Models of curves and abelian varieties
Organisation:
Time: Tuesdays 10h00-12h00
Room: SR 210 Arnimallee 3
Course notes:
Comments and feedback about these notes is highly encouraged!
Weierstrass models of elliptic curves
Intersection theory on regular surfaces - Note that the last section is not complete.
Resolution of singularities of surfaces
Abelian varieties and line bundles
Summary:
Curves and abelian varieties are among the most important examples of algebraic varieties. In complex algebraic geometry, they are very well understood and are used extensively to study more complicated varieties, while in arithmetic geometry, they are an almost endless source of deep theorems and wide open conjectures.
The goal of this course is to explain how curves and abelian varieties naturally degenerate in one-dimensional families. This is an important topic on its own, for instance for the formulation and study of the Birch-Swinnerton-Dyer conjecture, and also a first step towards the general theory of moduli of curves and abelian varieties. More precisely, given a curve or an abelian variety over a local or global field, we will establish the existence of the minimal regular model of the curve and the existence of the Néron model of the abelian variety. We will also briefly discuss the relationship between the minimal regular model of a curve and the Néron model of its Jacobian.
Along the way, we will encounter many basic tools of algebraic geometry: resolution of singularities of surfaces, intersection theory on surfaces, dualizing complexes, the Picard functor…
Prerequisites:
Algebraic geometry, the language of schemes, the theory of algebraic curves over a field; roughly speaking, Hartshorne’s Algebraic Geometry chapters 1-4 or Liu’s Algebraic Geometry and Arithmetic Curves chapters 1-7. Familiarity with abelian varieties is useful but not assumed, we will review the fundamentals in the course.
Main references:
- Liu, Algebraic Geometry and Arithmetic Curves
- Bosch, Lüktebohmert, Raynaud, Néron Models