🌿Computational Algebraic Geometry Unit 9 – Intersection Theory & Bézout's Theorem

Key Concepts

• Intersection theory studies the intersection of algebraic varieties in algebraic geometry
• Bézout's theorem is a fundamental result in intersection theory that relates the number of intersection points of two plane curves to their degrees
• Algebraic varieties are geometric objects defined by polynomial equations
  • Can be thought of as higher-dimensional analogues of curves and surfaces
• Intersection multiplicity measures the complexity of the intersection between two varieties at a point
• Divisors are formal linear combinations of subvarieties of codimension one, used to study the geometry of varieties
• Chow rings are algebraic structures that encode information about the intersection theory of a variety
• Computational algebraic geometry applies algorithmic and computational techniques to solve problems in algebraic geometry

Historical Context

• Intersection theory has its roots in the work of 19th-century mathematicians such as Bézout, Plücker, and Cayley
• Bézout's theorem, named after Étienne Bézout, was first published in 1779
• The development of modern intersection theory is closely tied to the rise of algebraic geometry in the 20th century
  • Mathematicians such as Severi, Weil, and Grothendieck made significant contributions
• The introduction of schemes by Grothendieck in the 1960s provided a unified framework for intersection theory
• Computational methods in algebraic geometry gained prominence in the late 20th century with the advent of powerful computers
• Recent developments include the use of tropical geometry and toric varieties in intersection theory

Geometric Interpretation

• Intersection theory studies the intersection of algebraic varieties, which are geometric objects defined by polynomial equations
• The intersection of two curves in the plane is a set of points, while the intersection of higher-dimensional varieties is more complex
• Bézout's theorem states that the number of intersection points of two plane curves (counted with multiplicity) is equal to the product of their degrees
  • For example, a line (degree 1) and a conic (degree 2) intersect in two points
• Intersection multiplicity measures the complexity of the intersection between two varieties at a point
  • A simple intersection has multiplicity 1, while a tangency has multiplicity greater than 1
• The intersection product of two subvarieties is a subvariety of codimension equal to the sum of the codimensions of the original subvarieties
• The self-intersection of a subvariety is related to its normal bundle and can provide information about its geometry

Algebraic Foundations

• Intersection theory relies on the algebraic structure of polynomial rings and their quotients
• Polynomial rings $k[x_1, \ldots, x_n]$ over a field $k$ are the basic building blocks of algebraic geometry
• Ideals in polynomial rings define algebraic varieties as their zero sets
  • For example, the ideal $\langle x^2 + y^2 - 1 \rangle$ defines the unit circle in the plane
• The quotient ring $k[x_1, \ldots, x_n]/I$ corresponds to the functions on the variety defined by the ideal $I$
• Gröbner bases are a key computational tool for working with ideals and their quotients
• The dimension of a variety is related to the Krull dimension of its coordinate ring
• The degree of a variety can be defined in terms of the Hilbert polynomial of its coordinate ring

Intersection Theory Basics

• Intersection theory deals with the intersection product of subvarieties of a given variety
• The intersection product is a bilinear operation on the Chow ring of a variety, which is a graded ring encoding information about the variety's subvarieties
• The Chow ring is constructed using the group of divisors modulo rational equivalence
  • Divisors are formal linear combinations of subvarieties of codimension one
• The intersection product satisfies the properties of commutativity, associativity, and the projection formula
• The degree of the intersection product of two subvarieties is equal to the product of their degrees (Bézout's theorem)
• The intersection product is used to define important invariants such as the intersection number and the Euler characteristic
• The Chern classes of vector bundles on a variety can be used to compute intersection products via the Grothendieck-Riemann-Roch theorem

Bézout's Theorem

• Bézout's theorem is a fundamental result in intersection theory, relating the number of intersection points of two plane curves to their degrees
• The theorem states that the number of intersection points of two plane curves (counted with multiplicity) is equal to the product of their degrees
  • For example, a line (degree 1) and a cubic curve (degree 3) intersect in three points
• The intersection points are counted with multiplicity, which means that tangencies and higher-order intersections contribute more than one point to the total count
• Bézout's theorem can be generalized to higher-dimensional varieties, where it relates the degree of the intersection product to the degrees of the intersecting varieties
• The theorem is a consequence of the properties of the Chow ring and the intersection product
• Bézout's theorem has numerous applications in algebraic geometry, including the study of linear systems, the calculation of genus, and the classification of varieties

Applications and Examples

• Intersection theory has applications in various areas of mathematics, including enumerative geometry, topology, and number theory
• In enumerative geometry, intersection theory is used to count the number of geometric objects satisfying certain conditions
  • For example, the number of lines intersecting four given lines in 3D space (answer: 2)
• In topology, intersection theory is used to define and study characteristic classes of vector bundles, such as Chern classes and Stiefel-Whitney classes
• The intersection product on the Chow ring of a variety is related to the cup product in cohomology via the cycle class map
• In number theory, intersection theory is used in the study of Diophantine equations and arithmetic geometry
  • For example, the Mordell-Weil theorem on the group of rational points on an elliptic curve
• Intersection theory also has applications in physics, particularly in string theory and the study of moduli spaces of instantons

Computational Techniques

• Computational algebraic geometry provides algorithmic tools for working with varieties, ideals, and intersection theory
• Gröbner bases are a key computational tool for solving polynomial equations and performing computations in quotient rings
  • Buchberger's algorithm is used to compute Gröbner bases
• Resultants and subresultants are used to eliminate variables from polynomial equations and to compute the intersection of varieties
• The Macaulay2 and Singular software systems are powerful tools for computational algebraic geometry, with built-in support for intersection theory
• Toric varieties and their associated combinatorial objects, such as polytopes and fans, provide a rich source of examples and computational techniques in intersection theory
• The Schubert calculus is a computational framework for intersection theory in Grassmannians and flag varieties, with applications to enumerative geometry
• Tropical geometry is a recent development that uses piecewise-linear approximations of algebraic varieties to simplify computations in intersection theory