Counting Rational Points on Del Pezzo Surfaces with a Conic Bundle Structure
نویسندگان
چکیده
For any number field k, upper bounds are established for the number of k-rational points of bounded height on non-singular del Pezzo surfaces defined over k, which are equipped with suitable conic bundle structures over k.
منابع مشابه
Manin’s Conjecture for Quartic Del Pezzo Surfaces with a Conic Fibration
— An asymptotic formula is established for the number of Q-rational points of bounded height on a non-singular quartic del Pezzo surface with a conic bundle structure.
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— We discuss Manin’s conjecture concerning the distribution of rational points of bounded height on Del Pezzo surfaces, and its refinement by Peyre, and explain applications of universal torsors to counting problems. To illustrate the method, we provide a proof of Manin’s conjecture for the unique split singular quartic Del Pezzo surface with a singularity of type D4.
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— Manin’s conjecture for the asymptotic behavior of the number of rational points of bounded height on del Pezzo surfaces can be approached through universal torsors. We prove several auxiliary results for the estimation of the number of integral points in certain regions on universal torsors. As an application, we prove Manin’s conjecture for a singular quartic del Pezzo surface.
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— Manin’s conjecture for the asymptotic behavior of the number of rational points of bounded height on del Pezzo surfaces can be approached through universal torsors. We prove several auxiliary results for the estimation of the number of integral points in certain regions on universal torsors. As an application, we prove Manin’s conjecture for a singular quartic del Pezzo surface.
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As another application of the degeneration methods of [V3], we count the number of irreducible degree d geometric genus g plane curves, with fixed multiple points on a conic E, not containing E, through an appropriate number of general points in the plane. As a special case, we count the number of irreducible genus g curves in any divisor class D on the blow-up of the plane at up to five points...
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تاریخ انتشار 2014