9.3 Intersection theory in projective space
7 min read•july 30, 2024
Intersection theory in is a powerful tool for understanding how geometric objects intersect. It extends concepts from affine space, allowing us to analyze intersections more comprehensively. This theory is crucial for solving complex geometric problems and counting intersections accurately.
is a key result in this field, relating the degrees of varieties to their intersection behavior. It's super useful for solving counting problems in geometry and has applications in various areas of math and physics. Understanding this theorem is essential for mastering intersection theory.
Intersection Multiplicity and Degree in Projective Space
Measuring Intersection Complexity
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- measures the complexity of the intersection between two varieties at a point in projective space
- Local invariant captures the number of times the varieties intersect, counted with appropriate multiplicity
- Extends the concept of intersection multiplicity from affine space to projective space
- Allows for a more comprehensive analysis of the intersection behavior of projective varieties
- Example: The intersection multiplicity of two projective curves at a point can be determined by analyzing their local equations in a suitable coordinate chart
Computing and Properties of Degree
- The degree of a projective variety is the number of points of intersection between the variety and a general linear subspace of complementary dimension
- Global invariant measures the size or complexity of the variety
- Can be computed using the Hilbert polynomial, which encodes information about the dimensions of the homogeneous coordinate ring of the variety in each degree
- Satisfies properties such as additivity, multiplicativity, and invariance under linear transformations
- Example: The degree of a projective hypersurface (codimension 1 subvariety) is equal to the degree of its defining homogeneous polynomial
- The projective version of Bézout's theorem relates the intersection multiplicity and degree of projective varieties
- States that the sum of the intersection multiplicities of two varieties at all their points of intersection is equal to the product of their degrees
- Provides a powerful tool for understanding the global intersection behavior of projective varieties
- Generalizes the affine version of Bézout's theorem to the projective setting
- Example: Two projective curves of degrees and intersect in points counted with multiplicity, assuming they intersect properly
Intersection Theory for Subvarieties
Subvarieties and Chow Ring
- Subvarieties of projective space are locally closed subsets defined by homogeneous polynomial equations
- Include projective varieties (irreducible subvarieties) and schemes (possibly reducible subvarieties with nilpotent elements)
- Provide a rich class of objects for studying intersection theory in projective space
- Allow for the development of a comprehensive intersection theory that goes beyond just varieties
- Example: A projective curve (1-dimensional subvariety) in projective 3-space can be defined by the vanishing of two homogeneous polynomials
- The Chow ring of projective space is a graded ring that encodes information about the intersection theory of subvarieties
- Elements are formal linear combinations of subvarieties modulo rational equivalence
- Product in the Chow ring corresponds to the intersection of subvarieties
- Provides an algebraic framework for studying intersections and their properties
- Allows for the computation of intersection products and intersection numbers
- Example: In the Chow ring of projective 2-space, the class of a line times the class of a point is equal to the class of a point
Intersection Product and Functoriality
- The intersection product of subvarieties in projective space is a bilinear operation that associates to two subvarieties another subvariety representing their intersection
- Well-defined on equivalence classes in the Chow ring and satisfies properties such as commutativity, associativity, and the projection formula
- Allows for the computation of intersections between subvarieties of complementary codimension
- Can be used to study the geometry and topology of subvarieties through their intersection behavior
- Example: The intersection product of two distinct lines in projective 2-space is a single point
- The intersection multiplicity of two subvarieties at a point can be computed using the Serre intersection formula
- Involves the Tor functor applied to the local rings of the subvarieties at the point
- Provides a way to compute intersection multiplicities in terms of algebraic invariants
- Generalizes the classical definition of intersection multiplicity for varieties
- Example: The intersection multiplicity of two curves at a point can be computed by analyzing their local equations and applying the Serre intersection formula
- The intersection theory of subvarieties is functorial with respect to morphisms of projective varieties
- Intersections are preserved by proper pushforwards and pullbacks
- Allows for the study of intersection theory in families and under morphisms
- Provides a powerful tool for understanding the behavior of intersections under geometric transformations
- Example: The pullback of the intersection of two subvarieties under a morphism is equal to the intersection of their pullbacks
Bézout's Theorem in Projective Space
Statement and Applications
- Bézout's theorem in projective space states that the intersection of two projective varieties of degrees and consists of points counted with multiplicity, assuming the varieties intersect properly (i.e., in the expected dimension)
- Provides a fundamental result in intersection theory relating the degrees of varieties to their intersection behavior
- Can be used to determine the number of intersection points between two projective curves or surfaces
- Allows for the solution of various enumerative geometry problems
- Example: Two projective plane curves of degrees 3 and 4 intersect in 12 points counted with multiplicity, assuming they have no common components
