Glossary

8.1 Regular and singular points

4 min read•july 30, 2024

Regular and singular points are crucial concepts in algebraic geometry. They help us understand the and behavior of curves and surfaces at specific points. This knowledge is essential for analyzing the overall structure and properties of geometric objects.

Identifying regular and singular points involves examining the tangent lines or planes at each point. We use tools like the and partial derivatives to determine regularity. Understanding these concepts is key to grasping the broader topic of singularities in algebraic geometry.

Regular vs Singular Points

Defining Regular and Singular Points

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  • A point on an algebraic curve or surface is regular if the curve or surface is smooth at that point, meaning it has a well-defined tangent line (for curves) or plane (for surfaces)
  • A point on an algebraic curve or surface is singular if the curve or surface is not smooth at that point, meaning it does not have a well-defined tangent line or plane
  • Regular points are also called simple or non-singular points, while singular points are also called multiple points or singularities

Determining Regularity using the Jacobian Matrix

  • The Jacobian matrix of partial derivatives can be used to determine the regularity of a point on an algebraic curve or surface
    • If the Jacobian matrix has full rank at a point, then the point is regular
    • If the Jacobian matrix does not have full rank at a point, then the point is singular
  • The set of all singular points on an algebraic curve or surface is called the

Singularities: Geometric Properties

Types of Singularities

  • A is a point where two or more branches of the curve intersect transversally, forming distinct tangent lines (e.g., the origin in the curve y2=x2(x+1)y^2 = x^2(x+1))
  • A is a point where two branches of the curve meet tangentially, forming a sharp point with a single tangent line (e.g., the origin in the curve y2=x3y^2 = x^3)
  • A is a point where two branches of the curve meet with the same tangent line and have at least second-order contact (e.g., the origin in the curve y2=x4y^2 = x^4)
  • An is a point where several branches of the curve intersect, each with its own distinct tangent line (e.g., the origin in the curve y2=x2(x−1)2y^2 = x^2(x-1)^2)

Special Cases of Singularities

  • An or is a point that lies on the curve but has no other points of the curve in its neighborhood (e.g., the origin in the curve x2+y2=0x^2 + y^2 = 0)
  • A is a point where a curve crosses itself, forming a node or a tacnode (e.g., the origin in the curve y2=x2(x−1)y^2 = x^2(x-1))
  • A is a point where the curve has a more complicated structure, such as a higher-order cusp or a point with an infinite number of tangent lines (e.g., the origin in the curve y2=x5y^2 = x^5)

Identifying Regularity: Algebraic Techniques

Plane Algebraic Curves

  • For a plane algebraic curve defined by a polynomial equation f(x,y)=0f(x, y) = 0, a point (a,b)(a, b) is singular if and only if the partial derivatives fx(a,b)f_x(a, b) and fy(a,b)f_y(a, b) are both zero
    • Example: For the curve y2=x3y^2 = x^3, the origin (0,0)(0, 0) is singular because fx(0,0)=0f_x(0, 0) = 0 and fy(0,0)=0f_y(0, 0) = 0

Space Algebraic Curves

  • For a space algebraic curve defined by the intersection of two polynomial equations f(x,y,z)=0f(x, y, z) = 0 and g(x,y,z)=0g(x, y, z) = 0, a point (a,b,c)(a, b, c) is singular if and only if the Jacobian matrix [fxfyfz;gxgygz][f_x f_y f_z; g_x g_y g_z] evaluated at (a,b,c)(a, b, c) has rank less than 2
    • Example: For the curve defined by x2+y2−z2=0x^2 + y^2 - z^2 = 0 and x2−y2=0x^2 - y^2 = 0, the origin (0,0,0)(0, 0, 0) is singular because the Jacobian matrix at (0,0,0)(0, 0, 0) has rank 1

