8.3 Classification of singularities

Singularities in algebraic geometry can be tricky, but don't worry! We'll break down the main types like nodes, cusps, and tacnodes. These quirky points on curves where things get weird are key to understanding the shape and behavior of algebraic curves.

We'll also look at ways to measure how complex singularities are, like the delta invariant and Milnor number. Plus, we'll explore techniques like blow-ups to analyze and resolve singularities, turning messy curves into smoother ones. It's like untangling mathematical knots!

Singularities Classification

Common Types of Singularities

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• Singularity is a point on an algebraic curve where the curve is not smooth
  • Has a sharp point, self-intersection, or other irregular behavior
• Node (ordinary double point) is a singularity where the curve crosses itself transversely
  • Forms a point with two distinct tangent lines (crunode)
• Cusp is a singularity where the curve has a sharp point
  • Two branches of the curve have the same tangent line at the singular point (spinode)
• Tacnode is a singularity where the curve touches itself
  • Forms a point with two coincident tangent lines and a higher order of contact than a node
• Other types of singularities
  • Triple points where three branches intersect at a point
  • Ordinary multiple points where multiple branches intersect transversely
  • Higher order cusps

Measuring Singularity Complexity

• Delta invariant measures the complexity of a singularity
  • Related to the genus of the curve after resolving the singularity
  • Higher delta invariant indicates a more complex singularity
• Milnor number is another invariant that measures the complexity of a singularity
  • Defined as the degree of the gradient map of the curve at the singular point
  • Provides information about the topological type of the singularity

Singularity Structure Analysis

Blow-up Technique

• Blow-up is a technique used to study the local structure of a singularity
  • Transforms the curve into a new one with simpler singularities or no singularities
• Blow-up of a point on a curve replaces the point with the set of all tangent lines to the curve at that point
  • Forms a projective line called the exceptional divisor
• Multiplicity of a singularity is the minimum number of blow-ups required to resolve the singularity
  • Resolves into a smooth point or a collection of points

Analyzing Different Singularities with Blow-ups

• Blow-up of a node results in two smooth points
• Blow-up of a cusp results in a smooth point and a projective line tangent to the curve
• Blow-up of a tacnode may result in a node, a cusp, or a smooth point
  • Depends on the nature of the tangency between the two branches
• Intersection multiplicity of two curves at a point can be determined by analyzing the exceptional divisors
  • Obtained from successive blow-ups at the point

Singularity Resolution

Resolution Process

• Resolution of a singularity transforms the singular curve into a smooth curve or a collection of smooth curves
  • Achieved through a sequence of blow-ups
• Minimal resolution of a singularity is the resolution obtained with the least number of blow-ups
  • Equal to the multiplicity of the singularity
• Resolution graph is a tree-like diagram that represents the exceptional divisors and their intersections
  • Obtained from the blow-up process

Computing Invariants from the Resolution

• Self-intersection number of an exceptional divisor in the resolution graph
  • Negative of the number of other exceptional divisors it intersects
• Genus of the resolved curve can be computed using the genus formula
  • Involves the delta invariants of the singularities and the self-intersection numbers of the exceptional divisors
• Canonical divisor of the resolved curve can be expressed in terms of the exceptional divisors
  • Also involves the pullback of the canonical divisor of the original curve