Density of rational points on del Pezzo surfaces of degree 1

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• Rosa Winter (King's College London)
• Tuesday 22 November 2022, 14:30-15:30
• Centre for Mathematical Sciences, MR13.

If you have a question about this talk, please contact Rong Zhou.

Let X be an algebraic variety over an infinite field k. In arithmetic geometry we are interested in the set X(k) of k-rational points on X. For example, is X(k) empty or not? And if it is not empty, is X(k) dense in X with respect to the Zariski topology? Del Pezzo surfaces are surfaces classified by their degree d, which is an integer between 1 and 9 (for d ≥ 3, these are the smooth surfaces of degree d in P^d ). For del Pezzo surfaces of degree at least 2 over a field k, we know that the set of k-rational points is Zariski dense provided that the surface has one k-rational point to start with (that lies outside a specific subset of the surface for degree 2). However, for del Pezzo surfaces of degree 1 over a field k, even though we know that they always contain at least one k-rational point, we do not know if the set of k-rational points is Zariski dense in general. I will talk about density of rational points on del Pezzo surfaces, state what is known so far, and show a result that is joint work with Julie Desjardins, in which we give sufficient and necessary conditions for the set of k-rational points on a specific family of del Pezzo surfaces of degree 1 to be Zariski dense, where k is finitely generated over Q.

This talk is part of the Number Theory Seminar series.

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