Rational points on a family of del Pezzo surfaces of degree one

This thesis is about rational points on so-called del Pezzo surfaces, which are a certain type of surfaces with relatively simple geometry. The lower the degree of a del Pezzo surface, the more intricate is its geometry. Let S be a del Pezzo surface over a field k. It is known that if the degree of S is not 1 and S(k) is non-empty (with a mild extra condition for degree 2), the surface S is k-unirational, meaning that there is a dominant rational map from some projective space to S. If k is an infinite field, this implies that the set S(k) of k-rational points lies Zariski dense in S. But in general, we do not know whether unirationality holds when the degree is 1, and the answer to this question seems way out of reach. So if we want to prove the density of the k-rational points of del Pezzo surfaces of degree 1, we have to search for alternative methods. In this thesis, a result is proven that gives sufficient and necessary conditions for the Zariski density of the rational points...Show moreThis thesis is about rational points on so-called del Pezzo surfaces, which are a certain type of surfaces with relatively simple geometry. The lower the degree of a del Pezzo surface, the more intricate is its geometry. Let S be a del Pezzo surface over a field k. It is known that if the degree of S is not 1 and S(k) is non-empty (with a mild extra condition for degree 2), the surface S is k-unirational, meaning that there is a dominant rational map from some projective space to S. If k is an infinite field, this implies that the set S(k) of k-rational points lies Zariski dense in S. But in general, we do not know whether unirationality holds when the degree is 1, and the answer to this question seems way out of reach. So if we want to prove the density of the k-rational points of del Pezzo surfaces of degree 1, we have to search for alternative methods. In this thesis, a result is proven that gives sufficient and necessary conditions for the Zariski density of the rational points on a certain family of del Pezzo surfaces of degree 1.Show less

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Rational points on a family of del Pezzo surfaces of degree one