Manin’s conjecture for a quartic del Pezzo surface with A3 singularity and four lines

Boudec, Pierre Le

doi:10.1007/s00605-011-0315-3

Cited by 1 publication

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“…They should also be enough for some del Pezzo surfaces of higher degree (e.g., the ones treated over Q in [Der07b] of type A 2 in degree 5, [Lou10] of type A 2 and [Bro09] of type A 1 with three lines in degree 6). We expect that an application to the cubic cases mentioned above or to other quartic del Pezzo surfaces (such as the ones treated over Q in [LB12a] of type A 3 with four lines, [LB12b] of types 3A 1 and A 2 + A 1 , [BBP12], [Lou12a] of types 2A 1 with eight lines, and the smooth quartic del Pezzo surfaces of [BB11b]) would require additional work.…”

Section: Introductionmentioning

confidence: 99%

Counting imaginary quadratic points via universal torsors

Derenthal

Frei²

2014

Compositio Math.

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A conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin's conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin's conjecture over imaginary quadratic fields K for the quartic del Pezzo surface S of singularity type A 3 with five lines given in P 4 K by the equationsQ, and [FMT89, Appendix] and [PT01] for computational evidence in degree 3 over Q).The most important technique is the use of universal torsors, which were invented by Colliot-Thélène and Sansuc (see [CTS87], for example) and first applied to Manin's conjecture by Salberger (see [Pey98, Sal98]). The testing ground was a new proof in the case of split toric varieties over Q (see [Sal98]). 1 After submission of the present article, Loughran [Lou13] showed how to derive this over arbitrary number fields from the work of Skinner [Ski97].

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