# Blow ups and Deformations ## Content

A variety is smooth if the tangent space is well-defined 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

## 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