9.1 Plane curves and their singularities

Plane curves are the building blocks of algebraic geometry, defined by polynomial equations in two variables. They come in various degrees and can be smooth or have singularities. Understanding their properties is key to grasping more complex geometric concepts.

Singularities on curves are special points where the curve behaves unusually. These can be nodes, cusps, or more complex types. Analyzing singularities helps us understand the curve's shape and properties, and is crucial for classifying and studying algebraic curves.

Plane algebraic curves

Definition and basic properties

Top images from around the web for Definition and basic properties

• 

  Graphs of Polynomial Functions | College Algebra Corequisite View original

  Is this image relevant?

• 

  Category:Plane algebraic curves - Wikimedia Commons View original

  Is this image relevant?

• 

  Asymptote - Wikipedia View original

  Is this image relevant?

• 

  Graphs of Polynomial Functions | College Algebra Corequisite View original

  Is this image relevant?

• 

  Category:Plane algebraic curves - Wikimedia Commons View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and basic properties

• 

  Graphs of Polynomial Functions | College Algebra Corequisite View original

  Is this image relevant?

• 

  Category:Plane algebraic curves - Wikimedia Commons View original

  Is this image relevant?

• 

  Asymptote - Wikipedia View original

  Is this image relevant?

• 

  Graphs of Polynomial Functions | College Algebra Corequisite View original

  Is this image relevant?

• 

  Category:Plane algebraic curves - Wikimedia Commons View original

  Is this image relevant?

1 of 3

• A plane algebraic curve is the set of points in the affine or projective plane whose coordinates satisfy a polynomial equation in two variables
• The degree of a plane curve is the degree of the defining polynomial equation
  • Curves of degree 1 are lines, degree 2 are conics (circles, ellipses, parabolas, hyperbolas), degree 3 are cubics, and so on
• A plane curve is irreducible if its defining polynomial cannot be factored into lower degree polynomials
  • An irreducible plane curve is also called a plane curve
• The intersection of two plane curves is a finite set of points
  • The number of intersection points (counting multiplicity) is equal to the product of the degrees of the curves by Bézout's theorem

Rationality and smoothness

• A plane curve is rational if it can be parameterized by rational functions
  • Every line and conic is rational, but not all curves of higher degree are rational
  • For example, the cubic curve $y^2 = x^3 - x$ (elliptic curve) is not rational
• A plane curve is smooth or non-singular if it has no singular points
  • Otherwise, it is called singular
  • For example, the curve $y^2 = x^2(x+1)$ has a singular point at $(0,0)$

Singularities on curves

Types of singularities

• A point on a plane curve is singular if the partial derivatives of the defining polynomial vanish at that point
• A node or ordinary double point is a singular point where the curve crosses itself transversally
  • The tangent lines at a node are distinct
  • For example, the curve $y^2 = x^2(x+1)$ has a node at $(0,0)$
• A cusp is a singular point where the curve crosses itself tangentially
  • The tangent lines at a cusp coincide
  • For example, the curve $y^2 = x^3$ has a cusp at $(0,0)$
• A tacnode is a singular point where the curve crosses itself with multiplicity 2, meaning it looks like two branches of the curve touching each other
  • For example, the curve $y^2 = x^4$ has a tacnode at $(0,0)$
• An ordinary multiple point of multiplicity $m$ is a singular point where $m$ smooth branches of the curve intersect transversally
  • For example, the curve $(y^2-x^2)(y-x^2) = 0$ has an ordinary triple point at $(0,0)$

Higher order singularities

• Other higher order singularities can occur, such as triple points, inflection points, and more complicated singularities
• These singularities can be analyzed using more advanced techniques such as blow-ups and resolution of singularities
• The classification of higher order singularities is a deep and active area of research in algebraic geometry

Local behavior of curves

Tangent cones and multiplicities

• The tangent cone of a curve at a point is the set of tangent lines to the curve at that point
  • It can be computed by taking the lowest degree terms of the Taylor expansion of the curve at the point
• For a node, the tangent cone consists of two distinct lines
  • For a cusp, the tangent cone is a double line
  • For a tacnode, the tangent cone is a double line
• The multiplicity of a singular point can be determined by the degree of the lowest degree terms in the Taylor expansion of the curve at the point

Puiseux series and branches

• Puiseux series can be used to analyze the local behavior of a curve near a singular point
  • They provide a parametrization of the curve in terms of fractional power series
  • For example, the curve $y^2 = x^3$ can be parametrized near $(0,0)$ by $x = t^2, y = t^3$
• The branches of a curve at a singular point correspond to the Puiseux series expansions at that point
  • The number of branches is equal to the number of distinct Puiseux series
• The intersection multiplicity of two branches at a singular point can be computed using the Puiseux series expansions of the branches
  • For example, the curve $y^2 = x^2(x+1)$ has two branches at $(0,0)$ with intersection multiplicity 2

Multiplicity of singular points

Definition and computation

• The multiplicity of a point on a plane curve is the order of vanishing of the defining polynomial at that point
  • It is the lowest degree of a monomial in the Taylor expansion of the polynomial at the point
• For a singular point, the multiplicity is at least 2
  • For a non-singular point, the multiplicity is 1
• The multiplicity of a point can be computed by successively taking derivatives of the defining polynomial and evaluating at the point until a nonzero value is obtained
  • For example, for the curve $y^2 = x^3$, the multiplicity of $(0,0)$ is 2 since $f(0,0) = f_x(0,0) = 0$ but $f_y(0,0) \neq 0$
• The multiplicity of a singular point is equal to the degree of the tangent cone at that point

Relations to other invariants

• The sum of the multiplicities of all singular points on a plane curve is bounded by the genus of the curve
  • The genus can be computed using the degree-genus formula $g = \frac{(d-1)(d-2)}{2} - \sum_P \frac{m_P(m_P-1)}{2}$, where $d$ is the degree and $m_P$ is the multiplicity of a singular point $P$
• The delta invariant of a singular point is the difference between its multiplicity and the number of branches passing through it
  • It measures the complexity of the singularity
  • For example, a node has delta invariant 1, a cusp has delta invariant 1, and a tacnode has delta invariant 2