8.3 Classification of singularities
3 min read•july 30, 2024
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 and . 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
- (ordinary double point) is a singularity where the curve crosses itself transversely
- Forms a point with two distinct tangent lines (crunode)
- 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)
- 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
- where three branches intersect at a point
- where multiple branches intersect transversely
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
- 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
- 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
- of a singularity is the resolution obtained with the least number of blow-ups
- Equal to the multiplicity of the singularity
- 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
Key Terms to Review (14)
Cusp: A cusp is a point on a curve where the curve is not smooth; it usually occurs when the tangent to the curve is not well-defined or when two branches of the curve meet. Cusps can significantly affect the shape and behavior of curves, making them interesting for classification and analysis. Understanding cusps helps in studying regular and singular points, recognizing their importance in determining the characteristics of plane curves and their singularities.
Delta invariant: The delta invariant is a numerical quantity associated with a singularity of a variety, providing insight into its geometric and topological properties. It serves as a crucial tool in classifying singularities and understanding their behavior, particularly in the context of plane curves where singular points can significantly affect the structure and properties of the curve.
Exceptional Divisor: An exceptional divisor is a type of divisor that appears in the context of blow-ups and resolutions of singularities in algebraic geometry. It represents the preimage of a point that has been transformed during the blowing-up process, often serving to help resolve or control singularities within a space. Exceptional divisors play a crucial role in understanding the structure of varieties and how singularities can be managed or classified.
Higher order cusps: Higher order cusps refer to singular points on a curve where the behavior is more complicated than simple cusps. They are characterized by a more pronounced sharpness or a tangential direction that varies in the vicinity of the point, leading to multiple branches emerging or converging at these points. Understanding higher order cusps is crucial for classifying singularities, as they reveal deeper geometric and topological properties of the curves.
Intersection multiplicity: Intersection multiplicity is a concept that quantifies the 'number of times' two varieties intersect at a given point, taking into account the geometric and algebraic properties of the varieties involved. This term helps in understanding how two projective varieties meet, providing insights into their local behavior at singularities and contributing to the classification of singularities, especially in the context of plane curves and their interactions.
Milnor Number: The Milnor number is an important invariant in algebraic geometry that measures the complexity of singularities of a smooth function or a variety. It is defined as the dimension of the local ring of the singularity divided by the dimension of the ambient space, providing insight into how singular the point is. This number helps classify different types of singularities and plays a critical role in understanding their geometric and topological properties.
Minimal Resolution: Minimal resolution refers to the process of resolving singularities in algebraic varieties by finding a minimal model that retains essential geometric features while eliminating or simplifying the singular points. This process is crucial in the classification of singularities, as it allows for a clearer understanding of the geometric properties and behavior of the variety near these problematic points.
Multiplicity: Multiplicity refers to the number of times a particular root or point appears in the context of algebraic geometry, particularly when discussing curves and their singularities. It provides valuable information about the behavior of a curve at a given point, such as how many times it intersects itself or how it behaves near a singularity. Understanding multiplicity is crucial for analyzing projective closures, classifying singularities, and studying plane curves.
Node: A node is a type of singularity in algebraic geometry that typically occurs in a plane curve. It is characterized by a point where two branches of the curve intersect and have a distinct tangential direction, making the point look like a 'bump' or 'kink'. Nodes can be thought of as specific points where the curve fails to be smooth, highlighting the differences between regular points and singular points.
Ordinary multiple points: Ordinary multiple points are points on a curve where the curve intersects itself in a specific way, allowing for multiple tangents at that point. These points have a certain multiplicity, indicating how many times the curve touches or crosses itself. Recognizing these points is crucial for understanding the overall behavior of the curve and analyzing its singularities.
Resolution Graph: A resolution graph is a combinatorial tool used in algebraic geometry to represent the resolutions of singularities, illustrating how singular points can be resolved into non-singular structures. It captures the relationship between different resolutions, which often involves sequences of blow-ups, helping to classify the singularities by showing how they can be transformed into simpler forms.
Resolution of Singularity: Resolution of singularity refers to a mathematical process used in algebraic geometry to transform a singular space into a non-singular (smooth) one. This technique is crucial for understanding the structure of varieties and allows mathematicians to classify and study singularities, leading to a deeper insight into their properties and behaviors.
Tacnode: A tacnode is a specific type of singularity that occurs on a plane curve where two branches of the curve meet tangentially at a single point, resulting in a higher-order contact than a regular intersection. This phenomenon is characterized by having both branches of the curve share a common tangent line at that point. Tacnodes can provide insights into the local behavior of curves, particularly how they intersect or touch each other, and are important in the classification of singularities as they signify more complex interactions than simple crossings.
Triple points: Triple points refer to a specific type of singularity in algebraic geometry where three curves intersect at a single point, resulting in a point of higher complexity than ordinary intersections. This concept highlights the behavior of plane curves and their singularities, providing insight into the nature and classification of these critical points. Understanding triple points is essential for studying how curves interact and the implications these interactions have on their geometric properties.