Smooth Surfaces with Maximally Many Lines

Abstract: How many lines can lie on a smooth surface of degree d? This classical question in algebraic geometry has been studied since at least the mid 1800s, when Clebsch gave an upper bound of d(11d-24) for the number of lines on a smooth surface of degree d over the complex numbers. Since then, Segre and then Bauer and Rams have given sharper upper bounds, the latter of which also holds over fields of characteristic p > d. However, over a field of characteristic p < d, there are smooth projective surfaces of degree d which break these upper bounds. In this talk, I’ll give a new upper bound for the number of lines which can lie on a smooth surface of degree d which holds over any field. In addition, we’ll fully classify those surfaces which attain this upper bound and talk about some of their other surprising properties. This talk is based on joint work with Tim Ryan and Karen Smith.

algebraic geometrynumber theory

Audience: researchers in the discipline

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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.

We normally meet in-person in the indicated room. For online editions, we use Zoom and distribute the link through the mailing list. If you wish to be put on the mailing list, please subscribe to ntag-external using lists.sfu.ca

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