Algebraic Surfaces

  • For an algebraic surface defined by a polynomial equation f(x,y,z)=0f(x, y, z) = 0, a point (a,b,c)(a, b, c) is singular if and only if the gradient vector ∇f(a,b,c)=(fx(a,b,c),fy(a,b,c),fz(a,b,c))\nabla f(a, b, c) = (f_x(a, b, c), f_y(a, b, c), f_z(a, b, c)) is the zero vector
    • Example: For the surface x2+y2−z2=0x^2 + y^2 - z^2 = 0, the origin (0,0,0)(0, 0, 0) is singular because ∇f(0,0,0)=(0,0,0)\nabla f(0, 0, 0) = (0, 0, 0)

Advanced Techniques

  • The multiplicity of a can be determined by the order of vanishing of the defining polynomials and their partial derivatives at that point
  • Blow-up techniques can be used to resolve singularities and study their local structure by introducing new coordinates and transforming the equation of the curve or surface

Key Terms to Review (25)

Acnode: An acnode is a type of point on an algebraic curve that is characterized by being a singular point with a certain geometric property: it does not lie on any tangent line to the curve. Acnodes represent interesting cases in the study of curves, as they can help identify unique behaviors and properties of the curve's shape and structure.
Affine Space: An affine space is a geometric structure that generalizes the properties of Euclidean spaces by allowing for the study of points and vectors without a fixed origin. It serves as a foundation for understanding how geometric figures can be manipulated and described in algebraic terms, linking concepts like vector spaces to polynomial rings and ideals.
Coordinate chart: A coordinate chart is a mathematical tool that assigns coordinates to points in a space, allowing for the description and analysis of geometric and algebraic structures. By providing a systematic way to map local regions of a manifold or algebraic variety to simpler Euclidean spaces, coordinate charts play a crucial role in understanding isomorphisms and embeddings, as well as identifying regular and singular points within these structures.
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.
Deformation theory: Deformation theory is a branch of mathematics that studies how geometric objects can change or deform while retaining certain structural properties. It plays a crucial role in understanding the relationships between different varieties and their singularities, highlighting how small changes can lead to significant differences in the underlying geometry.
Degenerate Singularity: A degenerate singularity is a point on a geometric object where the object fails to be smooth or regular, typically resulting from the loss of dimensionality or distinctiveness. Such singularities can complicate the study of the geometry and algebraic properties of varieties, as they often lead to the breakdown of local behavior that is otherwise expected in regular points.
Derivative test: The derivative test is a method used to determine the local extrema of a function by analyzing its first and second derivatives. By evaluating the sign of the first derivative, one can identify critical points where the function's slope changes, indicating potential maxima or minima. The second derivative further helps classify these points as local maxima, local minima, or saddle points, offering a comprehensive understanding of the function's behavior around those points.
Dimension: Dimension is a fundamental concept in geometry and algebra that refers to the number of independent directions or parameters needed to describe a space or object. In algebraic geometry, it helps classify varieties based on their geometric properties, influencing how they are represented and understood in terms of both affine and projective spaces.
Field Extension: A field extension is a way to create a new field from an existing field by adding elements that do not already belong to it, allowing for the expansion of the field's structure. This concept is crucial as it enables the study of polynomials and rational functions, linking algebraic structures to geometric properties and offering insights into how different varieties relate to each other.
Isolated Point: An isolated point is a point in a topological space that is not a limit point of any subset of the space, meaning there exists a neighborhood around it that contains no other points from the set. This concept is important when analyzing both regular and singular points, as isolated points can indicate a lack of nearby points affecting the local properties of curves or surfaces.
Jacobian Matrix: The Jacobian matrix is a matrix of first-order partial derivatives of a vector-valued function. It plays a crucial role in understanding how changes in input variables affect output variables, particularly in multivariable calculus and algebraic geometry. The Jacobian is essential for identifying regular and singular points, as well as for analyzing tangent spaces and applying the Jacobian criterion for determining the nature of critical points.
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 point: An ordinary multiple point is a type of singular point on a curve where the curve intersects itself with a certain multiplicity. At this point, the derivatives of the curve up to a certain order vanish, indicating that multiple branches of the curve meet at a single point. Understanding ordinary multiple points is crucial when analyzing the behavior of curves and their singularities, especially in determining how these curves can be deformed and what their local structure looks like.
