Blow ups and Deformations
Content
A variety is smooth if the tangent space is welldefined and has everywhere the same dimension.
If this is not the case, the points where the tangent space is too large are called
singularities. In general singular spaces are much harder to study than smooth spaces and therefore it is important
to turn a singular space into a smooth one. In this course we will study two different ways to smoothen singularities:
deforming and blowing up. We will study these two ways in detail and apply them to an important
class of singularities: the simple singularities. For these singularities there is a nice correspondence between these
two types of smoothening, which we will study using representations of quivers.
Topics

Smooth versus Singular

Hypersurface singularities and deformations

Quotient singularities

Resolutions of Singularities

Quivers and McKay correspondence

Braiding it all together
Study Material

Syllabus
A syllabus is available
Learning Objectives
 Calculate the Jacobi algebra of a singularity.
 Deform a simple singularity
 Use representation theory to resolve a singularity
 Determine all singular points of a hypersurface.
 Blow up a simple singularity
 Describe the classification of simple singularities