# 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**