Bertini's theorem over finite field and Frobenius nonclassical varieties

Abstract: Let X be a smooth subvariety of $\mathbb{P}^n$ defined over a field k. Suppose k is an infinite field, then the classical theorem of Bertini asserts that X admits a smooth hyperplane section. However, if k is a finite field, there are examples of X such that every hyperplane H in $\mathbb{P}^n$ defined over k is tangent to X. One of the remedies in this situation is to extending the ground field k to its finite extension, and considering all the hyperplanes defined over the extension field. Then one can ask: Knowing the invariants of X (e.g. the degree of X), how much one needs to extend k in order to guarantee at least one transverse hyperplane section? In this talk we will report several results regarding to this type of questions. We also want to talk about a special type of varieties (Frobenius nonclassical varieties) that appear naturally in our research. This is a joint work with Shamil Asgarli and Kuan-Wen Lai.

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.

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