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AbundantPtsDelPezzo

Financiado

Abundance of rational points on low-degree del Pezzo surfaces

Describing the set of rational solutions to polynomial equations is the oldest field in mathematics and one of the fundamental goals in number theory. In arithmetic geometry such solutions are studied using geometric tools: ration... ver más

31/07/2027

ALU-FR

174K€

Presupuesto del proyecto: 174K€

Líder del proyecto

ALBERTLUDWIGSUNIVERSITAET FREIBURG No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Financiación concedida El organismo HORIZON EUROPE notifico la concesión del proyecto el día 2024-03-14

HORIZON-MSCA-2023-PF-01-01: MSCA Postdoctoral Fellowships 2023 ExpectedOutcome:Project results are expected to contribute to the following outcomes:

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ALU-FR

173.85K€ | Lider

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Duración del proyecto: 40 meses Fecha Inicio: 2024-03-14
Fecha Fin: 2027-07-31

Líder del proyecto

ALBERTLUDWIGSUNIVERSITAET FREIBURG No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 174K€

Descripción del proyecto Describing the set of rational solutions to polynomial equations is the oldest field in mathematics and one of the fundamental goals in number theory. In arithmetic geometry such solutions are studied using geometric tools: rational solutions to equations correspond to rational points on the corresponding geometric object. Fano varieties are among the simplest geometric objects, but still far from fully understood. This makes them a great class to test conjectures and develop new techniques. It is generally believed that Fano varieties, if they have a rational point, should have many, and they should be well-distributed. For curves (dimension 1), this is the case. Fano varieties in dimension 2 are del Pezzo surfaces, and already here there are many open questions. These surfaces have been a very active area of research in the last 50 years. Del Pezzo surfaces are classified by their degree, an integer between 1 and 9. The lower the degree, the more complex these surfaces become, and especially del Pezzo surfaces of degree 1 are notoriously difficult. Current results on the rational points on these surfaces make use of ad-hoc constructions, and a general geometric approach is missing. This forms a sharp contrast with del Pezzo surfaces of higher degree, and leaves a big gap in the understanding of rational points on Fano varieties. This project proposes to create a systematic approach to construct rational points on del Pezzo surfaces of degree 1, and use this to prove several different results on ‘abundant’ rational points for new families of surfaces. This will lead to answering big open questions (unirationality, Hilbert property and weak weak approximation for del Pezzo surfaces of degree 1), and providing evidence towards a long-standing conjecture on rational points on rationally connected varieties. Recent developments on the construction of rational points and low-genus curves on del Pezzo surfaces make this the perfect time to tackle the proposed objectives.

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