Polynomial Ring: A polynomial ring is a mathematical structure formed by the set of all polynomials in one or more variables with coefficients from a given ring. This concept allows for the manipulation and study of polynomials, making it foundational for various areas in algebra, particularly when exploring ideals, algebraic sets, and geometric properties.
Regular Point: A regular point refers to a point on an algebraic variety where the local behavior of the variety resembles that of a smooth manifold. At a regular point, the dimension of the tangent space equals the dimension of the variety itself, ensuring that there are no singularities or discontinuities in the geometry at that location.
Riemann-Roch Theorem: The Riemann-Roch Theorem is a fundamental result in algebraic geometry that relates the dimension of a space of meromorphic functions on a curve to the degree of the divisor associated with those functions. It provides powerful tools for calculating dimensions of certain vector spaces and has deep implications in the study of curves, their function fields, and intersections.
Self-intersection point: A self-intersection point occurs when a curve intersects itself at a certain point, meaning that the same point on the curve can be reached by two different paths within the curve. This phenomenon indicates a type of singularity, revealing important information about the geometric and topological properties of the curve. Understanding self-intersection points is crucial when analyzing the regularity or singularity of a curve.
Singular Locus: The singular locus of an algebraic variety is the set of points where the variety fails to be smooth, meaning that the local ring of functions at these points does not have a regular sequence. In other words, these points are where the variety exhibits 'singularities' or irregular behavior, which can affect the structure and properties of the variety significantly. Understanding the singular locus is crucial for studying the geometry and topology of algebraic varieties.
Singular point: A singular point is a point on a geometric object where the object fails to be well-behaved in some way, such as having a cusp or a node. These points often indicate a breakdown in the smoothness or differentiability of the object, making them essential for understanding the overall structure and behavior of curves and surfaces. Recognizing and analyzing singular points is crucial for determining the properties and classifications of various geometric forms.
Smoothness: Smoothness refers to a property of a space where it behaves nicely in terms of differentiability, meaning that it has no abrupt changes, singularities, or 'sharp points.' In algebraic geometry, smoothness implies that the variety is well-behaved at every point, allowing for the application of calculus and differential geometry concepts. This property is essential for understanding how varieties can be manipulated and transformed without encountering issues that arise from singular points.
Subvariety: A subvariety is a subset of a variety that inherits the structure of the larger variety and is defined by the vanishing of certain polynomials. Subvarieties can be seen as the 'smaller' pieces within a larger geometric object, allowing for a deeper understanding of the overall structure and properties of varieties. They play a crucial role in algebraic geometry, linking algebraic concepts like ideals to geometric notions.
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.
Tangency: Tangency refers to the condition where a curve touches another curve or line at a single point without crossing it. This concept is essential in understanding the behavior of curves, particularly when analyzing regular and singular points, as well as intersections between plane curves. The properties of tangents at points of tangency provide insights into the local structure of the curves involved.
Variety: In algebraic geometry, a variety is a fundamental geometric object that can be defined as the solution set of one or more polynomial equations over a given field. This concept connects to the study of polynomial rings and ideals, where varieties correspond to the zeros of polynomials, highlighting their geometric significance in higher-dimensional spaces. Varieties can also be connected to singularities and the resolution of these points, offering insight into their structure and behavior in algebraic contexts.
Whitney's Theorem: Whitney's Theorem refers to a fundamental result in algebraic geometry that relates to the behavior of regular and singular points on algebraic varieties. It provides insights into the nature of these points, indicating that the singular points of a variety can be understood through the topology of the variety, specifically regarding its tangent spaces. The theorem helps differentiate between regular points, where the variety behaves nicely, and singular points, where it exhibits more complex or problematic characteristics.
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