Project description
Finding rational points on complex geometric surfaces
For centuries, mathematicians have studied rational solutions to polynomial equations, which correspond to specific points on geometric objects. Fano varieties, a class of these objects, are relatively simple but not fully understood, making them ideal for testing conjectures and developing new methods. Among these varieties, del Pezzo surfaces are particularly intriguing. While higher-degree del Pezzo surfaces are better understood, those of degree 1 remain complex and challenging. With the support of the Marie Skłodowska-Curie Actions programme, the AbundantPtsDelPezzo project will develop a systematic way to construct rational points on degree-1 del Pezzo surfaces, addressing open questions and advancing understanding of rational points on these objects. The proposed research could also provide key insights into broader mathematical conjectures about rationally connected varieties.
Objective
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.
Fields of science (EuroSciVoc)
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CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
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Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.2 - Marie Skłodowska-Curie Actions (MSCA)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
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Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
HORIZON-TMA-MSCA-PF-EF - HORIZON TMA MSCA Postdoctoral Fellowships - European Fellowships
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Call for proposal
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(opens in new window) HORIZON-MSCA-2023-PF-01
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79098 Freiburg
Germany
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