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: Description: Research/departmental seminar

Seminar usually meets in-person. For online editions, the Zoom link is shared via mailing list.