Birational geometry for number theorists

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.

CMI summer school in arithmetic geometry, Gottingen, 2006. Numerous corrections following referee's report

References in corpus (5)

• Invariance of Plurigenera
• Existence of minimal models for varieties of log general type
• A high fibered power of a family of varieties of general type dominates a variety of general type
• A refined Kodaira dimension and its canonical fibration
• Celestial integration, stringy invariants, and Chern-Schwartz-MacPherson classes

Cited by in corpus (7)

• Campana points and powerful values of norm forms
• Campana points on biequivariant compactifications of the Heisenberg group
• Failure of the Brauer-Manin principle for a simply connected fourfold over a global function field, via orbifold Mordell
• Weak approximation and the Hilbert property for Campana points
• Sums of four squareful numbers
• One construction of a K3 surface with a dense set of rational points
• Insufficiency of the étale Brauer-Manin obstruction: towards a simply connected example