Glossary

9.1 Plane curves and their singularities

5 min read•july 30, 2024

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?

  • Asymptote - Wikipedia View original

    Is this image relevant?

  • Graphs of Polynomial Functions | College Algebra Corequisite 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?

  • Asymptote - Wikipedia View original

    Is this image relevant?

  • Graphs of Polynomial Functions | College Algebra Corequisite View original

    Is this image relevant?

1 of 3
  • A is the set of points in the affine or projective plane whose coordinates satisfy a polynomial equation in two variables
  • The 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 is also called a plane curve
  • The intersection of two plane curves is a finite set of points
    • The number of (counting ) is equal to the product of the degrees of the curves by

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 y2=x3−xy^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 y2=x2(x+1)y^2 = x^2(x+1) has a at (0,0)(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 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 y2=x2(x+1)y^2 = x^2(x+1) has a node at (0,0)(0,0)
  • A is a singular point where the curve crosses itself tangentially
    • The tangent lines at a cusp coincide
    • For example, the curve y2=x3y^2 = x^3 has a cusp at (0,0)(0,0)
  • A is a singular point where the curve crosses itself with multiplicity 2, meaning it looks like two of the curve touching each other
    • For example, the curve y2=x4y^2 = x^4 has a tacnode at (0,0)(0,0)
  • An of multiplicity mm is a singular point where mm smooth branches of the curve intersect transversally
    • For example, the curve (y2−x2)(y−x2)=0(y^2-x^2)(y-x^2) = 0 has an ordinary triple point at (0,0)(0,0)

Higher order singularities

  • Other can occur, such as , , and more complicated singularities
  • These singularities can be analyzed using more advanced techniques such as blow-ups and
  • 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 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

  • 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 y2=x3y^2 = x^3 can be parametrized near (0,0)(0,0) by x=t2,y=t3x = 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 of two branches at a singular point can be computed using the Puiseux series expansions of the branches
    • For example, the curve y2=x2(x+1)y^2 = x^2(x+1) has two branches at (0,0)(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 y2=x3y^2 = x^3, the multiplicity of (0,0)(0,0) is 2 since f(0,0)=fx(0,0)=0f(0,0) = f_x(0,0) = 0 but fy(0,0)≠0f_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 of the curve
    • The genus can be computed using the degree-genus formula g=(d−1)(d−2)2−∑PmP(mP−1)2g = \frac{(d-1)(d-2)}{2} - \sum_P \frac{m_P(m_P-1)}{2}, where dd is the degree and mPm_P is the multiplicity of a singular point PP
  • The 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

Key Terms to Review (26)

