9.3 Intersection theory for plane curves
4 min read•july 30, 2024
Intersection theory for plane curves is a powerful tool in algebraic geometry. It helps us understand how curves intersect, counting their meeting points and measuring how they touch. This theory connects deeply to curve degrees and shapes.
is the star of intersection theory. It tells us that two curves' total intersections equal their degree product. This simple idea unlocks complex geometric problems and helps classify curves by their and singularities.
Intersection multiplicity of curves
Definition and properties
Top images from around the web for Definition and properties
OsculatingPlane | Wolfram Function Repository View original
Is this image relevant?
File:Tangency Example 1.svg - Wikimedia Commons View original
Is this image relevant?
OsculatingPlane | Wolfram Function Repository View original
Is this image relevant?
OsculatingPlane | Wolfram Function Repository View original
Is this image relevant?
File:Tangency Example 1.svg - Wikimedia Commons View original
Is this image relevant?
1 of 3
Top images from around the web for Definition and properties
OsculatingPlane | Wolfram Function Repository View original
Is this image relevant?
File:Tangency Example 1.svg - Wikimedia Commons View original
Is this image relevant?
OsculatingPlane | Wolfram Function Repository View original
Is this image relevant?
OsculatingPlane | Wolfram Function Repository View original
Is this image relevant?
File:Tangency Example 1.svg - Wikimedia Commons View original
Is this image relevant?
1 of 3
- The of two plane curves C and D at a point P is a non-negative integer that measures the degree of between the curves at P
- If the curves C and D do not pass through P, the intersection multiplicity at P is 0
- If P is a simple point (transversal intersection) of both curves, the intersection multiplicity is 1
- Higher intersection multiplicities occur when the curves are tangent or have higher-order contact at P (osculating curves, curves with a common cusp)
Computation methods
- The intersection multiplicity can be computed using the dimensions of certain local ring quotients ()
- The intersection multiplicity can also be determined by the of resultants ()
- The sum of intersection multiplicities at all points of intersection of C and D is equal to the product of their degrees, assuming C and D have no common components (Bézout's theorem)
Bézout's theorem for intersections
Statement and proof
- Bézout's theorem states that if C and D are two plane curves of degrees m and n, respectively, with no common component, then the total number of intersections of C and D, counted with multiplicity, is equal to mn
- The proof of Bézout's theorem can be done using resultants, by considering the intersection of C and D as a divisor on the complex (divisor-based proof)
- Bézout's theorem is a fundamental result in intersection theory and has numerous applications in algebraic geometry and related fields (enumerative geometry, solving systems of polynomial equations)
Applications and examples
- Bézout's theorem can be used to determine the maximum number of intersections between two curves of given degrees (degree bounds)
- The theorem helps analyze the behavior of intersections under certain transformations, such as projective transformations or birational maps (invariance properties)
- Examples of Bézout's theorem include finding the number of intersections between a line and a conic (2 intersections), or between two conics (4 intersections, counting multiplicity)
Intersection theory in geometry
Solving geometric problems
- Intersection theory can be used to determine the number and nature of intersections between plane curves, which is useful in solving various geometric problems
- By applying Bézout's theorem, one can find the total number of intersections (real and complex) between two curves, given their degrees
- Intersection multiplicities can help identify tangency points, multiple points, or other special intersection points between curves (singularities, inflection points)
Studying families of curves
- Intersection theory can be used to study the geometry of families of curves, such as pencils of curves or linear systems (, multiplicities)
- Applications of intersection theory include finding common tangents to curves, studying singular points of curves, and analyzing the behavior of curves under certain transformations (projective, birational)
- Intersection theory helps classify special families of curves, such as , elliptic curves, or curves with prescribed singularities (, )
Intersection theory and curve genus
Genus and its properties
- The genus of a smooth, irreducible, projective plane curve C is a non-negative integer that measures the complexity of the curve's topology (number of holes or handles)
- The genus can be computed using the : g = (d-1)(d-2)/2, where d is the degree of the curve C
- The degree-genus formula can be derived using intersection theory, specifically by considering the intersection of C with a line and applying the adjunction formula (, )
Relation to singularities and classification
- The genus is related to the number and types of singularities (multiple points, cusps, etc.) that a curve possesses; intersection theory helps analyze these singularities (, )
- The genus is an important invariant in the classification of algebraic curves and surfaces, and intersection theory plays a crucial role in studying the geometry and arithmetic of curves (moduli spaces, )
- Examples of curves with different genera include rational curves (genus 0), elliptic curves (genus 1), and hyperelliptic curves (genus ≥ 2)
Key Terms to Review (28)
Base Points: Base points refer to specific points in the projective space where certain algebraic varieties, such as curves, can be defined or encountered. These points are significant in intersection theory because they indicate where two or more curves meet or intersect, impacting their properties and classifications. Understanding base points is crucial for analyzing the behavior of curves under various conditions and establishing relationships between different varieties.
