🕴🏼Elementary Algebraic Geometry Unit 8 – Singularities

What Are Singularities?

• Points on an algebraic variety where the geometry or topology behaves differently than at regular points
• Occur when the tangent space at a point has higher dimension than expected, indicating a non-smooth point
• Arise from the intersection of multiple components of a variety or from self-intersections
• Can be classified based on their local properties and how they affect the geometry of the variety
• Play a crucial role in understanding the global structure and properties of algebraic varieties
• Provide insights into the complexity and degeneracy of algebraic equations and geometric objects
• Studying singularities helps in resolving and simplifying complex algebraic varieties

Types of Singularities

• Isolated singularities are singular points surrounded by regular points in a neighborhood (nodes, cusps)
• Non-isolated singularities form a subvariety of singular points within the algebraic variety (lines, curves)
• Normal singularities have a well-defined tangent cone and can be resolved by a finite sequence of blowups
  • Examples include double points, pinch points, and simple surface singularities (A-D-E singularities)
• Non-normal singularities lack a well-defined tangent cone and may require more complex resolution procedures
• Rational singularities are characterized by the vanishing of higher cohomology groups and have good properties
  • Rational double points and quotient singularities are examples of rational singularities
• Canonical singularities are a generalization of rational singularities and play a role in the minimal model program
• Degenerate singularities occur when the Jacobian matrix of the defining equations has rank less than expected

Detecting Singularities

• Compute the Jacobian matrix of the defining equations and check its rank at each point
  • Singular points have a Jacobian matrix with rank less than the codimension of the variety
• Use the Jacobian criterion, which states that a point is singular if and only if all partial derivatives vanish simultaneously
• Calculate the tangent cone at a point by taking the lowest degree terms of the defining equations
  • The point is singular if the tangent cone is not a linear subspace of the expected dimension
• Analyze the local ring at a point and check if it is regular (a regular local ring corresponds to a non-singular point)
• Apply the Nullstellensatz to relate the singularities to the properties of the ideal of defining equations
• Utilize computational tools and algorithms, such as Gröbner bases and resolution of singularities, to detect and analyze singularities

Local Properties of Singularities

• The multiplicity of a singularity measures how many components of the variety intersect at that point
  • Computed using the Hilbert-Samuel multiplicity or the length of the local ring
• The embedding dimension is the minimal number of generators needed for the maximal ideal of the local ring
  • Provides information about the complexity and structure of the singularity
• Milnor number quantifies the degeneracy of a singularity and is related to the topology of the variety near the point
• Tjurina number is another invariant that captures the properties of the singularity and its deformations
• The link of a singularity is the intersection of the variety with a small sphere centered at the singular point
  • Provides topological information about the singularity and its resolution
• Monodromy action describes how the local topology changes when circling around the singularity
• The resolution graph encodes the combinatorial data of the exceptional divisors in a resolution of the singularity

Resolution of Singularities

• The process of transforming a singular variety into a non-singular one while preserving essential properties
• Achieved through a sequence of birational morphisms, such as blowups and normalizations
• Hironaka's theorem guarantees the existence of a resolution for varieties over fields of characteristic zero
  • The resolution is not unique, but minimal resolutions are often considered
• Blowups replace singular points with exceptional divisors that have simpler singularities or are non-singular
  • Iterated blowups can resolve singularities step by step
• Normalizations resolve non-normal singularities by separating irreducible components and removing self-intersections
• Embedded resolutions consider the variety as a subvariety of a larger ambient space and resolve singularities in that context
• Algorithmic approaches, such as the Bierstone-Milman and Villamayor algorithms, provide constructive methods for resolution
• The resolution of singularities has applications in the minimal model program, the study of log canonical thresholds, and the classification of algebraic varieties

Singularities in Algebraic Curves

• Algebraic curves are one-dimensional algebraic varieties defined by polynomial equations in two variables
• Singular points on curves are classified as nodes (ordinary double points), cusps, or higher-order singularities
  • Nodes have two distinct tangent lines, while cusps have a single tangent line with higher contact order
• The genus of a curve is related to the number and types of singularities it possesses
  • The genus formula $g = \frac{(d-1)(d-2)}{2} - \sum_{P} \delta_P$ relates the degree $d$, genus $g$, and the delta invariants $\delta_P$ of the singularities
• Resolving singularities of curves is achieved by a finite sequence of blowups and normalizations
  • The resolution process can be visualized using the Enriques diagram or the dual graph
• The Milnor number of a curve singularity is equal to the delta invariant and measures the difference between the genus of the curve and its normalization
• Singular curves play a role in the study of algebraic surfaces, as they often arise as special fibers of fibrations or as branch loci of coverings
• The moduli space of curves with prescribed singularities is an important object in algebraic geometry and has connections to enumerative geometry and Gromov-Witten theory

Applications and Examples

• Singularity theory has applications in various fields, including algebraic geometry, topology, differential equations, and physics
• The study of singularities is crucial for understanding the geometry and topology of algebraic varieties and their moduli spaces
• Singularities arise naturally in the classification of algebraic surfaces and higher-dimensional varieties
  • The minimal model program aims to transform varieties into simpler models with mild singularities
• Singularities play a role in the study of algebraic curves, their moduli spaces, and their enumerative properties
  • The Deligne-Mumford compactification of the moduli space of curves involves stable curves with nodal singularities
• In differential equations, singularities of solutions often correspond to critical points or phase transitions
  • The study of singularities helps in understanding the qualitative behavior of dynamical systems
• Singularity theory has applications in string theory and quantum field theory, where singular spaces and their resolutions are used to describe physical phenomena
  • Orbifold singularities and their resolutions are important in the study of string compactifications and gauge theories
• Examples of singularities include the origin in the curve $y^2 = x^3$ (cusp), the vertex of the cone $z^2 = x^2 + y^2$ (isolated singularity), and the singular point of the surface $x^2 + y^2 = z^3$ (rational double point)

Advanced Topics and Open Problems

• Equisingularity theory studies families of singularities and their deformations, aiming to classify singularities up to topological or analytical equivalence
• The Nash problem asks whether every singularity can be resolved by a sequence of Nash blowups (blowups preserving the Nash structure)
  • The problem is open in general but has been solved in some special cases
• The log minimal model program extends the minimal model program to pairs $(X, D)$ consisting of a variety $X$ and a divisor $D$, allowing for certain controlled singularities
• The study of singularities in positive characteristic poses additional challenges due to the failure of Hironaka's resolution theorem
  • Resolution of singularities in positive characteristic is an active area of research with progress in low dimensions
• The connection between singularities and arc spaces provides a framework for studying singularities using formal arcs and jet schemes
  • The Nash blow-up and the jet schemes are used to analyze the structure of singularities
• Motivic integration is a powerful tool that assigns motivic measures to subsets of arc spaces and allows for the computation of invariants of singularities
• The Bernstein-Sato polynomial and the $b$-function are important invariants associated with singularities of hypersurfaces and have connections to $D$-module theory and representation theory
• The study of singularities in the context of non-commutative algebraic geometry and derived categories is an active area of research, providing new perspectives and tools for understanding singularities