Unbounded negativity on rational surfaces in positive characteristic

Abstract: Fix your favourite smooth projective surface S and wonder: how negative can the self-intersection of a curve in S be? Apparently, there are situations in which curves might not actually get so negative: an old folklore conjecture, nowadays known as the Bounded Negativity Conjecture, predicts that if S were defined over the complex numbers, then the self-intersection of any curve in S is bounded below by a constant depending only on S. If, however, S were defined over a field of positive characteristic, then it is known that the Bounded Negativity Conjecture as stated cannot hold. For a long time, however, it was not known whether the Conjecture failed for rational surfaces in positive characteristic. In this talk, I describe the first examples of rational surfaces failing Bounded Negativity which I constructed with Remy van Dobben de Bruyn earlier this year.

algebraic geometrynumber theory

Audience: researchers in the topic

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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