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.
Blow-ups: A blow-up is a geometric transformation that replaces a point on a variety with an entire projective space, effectively 'resolving' singularities by introducing new structure. This technique allows for a clearer understanding of the behavior of curves at intersection points and helps in studying their properties more comprehensively.
Canonical divisor: A canonical divisor is a divisor that represents the class of differential forms on a smooth projective variety, reflecting its geometric properties. In algebraic geometry, it serves as an essential tool to analyze the properties of varieties, particularly in the context of Riemann-Roch theorem and intersection theory. Understanding canonical divisors helps in studying the behavior of curves and their intersections in projective space.
Cubic curves intersecting: Cubic curves intersecting refers to the points where two cubic plane curves meet in the projective plane. These intersections are significant in algebraic geometry because they reveal properties about the curves, such as their multiplicities and how they relate to each other through their defining equations.
Cusp Points: Cusp points are singular points on a curve where the curve has a distinct, pointed shape, resembling a sharp corner or tip. These points are critical in the study of plane curves as they indicate where the curve fails to be smooth, often leading to interesting behaviors in intersections and tangents.
Cuspidal curves: Cuspidal curves are algebraic curves characterized by having a singularity called a cusp, where the curve intersects itself at a point and is not smooth at that point. These curves are significant in understanding the geometry of plane curves, particularly in relation to their intersections and singularities. The study of cuspidal curves reveals important properties about the overall behavior and classification of algebraic curves in the plane.
Degree of a curve: The degree of a curve is a fundamental concept in algebraic geometry that refers to the highest power of the variable in the polynomial equation that defines the curve. It provides critical information about the curve's geometric properties, such as its shape and intersection behavior with other curves. Understanding the degree is essential for analyzing how curves interact, especially when considering their intersections.
Degree-genus formula: The degree-genus formula is a fundamental result in algebraic geometry that relates the degree of a smooth projective curve to its geometric properties, specifically its genus. This formula is essential for understanding how the intersection theory for plane curves operates and provides crucial insights into the classification of algebraic surfaces.
Dimension of the Intersection: The dimension of the intersection refers to the geometric property that describes the dimensionality of the space where two or more geometric objects meet or overlap. In the context of plane curves, this concept becomes crucial as it helps to determine how these curves intersect, including the nature and number of intersection points they share, influencing the overall behavior and properties of the curves involved.
Finite intersection: A finite intersection refers to the scenario where a limited number of geometric objects intersect at a finite number of points. In the context of algebraic geometry, this concept is essential for understanding how curves interact and overlap in a plane. When examining intersections of plane curves, the finite intersection property helps in determining how many solutions or points exist where these curves meet, which is crucial for solving equations and understanding geometric relationships.
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.
Homogeneous coordinates: Homogeneous coordinates are a system of coordinates used in projective geometry that allows for the representation of points in a projective space. By adding an extra dimension, points can be expressed in a way that makes it easier to handle concepts such as infinity and intersections in geometric contexts. This representation plays a crucial role in various mathematical topics, connecting projective spaces, affine spaces, and the properties of projective varieties.
