6.1 Singular points and tangent cones

Singular points and tangent cones are crucial in understanding the geometry of algebraic varieties. They help us identify where a variety isn't smooth and give us a way to analyze these tricky spots.

By studying singular points, we can classify different types of singularities and measure their complexity. This knowledge is key for resolving singularities, a major focus of this chapter on singularities and resolution.

Singular Points of Algebraic Varieties

Definition and Properties of Singular Points

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• An algebraic variety is a set of points in affine or projective space that is the zero locus of a collection of polynomials
• A point on an algebraic variety is called a singular point if the variety is not smooth at that point and has some kind of singularity
• The dimension of the tangent space at a singular point is greater than the dimension of the variety
• The Jacobian matrix of the defining polynomials of the variety can be used to determine whether a point is singular, with the point being singular if and only if the Jacobian matrix has rank less than the codimension of the variety at that point

Definition and Properties of Regular Points

• A point on an algebraic variety is called a regular point if the variety is smooth at that point, meaning that the variety locally resembles a manifold around the point
• The dimension of the tangent space at a regular point is equal to the dimension of the variety
• If the Jacobian matrix of the defining polynomials has full rank at a point, then the point is a regular point
• Examples of regular points:
  • Every point on a non-singular curve (elliptic curve)
  • Every point on a smooth surface (sphere)

Constructing the Tangent Cone

Definition and Geometric Interpretation

• The tangent cone of an algebraic variety at a point is a geometric object that approximates the variety near the point and captures information about the singularity of the variety at that point
• The tangent cone can be thought of as the limit of secant lines through the point as the secant points approach the point along the variety
• If the point is a regular point, the tangent cone is a linear subspace of the same dimension as the variety
• If the point is singular, the tangent cone is a more complicated algebraic variety that captures the nature of the singularity

Construction Process

• To construct the tangent cone, consider the lowest degree homogeneous parts of the defining polynomials of the variety and evaluate them at the point
• The tangent cone is the zero locus of these lowest degree homogeneous polynomials in the affine or projective space of the same dimension as the ambient space of the variety
• Examples of tangent cones:
  • The tangent cone of the curve $y^2 = x^3$ at the origin is the zero locus of the polynomial $y^2$, which is a pair of intersecting lines
  • The tangent cone of the surface $z^2 = x^2 + y^2$ at the origin is the zero locus of $z^2$, which is a double plane

Multiplicity of Singular Points

Definition and Interpretation

• The multiplicity of a singular point is a measure of the complexity of the singularity and can be determined using the tangent cone
• The multiplicity is defined as the degree of the tangent cone, which is the degree of the lowest degree homogeneous part of the defining polynomials evaluated at the point
• If the multiplicity is 1, the point is a regular point, while if the multiplicity is greater than 1, the point is a singular point
• The multiplicity can also be interpreted as the number of tangent lines to the variety that pass through the point, counted with appropriate multiplicities

Relationship to Tangent Cone and Singularity Complexity

• Higher multiplicities indicate more complicated singularities, with different types of singularities possible for the same multiplicity
• The tangent cone of a singular point with multiplicity greater than 1 is not a linear subspace, but a more complicated algebraic variety
• Examples of singular points with different multiplicities:
  • A node of a curve has multiplicity 2 and its tangent cone consists of two distinct lines
  • A cusp of a curve has multiplicity 2 and its tangent cone is a single line with multiplicity 2
  • A ordinary double point of a surface has multiplicity 2 and its tangent cone is a pair of planes

Classifying Singular Points by Tangent Cone

Common Types of Singularities

• The type of a singular point can often be determined by the geometry of its tangent cone, which captures the local behavior of the variety near the point
• A node is a singular point whose tangent cone consists of a union of distinct linear subspaces, with the number of subspaces equal to the multiplicity of the point
• A cusp is a singular point whose tangent cone is irreducible but not a linear subspace, often with a characteristic shape like a sharp point or a self-intersecting curve
• A tacnode is a singular point formed by the intersection of two branches of the variety with distinct tangent directions, resulting in a tangent cone that is a union of two linear subspaces with multiplicity greater than one
• An ordinary multiple point is a singular point whose tangent cone consists of a union of linear subspaces with multiplicity equal to the multiplicity of the point, but which is not a node due to the subspaces not being distinct

Advanced Singularity Classification

• More complicated singularities can occur, such as higher order cusps, multiple points with non-linear tangent cones, and singularities with embedded components or non-reduced structure
• These singularities require more advanced techniques to classify, such as blowing up the variety at the singular point or considering the local ring of functions at the point
• Examples of more complicated singularities:
  • The Whitney umbrella, given by the equation $x^2 = y^2z$, has a pinch point singularity at the origin
  • The surface $x^3 + y^3 + z^3 = 0$ has a singularity at the origin whose tangent cone is the union of three coordinate planes, but the singularity is not an ordinary triple point