Concurrent exceptional curves on del Pezzo surfaces of degree one and torsion points on elliptic fibrations

Abstract: Let S be a del Pezzo surface of degree one. Then S contains 240 exceptional curves over an algebraically closed field. After blowing up a specific point one obtains an elliptic surface E, where the exceptional curves correspond to 240 sections. At most 16 exceptional curves gan go through the same point on S, and when this happens, the corresponding point on E is torsion on its fiber. In this talk I will consider the question how many exceptional curves can go through a point on S for which the corresponding point on E is non-torsion on its fiber. First of all I will explain how this question came up when studying the density of the set of rational points on del Pezzo surfaces of degree one. I will then show that if at least 9 exceptional curves intersect in a point on~S, the corresponding point on E is torsion on its fiber. This is less trivial than one might think by looking at the Mordell--Weil rank of E. Finally, in joint work with Julie Desjardins we show that 7 exceptional curves can go through a non-torsion point, and the question if 8 exceptional curves can go through a non-torsion point is still work in progress. I will show how one might try to tackle this.

algebraic geometry

Audience: researchers in the topic

ZAG (Zoom Algebraic Geometry) seminar

Series comments: Description: ZAG seminar

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