Deformation Theory

Deformation theory is the local study of deformations of schemes, i.e., it is the infinitesimal study of algebraic families of schemes around a specific element. More precisely, for a flat morphism of schemes, we want to understand the fibers in the neighborhood of an specific fiber. 

There are 4 main interesting situations:

• Subschemes of a fixed scheme X.
• Line bundles on a fixed scheme X.
• Coherent sheaves on a fixed scheme X.
• Deformations of abstract schemes.

A fundamental question in algebraic geometry is the existence of a moduli space parameterizing a given family of algebraic object, e.g., curves of genus g. This last quesiton is intimately related to the existence of auniversal family. Deformation theory techniques allows us to study local properties of families in the neighborhood of a point, even if a global moduli space does not exist.

The main goal of this group is to cover the most of the material presented in the book “Deformation Theory” by Robin Hartshorne, through 10 weekly meetings of two hours long. Eventually some completementary references can be used.

Organizer: Sebastián Fuentes Olguín