- Bézout's theorem can be applied to solve enumerative geometry problems, such as:
- Counting the number of curves or surfaces satisfying certain geometric conditions (e.g., passing through given points or tangent to given lines)
- Determining the number of lines that intersect four given lines in projective 3-space
- Calculating the number of conics tangent to five given lines in the projective plane
- Example: There are 2 conics passing through 5 given points in the projective plane, assuming the points are in general position
Refinements and Generalizations
- In some cases, Bézout's theorem may overcount the number of intersection points due to the presence of multiple components or non-reduced structure in the intersection
- Techniques such as excess intersection theory or the use of multiplicities can help refine the count
- Allows for a more accurate analysis of the intersection behavior in degenerate situations
- Provides a way to handle intersections with higher-dimensional components or non-transversal intersections
- Example: If two projective curves intersect non-transversely at a point, the intersection multiplicity at that point may be greater than 1
- Bézout's theorem can be generalized to the intersection of more than two varieties using the multi-homogeneous Bézout theorem
- Takes into account the multi-degrees of the varieties involved
- Allows for the study of intersections in multi-projective spaces or products of projective spaces
- Provides a more general framework for understanding the intersection behavior of multiple varieties
- Example: The multi-homogeneous Bézout theorem can be used to count the number of points in the intersection of three or more hypersurfaces in a product of projective spaces
Intersection Theory and Enumerative Geometry
Enumerative Problems and Schubert Calculus
- Enumerative geometry is the study of counting geometric objects satisfying certain conditions
- Examples include counting lines meeting four given lines in projective 3-space or conics tangent to five given lines in the projective plane
- Intersection theory provides a powerful tool for solving enumerative geometry problems by translating them into questions about the intersection of suitable subvarieties in projective space
- Allows for the systematic study and solution of a wide range of enumerative problems
- Example: The problem of counting lines on a cubic surface can be translated into a question about the intersection of certain subvarieties in the Grassmannian of lines in projective 3-space
- Schubert calculus is a framework within enumerative geometry that uses intersection theory to count geometric objects
- Introduces Schubert varieties, which are special subvarieties of Grassmannians (parametrizing linear subspaces of a vector space)
- Studies the intersections of Schubert varieties to solve enumerative problems
- The Schubert calculus formula expresses the number of geometric objects satisfying given conditions as the degree of a zero-dimensional intersection product of Schubert varieties in a Grassmannian
- Involves the cup product in the of the Grassmannian
- Example: The number of lines meeting four general lines in projective 3-space can be computed using the Schubert calculus formula in the Grassmannian of lines
Modern Approaches and Connections
- The Gromov-Witten theory is a modern approach to enumerative geometry that counts curves in algebraic varieties using intersection theory on moduli spaces of stable maps
- Has connections to string theory and quantum cohomology
- Provides a powerful framework for studying enumerative problems involving curves and their deformations
- Allows for the computation of invariants that capture the geometry and topology of the underlying variety
- Example: The Gromov-Witten invariants of a quintic threefold count the number of rational curves of given degree on the threefold and play a crucial role in mirror symmetry
- The Kontsevich formula is a celebrated result in enumerative geometry that counts the number of rational curves of given degree passing through prescribed points in the projective plane
- Proved using intersection theory on the moduli space of stable maps
- Provides a beautiful and deep connection between enumerative geometry and intersection theory
- Has applications in various areas of mathematics, including mathematical physics and integrable systems
- Example: The Kontsevich formula can be used to count the number of rational cubic curves passing through 8 general points in the projective plane, which is equal to 12
- Intersection theory and enumerative geometry have deep connections to other areas of mathematics, such as:
- Algebraic topology: The cohomology ring of a variety encodes information about its intersection theory and can be used to study enumerative problems
- Mathematical physics: Enumerative invariants, such as Gromov-Witten invariants, appear in the study of mirror symmetry and string theory
- Representation theory: The intersection theory of flag varieties and Schubert varieties is closely related to the representation theory of Lie groups and Lie algebras
- Example: The Schubert calculus in the Grassmannian is related to the representation theory of the special linear group and the combinatorics of Young tableaux
Key Terms to Review (16)
Algebraic Curves: Algebraic curves are one-dimensional varieties defined as the solution set of polynomial equations in two variables. These curves can represent a wide range of geometric and algebraic properties, and they play a crucial role in various areas such as geometry, computer vision, and cohomology. The study of algebraic curves includes their intersections, their use in algorithms for solving visual recognition problems, and the application of computational methods to analyze their properties through sheaf cohomology.
Bézout's Theorem: Bézout's Theorem states that for two projective varieties defined by homogeneous polynomials, the number of intersection points, counted with multiplicities, is equal to the product of their degrees. This principle connects algebraic geometry and polynomial equations, revealing deep relationships between the algebraic properties of varieties and their geometric behavior.
