Birational geometry for number theorists

TL;DR

This paper presents lecture notes from a summer school focusing on the intersection of birational geometry and number theory, emphasizing conjectures, Campana's program, and minimal model techniques.

Contribution

It introduces Campana's constellation framework and discusses their implications for arithmetic geometry, offering a new perspective on Kodaira dimension and related conjectures.

Findings

Campana's constellation framework for arithmetic geometry

Connections between Kodaira dimension and arithmetic conjectures

Speculative ideas on firmaments and their arithmetic significance

Abstract

Awfully idiosyncratic lecture notes from CMI summer school in arithmetic geometry July 31-August 4, 2006. Does not include: rationality problems, techniques of the minimal model problem and much of the rest. Includes: Lecture 0: geometry and arithmetic of curves Lecture 1: Kodaira dimension and properties, rational connectendess, Lang's and Campana's conjectures. Lecture 2: Campana's program; Campana constellations framed in terms of b-divisors, to allow for a definition of Kodaira dimension directly on the base. A speculative notion of firmaments which may deserve further investigation, especially the arithmetic side. Lecture 3: the minimal model program: very short discussion of bend-and-break; even shorter discussion of finite generation and the existence of flip. Lecture 4: Vojta's conjectures, Campana's conjectures, and ABC.