Glossary

3.3 Relationship between affine and projective varieties

4 min read•july 30, 2024

Affine and projective varieties are closely linked in algebraic geometry. transforms affine varieties into projective ones, adding points at infinity and preserving geometric properties. This process allows us to study affine varieties in a more complete setting.

Projective varieties can be seen as completions of affine ones. By using affine cones, patches, and , we can switch between these two perspectives. This relationship helps us understand both local and global properties of algebraic varieties.

Homogenization of Polynomials

Converting Affine Varieties to Projective Varieties

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  • Homogenization converts an into a by introducing an additional variable to each term of the defining polynomial(s)
  • The additional variable makes all terms have the same degree
  • To homogenize a polynomial f(x1,...,xn)f(x₁, ..., xₙ) of degree dd, multiply each term by the homogenizing variable x0x₀ raised to the power of dd minus the degree of the term
    • For example, homogenizing the polynomial f(x,y)=x2+xy+yf(x, y) = x^2 + xy + y of degree 2 results in F(x0,x,y)=x2+x0xy+x02yF(x₀, x, y) = x^2 + x₀xy + x₀^2y

Properties of Homogenized Varieties

  • The resulting homogeneous polynomial F(x0,x1,...,xn)F(x₀, x₁, ..., xₙ) defines a projective variety in Pnℙⁿ
  • The original affine variety is embedded as the subset where x0≠0x₀ ≠ 0
  • Homogenization preserves the geometric properties of the affine variety
  • Homogenization allows for the study of the affine variety's behavior "at infinity" in the projective space

Affine vs Projective Varieties

Embedding Affine Varieties into Projective Spaces

  • Every affine variety can be embedded into a projective space by homogenizing its defining polynomials
  • The resulting projective variety contains the original affine variety as a dense open subset
  • The affine variety can be recovered from the projective variety by dehomogenizing the defining polynomials and considering the subset where the homogenizing variable is non-zero

Projective Varieties as Completions of Affine Varieties

  • Projective varieties can be thought of as completions of affine varieties
  • Projective varieties add points "at infinity" to capture the limiting behavior of the affine variety
  • The of a projective variety is one less than the dimension of its associated
    • The affine cone is formed by taking the union of all scalar multiples of points in the projective variety

Affine Cone and Patch

Affine Cone

  • The affine cone of a projective variety V⊂PnV ⊂ ℙⁿ is the preimage of VV under the projection map from 𝔸n+1∖{0}𝔸ⁿ⁺¹ ∖ \{0\} to Pnℙⁿ
  • The affine cone is the set of all points (x0,x1,...,xn)(x₀, x₁, ..., xₙ) in 𝔸n+1𝔸ⁿ⁺¹ such that [x0:x1:...:xn]∈V[x₀ : x₁ : ... : xₙ] ∈ V
  • The affine cone is a union of lines through the origin in 𝔸n+1𝔸ⁿ⁺¹
  • The defining equations of the affine cone are the homogeneous polynomials that define the projective variety

Affine Patch

  • An of a projective variety VV is the intersection of VV with the subset of Pnℙⁿ where one of the homogeneous coordinates is non-zero
    • For example, Ui={[x0:x1:...:xn]∈V∣xi≠0}U_i = \{[x₀ : x₁ : ... : xₙ] ∈ V | x_i ≠ 0\}
  • Each affine patch is isomorphic to an affine variety
  • The affine variety is obtained by dehomogenizing the defining polynomials of VV with respect to the non-zero coordinate
  • The affine patches form an open cover of the projective variety
  • Affine patches allow for the study of local properties using affine coordinates

Dehomogenization and Homogenization

Dehomogenization

  • Dehomogenization converts a projective variety into an affine variety
  • To dehomogenize, set one of the homogeneous coordinates to 1 and consider the resulting affine equations in the remaining variables
  • To dehomogenize a homogeneous polynomial F(x0,x1,...,xn)F(x₀, x₁, ..., xₙ) with respect to xix_i, set xi=1x_i = 1 and simplify the equation
    • The simplified equation is a non-homogeneous polynomial f(x0,...,x^i,...,xn)f(x₀, ..., \hat{x}_i, ..., xₙ), where x^i\hat{x}_i denotes the omission of xix_i
  • The dehomogenized polynomial ff defines an affine variety, which is isomorphic to the affine patch of the projective variety where xi≠0x_i ≠ 0

Relationship between Homogenization and Dehomogenization

  • Homogenization and dehomogenization are inverse processes
  • These processes allow for the conversion between affine and projective varieties while preserving the underlying geometric structure
  • The choice of the coordinate for dehomogenization (or the homogenizing variable) affects the specific affine patch obtained
    • For example, dehomogenizing with respect to x0x₀ yields the affine patch U0U₀, while dehomogenizing with respect to x1x₁ yields U1U₁
  • The resulting affine varieties from different dehomogenizations are isomorphic and provide equivalent local descriptions of the projective variety

Key Terms to Review (20)

Affine cone: An affine cone is a geometric structure that consists of a set of points in affine space, along with the line segments connecting these points to the origin. It serves as a way to study varieties by translating properties of projective varieties into the context of affine varieties, highlighting the relationship between these two concepts and helping to understand their shared characteristics.
Affine patch: An affine patch is a subset of an algebraic variety that can be described using affine coordinates, making it easier to work with algebraic objects. These patches allow us to understand the local structure of varieties by translating geometric questions into algebraic ones, often simplifying the study of properties like dimension and intersection. Each affine patch corresponds to a specific coordinate system that captures part of the variety's structure in a more manageable form.
Affine Variety: An affine variety is a subset of affine space defined as the common zero set of a collection of polynomials. These varieties are fundamental objects in algebraic geometry, connecting geometric concepts with algebraic expressions through their coordinate rings and properties.
Algebraic Closure: An algebraic closure of a field is an extension field in which every non-constant polynomial has a root. This concept is crucial because it allows for the understanding of polynomial equations in a comprehensive manner, ensuring that solutions exist within this extended framework. It connects deeply to the study of varieties by providing a foundation for analyzing the solutions to polynomial equations, whether in affine or projective settings.
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.
Closure: Closure is a fundamental concept in algebraic geometry that refers to the smallest closed set containing a given set, where closed sets are defined with respect to a particular topology. In the context of varieties, closure helps us understand how points or subsets relate to the larger structure, particularly when connecting affine and projective varieties and analyzing their coordinate rings. It serves as a bridge between local and global properties, allowing for deeper insights into the behavior of these mathematical objects.
Coordinate Ring: The coordinate ring of an affine variety is a way to represent the algebraic structure of the variety through polynomials. Specifically, it consists of all polynomial functions defined on the affine space corresponding to the variety, allowing for a bridge between geometric objects and algebraic expressions. This concept is crucial in understanding properties such as dimension, ideals, and relationships between different types of varieties.
Deformation theory: Deformation theory is a branch of mathematics that studies how geometric objects can change or deform while retaining certain structural properties. It plays a crucial role in understanding the relationships between different varieties and their singularities, highlighting how small changes can lead to significant differences in the underlying geometry.
Dehomogenization: Dehomogenization is the process of converting a projective variety, represented in homogeneous coordinates, into an affine variety by fixing one of the projective coordinates. This technique helps bridge the gap between projective and affine spaces, allowing for the study of properties and relationships of varieties in different contexts.
Dimension: Dimension is a fundamental concept in geometry and algebra that refers to the number of independent directions or parameters needed to describe a space or object. In algebraic geometry, it helps classify varieties based on their geometric properties, influencing how they are represented and understood in terms of both affine and projective spaces.
Embedding: An embedding is a mathematical mapping that allows one space to be represented within another, preserving the structure and properties of the original space. This concept is crucial in various areas of algebraic geometry, as it establishes relationships between different varieties, allowing us to study them in a unified way. By embedding affine varieties into projective spaces, for example, we can analyze their properties and behaviors in a broader context.
Field Extension: A field extension is a way to create a new field from an existing field by adding elements that do not already belong to it, allowing for the expansion of the field's structure. This concept is crucial as it enables the study of polynomials and rational functions, linking algebraic structures to geometric properties and offering insights into how different varieties relate to each other.
Homogeneous coordinate: Homogeneous coordinates are a system of coordinates used in projective geometry that allow for the representation of points in a way that facilitates the handling of infinity and transformations. By adding an extra dimension to traditional coordinates, homogeneous coordinates make it possible to represent points at infinity and simplify the equations of geometric objects, bridging the gap between affine and projective varieties.
Homogenization: Homogenization is the process of converting a given algebraic variety defined over a field into a projective variety by introducing an additional variable, allowing the variety to be studied in the projective setting. This process helps bridge the gap between affine and projective varieties, as it enables affine varieties to be expressed in a way that includes points at infinity, which are essential in projective geometry. By homogenizing equations, we can better understand the relationships between these two types of varieties and their geometric properties.
Intersection Theory: Intersection theory is a branch of algebraic geometry that studies the intersection of subvarieties within a given variety. It provides tools to count and analyze the intersection points of geometric objects, often taking into account their dimensions, multiplicities, and other geometric properties. This theory connects algebraic structures with geometric concepts, allowing for a deeper understanding of the relationships between affine and projective varieties, as well as their dimensions and the modern framework of schemes.
Irreducibility: Irreducibility refers to the property of a polynomial or algebraic variety that cannot be factored into simpler components over the given field or ring. This concept is vital in understanding the structure of varieties, as it determines whether a variety can be expressed as a union of smaller varieties, influencing how we analyze their geometric and algebraic properties.
Isomorphism: An isomorphism is a mathematical mapping that establishes a one-to-one correspondence between two structures, showing that they are essentially the same in terms of their properties and operations. This concept connects various algebraic structures and geometric entities, revealing how different representations can share the same underlying characteristics, such as affine varieties and projective varieties.
Morphism: A morphism is a structure-preserving map between two mathematical objects, typically within the context of algebraic geometry where it connects varieties. This concept allows for the exploration of relationships between different varieties, examining how their structures relate and translate through functions and coordinates.
Projective Variety: A projective variety is a type of geometric object that can be defined as the zero set of homogeneous polynomials in a projective space. It serves as a central concept in algebraic geometry, connecting the properties of geometric objects with algebraic representations and allowing for the study of both affine and projective spaces.
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.
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