Cohomology Ring: The cohomology ring is an algebraic structure that captures the topological features of a space through its cohomology groups, organized into a graded ring. It combines information from different dimensions and allows for operations such as cup products, which provide insight into how various cohomology classes interact. This concept plays a crucial role in understanding the intersection theory in projective spaces, linking geometric properties with algebraic invariants.
David Mumford: David Mumford is a renowned mathematician recognized for his significant contributions to algebraic geometry, particularly in intersection theory and the study of moduli spaces. His work laid the foundation for modern approaches to intersection multiplicity, birational geometry, and the understanding of rational maps between varieties.
Degeneration Techniques: Degeneration techniques are mathematical methods used to study the behavior of algebraic varieties by analyzing their simpler or limiting cases, often involving the transition from a complex variety to a simpler object, such as a singular variety or a lower-dimensional variety. These techniques are essential for understanding intersection theory in projective space, as they allow mathematicians to derive properties of complex varieties by observing how they degenerate into more manageable forms, thereby revealing insights about their geometric and topological features.
Degree of Intersection: The degree of intersection refers to the number of points at which two or more algebraic varieties meet in projective space, counting multiplicities. This concept is crucial in intersection theory as it provides a way to quantify the relationship between different geometric objects, revealing insights into their dimensional properties and the nature of their intersections.
Dimension of Intersection: The dimension of intersection refers to the dimensionality of the space formed by the intersection of two or more varieties in projective space. This concept is crucial in understanding how different geometric objects relate to one another, particularly in terms of their overlapping properties. It provides insight into the structure and complexity of intersections, allowing for deeper analysis of their geometric and algebraic characteristics.
Genericity Conditions: Genericity conditions refer to the properties or requirements that must be satisfied for certain results or theorems in algebraic geometry to hold generically, which means that they are true for 'most' cases within a given context. These conditions often ensure that specific configurations or arrangements do not occur, allowing for broader conclusions to be drawn from general cases. In the study of intersection theory in projective space, these conditions help in understanding the behavior of intersection points and the dimension of the varieties involved.
Hyperplane: A hyperplane is a flat, affine subspace of one dimension less than its ambient space. In simple terms, if you have an n-dimensional space, a hyperplane will be (n-1)-dimensional. Hyperplanes play a crucial role in intersection theory in projective space, as they can define the intersections of various algebraic varieties and help understand their geometric properties.
Intersection Multiplicity: Intersection multiplicity is a measure of how 'tangential' two geometric objects intersect at a point, quantifying the number of times the objects meet at that point. It provides a way to count intersections not just in terms of distinct points, but also considering their local behavior and how they are positioned with respect to one another. This concept is vital in understanding degrees of curves, their intersections in projective space, and the application of Bézout's theorem when studying the properties of homogeneous polynomials.
Jean-Pierre Serre: Jean-Pierre Serre is a renowned French mathematician known for his significant contributions to algebraic geometry, topology, and number theory. His work has had a profound impact on various mathematical fields, particularly in the development of sheaf theory and the study of projective varieties, linking many concepts together that are crucial for understanding modern algebraic geometry.
Local computations: Local computations refer to the techniques used to analyze geometric properties and behaviors of varieties or schemes in a small neighborhood around a point. This concept is crucial in understanding intersection theory, as it allows mathematicians to simplify complex global problems by focusing on simpler local ones, often utilizing tools like local rings and schemes to study the structure and properties of varieties at specific locations.
Projective Space: Projective space is a fundamental concept in algebraic geometry that extends the notion of Euclidean space by adding 'points at infinity' to allow for a more comprehensive study of geometric properties. This extension allows for the unification of various types of geometric objects, facilitating intersection theory, transformations, and various algebraic structures.
Pushforward: The pushforward is a fundamental operation in algebraic geometry that allows one to transfer geometric and algebraic data from one space to another via a morphism or a map. It enables the transformation of sheaves, functions, and other structures along a given map, capturing how properties and relationships are preserved or altered between different varieties. Understanding pushforward helps in analyzing intersections, rational maps, and their impact on the structures involved.
Smoothness Conditions: Smoothness conditions refer to the criteria that determine whether a mathematical object, like a variety or a manifold, has no singular points, meaning it behaves well and has nice geometric properties. In the context of intersection theory in projective space, these conditions are essential to ensure that intersections of varieties occur as expected, without unexpected singularities that can complicate their structure and analysis.
Transversality: Transversality is a geometric condition that occurs when two varieties intersect in a way such that their tangent spaces at the intersection points span the ambient space. This concept is crucial in understanding intersection theory and multiplicity, as it ensures that intersections are 'nice' and not overly complicated. Transversality implies that the intersection behaves well, leading to more predictable results in terms of intersection multiplicity and degree.