Bézout's Theorem: Bézout's Theorem is a fundamental result in algebraic geometry that states that the number of intersection points of two projective plane curves, counted with multiplicities, is equal to the product of their degrees. This theorem highlights the relationship between geometry and algebra and connects projective varieties with their intersections, making it essential for understanding various concepts like projective space, affine varieties, and singularities in plane curves.
Blowing up: Blowing up is a technique used in algebraic geometry to replace a point (or a subvariety) on a variety with a more complex structure, often a projective space, to resolve singularities or better understand the geometric properties of the variety. This process helps to analyze and visualize singular points on plane curves, allowing for clearer insights into their local behavior and interactions.
Branches: Branches are the distinct parts of a curve that can be defined mathematically as separate continuous paths. Each branch can represent different sections of a curve, especially when considering singularities, where the curve may intersect or overlap with itself. Understanding branches helps in analyzing the behavior and properties of curves near these critical points.
Conic Sections: Conic sections are the curves obtained by intersecting a cone with a plane. These curves include circles, ellipses, parabolas, and hyperbolas, each defined by unique geometric properties and equations. Understanding conic sections is crucial as they represent fundamental shapes in both geometry and algebra, with applications ranging from physics to engineering.
Cubic Curves: Cubic curves are algebraic plane curves defined by polynomial equations of degree three, typically expressed in the form $y = ax^3 + bx^2 + cx + d$. These curves can exhibit a variety of shapes and properties, including points of intersection, tangents, and singularities. Understanding cubic curves is essential as they can have intricate behaviors, including self-intersections and cusps, which lead to fascinating discussions about their geometry and algebraic properties.
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.
Degree: In algebraic geometry, the degree of a projective variety or an algebraic object is a numerical invariant that measures the number of intersections with hyperplanes in a projective space. This concept helps to classify and understand the geometric properties of varieties, including their complexity and singularities, and plays a significant role in the dimension theory of projective varieties and the classification of algebraic surfaces.
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.
Genus: Genus refers to a topological invariant that measures the number of 'holes' in a surface or curve. In algebraic geometry, genus helps classify curves and surfaces based on their geometric properties, revealing important information about their structure and behavior. It connects various concepts such as singularities, intersection theory, and the classification of surfaces.
Higher Order Singularities: Higher order singularities refer to points on plane curves where the curve fails to be smooth, but where the failure is more complex than a simple cusp or node. These singularities occur when the derivative of the curve's defining equation vanishes to an order greater than one, indicating that multiple tangents can intersect at these points, leading to intricate local structures around the singularity. Understanding higher order singularities helps in analyzing the geometric properties and classifications of plane curves.
Inflection Points: Inflection points are specific points on a curve where the curvature changes sign, indicating a shift from concave up to concave down or vice versa. These points are important because they help identify where the graph of a function changes its bending behavior, which can relate to the behavior of the function itself, including local maxima and minima.
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.
Intersection points: Intersection points are specific locations in a geometric space where two or more curves, lines, or surfaces meet or cross each other. Understanding these points is crucial because they can reveal important information about the relationships between different curves, including how many times they intersect and the nature of those intersections, such as whether they are smooth or singular.
Irreducible Plane Curve: An irreducible plane curve is a type of algebraic curve that cannot be factored into simpler components over the complex numbers. This means that it cannot be expressed as a product of two non-constant polynomials, making it a fundamental object of study in algebraic geometry. The concept of irreducibility is crucial in understanding the behavior of curves and their singularities, as it indicates that the curve represents a 'whole' geometric entity rather than a combination of simpler ones.
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 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.
Plane algebraic curve: A plane algebraic curve is a one-dimensional variety defined by a polynomial equation in two variables, usually represented as $f(x, y) = 0$, where $f$ is a polynomial with coefficients in a field. These curves can exhibit a variety of shapes and properties, depending on the degree of the polynomial and the nature of its roots. Understanding the singularities of these curves is crucial as they can reveal important geometric and topological information about the curve's structure.
Puiseux series: A Puiseux series is a type of formal power series that allows for fractional powers of the variable, enabling the study of algebraic curves and their singularities in a more flexible way. This concept is especially significant when analyzing plane curves, as it provides a means to express local behavior near points of interest, particularly at singular points where traditional power series may not suffice. By using Puiseux series, one can uncover deeper geometric properties and singularity structures that are critical in algebraic geometry.
Rational Curve: A rational curve is a type of algebraic curve that can be parametrized by rational functions. This means there exists a parameter, typically denoted as 't', such that the coordinates of points on the curve can be expressed as rational expressions in terms of 't'. Rational curves are significant because they can serve as simple models for more complex geometrical structures and often have connections to singularities, especially when considering how they intersect or behave near those points.
Resolution of Singularities: Resolution of singularities is a process in algebraic geometry that aims to replace a singular algebraic variety with a non-singular one, allowing for the study of its geometric properties in a more manageable way. This process often involves techniques such as blowing up, which helps to resolve points where the variety fails to be well-defined or smooth. By resolving these singularities, mathematicians can better understand the behavior of curves and surfaces, and their intersections, in higher-dimensional spaces.
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.
Smooth curve: A smooth curve is a continuous curve that has no sharp corners or edges, meaning it can be drawn without lifting a pencil from the paper. This property implies that the curve is differentiable at every point, allowing for well-defined tangent lines. Smooth curves are important in studying plane curves and their singularities as well as understanding the properties of algebraic curves like genus and the Riemann-Roch theorem.
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.
Tangent Cone: A tangent cone is a geometric construct that describes the behavior of a curve or surface at a point, particularly in relation to singularities. It consists of all the directions in which you can approach the point and can be viewed as the 'best linear approximation' of the shape at that point. This concept is vital for analyzing how curves behave near singular points, providing insights into their local structure and geometry.
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.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide