's conjecture; see [Sal98,Pie16,FP16] and the references in [ADHL15,§6.4.1].…”

Section: Introductionmentioning

confidence: 99%

Cox rings over nonclosed fields

Derenthal

Pieropan

2018

J. London Math. Soc.

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“…Their arithmetic has been intensively studied. Rational points are very well distributed (the Batyrev-Manin-Peyre's principle [BM90], [Pey95]) thanks to the work of Batyrev & Tschinkel [BT98], [BT95], Salberger [Sal98] and later on Frei [Fre13] and Pieropan [Pie16].…”

Section: Introductionmentioning

confidence: 99%

“…The work of Salberger [Sal98] gives a first combinatorial way of parametrizing rational points on split toric varieties and he uses it to reprove the Batyrev-Manin-Peyre's principle over Q. His method, which we call universal torsor method for now on, is later generalized by Pieropan [Pie16] to imaginary quadratic fields. First introduced by Colliot-Thélène and Sansuc [CS87], universal torsors give us an effective way of parametrizing rational points via integral coordinates in some nice affine spaces as well as computing their heights.…”

Section: Introductionmentioning

confidence: 99%

Rational approximations on toric varieties

Huang

2018

Preprint

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Using the universal torsor method due to Salberger, we study the approximation of a general fixed point by rational points on split toric varieties. We prove that under certain geometric hypothesis the best approximations (in the sense of McKinnon-Roth's work) can be achieved on rational curves passing through the fixed point of minimal degree, confirming a conjecture of McKinnon. These curves correspond, according to Batyrev's terminology, to the centred primitive collections of the fan.

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Imaginary quadratic points on toric varieties via universal torsors

Abstract: Inspired by a paper of Salberger we give a new proof of Manin's conjecture for toric varieties over imaginary quadratic number fields by means of universal torsor parameterizations and elementary lattice point counting.Comment: 23 pages, minor revisio

Cited by 6 publications

References 30 publications

Campana points of bounded height on vector group compactifications

Campana points of bounded height on vector group compactifications

Cox rings over nonclosed fields

Rational approximations on toric varieties

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