Deligne's proof in Weil I (Lefschetz pencil)

Abstract: The main player in this talk is the notion of a Lefschetz pencil, a special kind of 1-parameter family of varieties with nice degeneration properties. Because we have discussed how the Riemann Hypothesis for varieties over finite fields reduces to studying the middle cohomology of an even dimensional variety, we will produce a Lefschetz fibration with odd dimensional fibers whose middle cohomology contains an "interesting piece," which we will show has big symplectic monodromy.

Reference: Milne, Lectures on Étale Cohomology, Section 31-32.

algebraic geometrynumber theory

Audience: advanced learners

STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Spring 2026 topic: The descent obstruction.

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.