Affine and projective varieties are closely linked in algebraic geometry. Homogenization 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 dehomogenization, 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

• Homogenization converts an affine variety into a projective variety 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 $f(x₁, ..., xₙ)$ of degree $d$, multiply each term by the homogenizing variable $x₀$ raised to the power of $d$ minus the degree of the term
  • For example, homogenizing the polynomial $f(x, y) = x^2 + xy + y$ of degree 2 results in $F(x₀, x, y) = x^2 + x₀xy + x₀^2y$

Properties of Homogenized Varieties

• The resulting homogeneous polynomial $F(x₀, x₁, ..., xₙ)$ defines a projective variety in $ℙⁿ$
• The original affine variety is embedded as the subset where $x₀ ≠ 0$
• 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 dimension of a projective variety is one less than the dimension of its associated affine cone
  • 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 $V ⊂ ℙⁿ$ is the preimage of $V$ under the projection map from $𝔸ⁿ⁺¹ ∖ {0}$ to $ℙⁿ$
• The affine cone is the set of all points $(x₀, x₁, ..., xₙ)$ in $𝔸ⁿ⁺¹$ such that $[x₀ : x₁ : ... : xₙ] ∈ V$
• The affine cone is a union of lines through the origin in $𝔸ⁿ⁺¹$
• The defining equations of the affine cone are the homogeneous polynomials that define the projective variety

Affine Patch

• An affine patch of a projective variety $V$ is the intersection of $V$ with the subset of $ℙⁿ$ where one of the homogeneous coordinates is non-zero
  • For example, $U_i = {[x₀ : x₁ : ... : xₙ] ∈ V | x_i ≠ 0}$
• Each affine patch is isomorphic to an affine variety
• The affine variety is obtained by dehomogenizing the defining polynomials of $V$ 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 $F(x₀, x₁, ..., xₙ)$ with respect to $x_i$, set $x_i = 1$ and simplify the equation
  • The simplified equation is a non-homogeneous polynomial $f(x₀, ..., \hat{x}_i, ..., xₙ)$, where $\hat{x}_i$ denotes the omission of $x_i$
• The dehomogenized polynomial $f$ defines an affine variety, which is isomorphic to the affine patch of the projective variety where $x_i ≠ 0$

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 $x₀$ yields the affine patch $U₀$, while dehomogenizing with respect to $x₁$ yields $U₁$
• The resulting affine varieties from different dehomogenizations are isomorphic and provide equivalent local descriptions of the projective variety