Student Algebraic Geometry

definition of crystalline cohomology, functoriality [BO, 5.4-19], and connections [BO, 2.1-2.10].

The need for divided powers + grothendieck's philosophy [BO, 1.11], PD-structures, PD-thickening, the cystalline site [BO, 3.1-5, 3.12-17, 5.1-3]

I will review the Picard-Fuchs equation for the Legendre family fixing erros from last time and review the theory of the Gauss-Manin connection.

This lecture will be independent of the following weeks.

**Note room change for this date only due to qualifying exams being held in 384H**

Abstract

We will give an introduction to the classification of surfaces and 2d minimal model programme with a focus on low characteristic.

Introduction to the topic and proof that there is no elliptic curve over the integers using Weierstrass equations.

The refined patchworking theorem (Section 2.5.10–11 of [IMS09])

A refinement of the tropical limit (Section 2.5.8–9 of [IMS09])

Construction of the tropical limit (Section 2.5.4–5 of [IMS09])

Statement of the problem (Section 2.5.1–3 of [IMS09]), small examples