8.1 Regular and singular points

Regular and singular points are crucial concepts in algebraic geometry. They help us understand the smoothness 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 Jacobian matrix 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 singular locus

Singularities: Geometric Properties

Types of Singularities

• A node is a point where two or more branches of the curve intersect transversally, forming distinct tangent lines (e.g., the origin in the curve $y^2 = x^2(x+1)$)
• A cusp 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 $y^2 = x^3$)
• A tacnode 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 $y^2 = x^4$)
• An ordinary multiple point is a point where several branches of the curve intersect, each with its own distinct tangent line (e.g., the origin in the curve $y^2 = x^2(x-1)^2$)

Special Cases of Singularities

• An isolated point or acnode 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 $x^2 + y^2 = 0$)
• A self-intersection point is a point where a curve crosses itself, forming a node or a tacnode (e.g., the origin in the curve $y^2 = x^2(x-1)$)
• A degenerate singularity 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 $y^2 = x^5$)

Identifying Regularity: Algebraic Techniques

Plane Algebraic Curves

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

Space Algebraic Curves

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

Algebraic Surfaces

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

Advanced Techniques

• The multiplicity of a singular point 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