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.
Jacobians: Jacobians are matrices that contain the first-order partial derivatives of a vector-valued function, playing a key role in various areas of mathematics, including algebraic geometry. In the context of intersection theory for plane curves, Jacobians help in studying the behavior of curves at their intersections and provide crucial information about their singular points and local properties. Understanding Jacobians is essential for analyzing how curves intersect and for determining multiplicities of intersections.
Local intersection number: The local intersection number is a concept in algebraic geometry that measures the intersection multiplicity of two curves at a particular point. It quantifies how many times the curves intersect at that point, taking into account their tangential behavior and how they approach each other. This number can indicate not just simple crossings, but also situations where the curves may be tangent or even meet at higher order intersections.
Multiple intersection: Multiple intersection refers to the situation where two or more curves intersect at a single point, but with higher multiplicity than one. This means that, instead of just crossing each other at that point, the curves 'touch' or have a shared behavior at that point, indicating a deeper connection between them. In the study of plane curves, understanding multiple intersections helps reveal the intricate relationships between curves and their geometric properties.
Nodal Curves: Nodal curves are a type of algebraic curve characterized by the presence of nodes, which are singular points where two branches of the curve intersect. These curves arise in various contexts, particularly in intersection theory, as they demonstrate interesting properties and behaviors when examining how curves intersect and relate to one another. The study of nodal curves helps in understanding more complex geometrical structures and their implications in algebraic geometry.
Order of Vanishing: The order of vanishing refers to the multiplicity with which a function or a curve intersects a point, measuring how many times the function approaches zero at that point. It plays a crucial role in intersection theory for plane curves, as it helps to determine how curves meet and interact at specific points. Understanding the order of vanishing is essential for analyzing the behavior of polynomial functions near their roots and has significant implications for the study of singularities in algebraic geometry.
Projective Plane: The projective plane is a geometric structure that extends the concept of the Euclidean plane by adding 'points at infinity' for parallel lines, allowing for a more unified view of geometric properties. In this setting, every pair of lines intersects at exactly one point, and every pair of points defines exactly one line. This concept is critical when discussing projective closure and homogenization, as it helps to understand how affine geometries can be transformed into projective ones, ensuring a consistent framework for analyzing intersections of curves.
Rational Curves: Rational curves are curves that can be parametrized by rational functions, meaning they can be expressed in terms of ratios of polynomials. These curves play a crucial role in algebraic geometry, particularly in the study of varieties and their intersections, as they provide insights into the geometric structure of algebraic sets and facilitate understanding of intersection theory for plane curves.
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.
Resultant-based approach: The resultant-based approach is a mathematical technique used to analyze the intersection of plane curves by calculating a resultant, which is a scalar value derived from the polynomials that define the curves. This approach provides a powerful method for determining the conditions under which two curves intersect, as well as the nature and multiplicity of the intersection points. By utilizing resultants, one can efficiently study the geometry of intersections and gain insights into the relationships between different algebraic entities.
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.
Smooth curves: Smooth curves are geometric representations that do not have any sharp corners or cusps, allowing for a continuous and differentiable structure throughout their entirety. They play a critical role in the study of algebraic geometry, particularly in analyzing the intersections and properties of plane curves, ensuring that such curves maintain a well-defined tangent at every point.
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.
Transversality: Transversality refers to a geometric condition where two or more subspaces intersect in a way that is neither tangent nor overlapping. In this context, it is important because it ensures that intersections behave nicely and allows for the application of intersection theory, particularly when studying how varieties interact in projective space and analyzing the intersection points of curves.
Two lines intersecting: Two lines intersecting refers to the scenario where two distinct lines in a plane meet at a single point. This point of intersection is significant in geometry and algebra, as it often represents a solution to a system of linear equations, highlighting relationships between different geometric entities, such as points and angles formed by the lines.