7.3 Homogenization and dehomogenization

Homogenization and dehomogenization are key techniques for moving between affine and projective spaces in algebraic geometry. They allow us to convert varieties and polynomials, bridging the gap between these two fundamental settings.

These processes help simplify complex geometric problems by leveraging the advantages of projective space. We can study behavior at infinity, apply powerful theorems, and gain deeper insights into the structure of algebraic varieties.

Homogenization vs Dehomogenization

The Process of Homogenization

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• Homogenization converts an affine algebraic variety into a projective algebraic variety by introducing an additional variable
  • Example: An affine variety $V(x^2 + y^2 - 1) \subseteq \mathbb{A}^2$ can be homogenized to a projective variety $V(x^2 + y^2 - z^2) \subseteq \mathbb{P}^2$
• To homogenize a polynomial $f(x_1, \ldots, x_n)$, introduce a new variable $x_0$ and multiply each monomial by an appropriate power of $x_0$ to make the degree of all monomials equal to the degree of $f$
  • Example: The polynomial $f(x, y) = x^2 + xy + y$ is homogenized to $f^*(x_0, x_1, x_2) = x_1^2 + x_1x_2 + x_0x_2$
• The homogenization of an affine variety $V(f_1, \ldots, f_s) \subseteq \mathbb{A}^n$ is the projective variety $V(f_1^*, \ldots, f_s^*) \subseteq \mathbb{P}^n$, where $f_i^*$ is the homogenization of $f_i$

The Process of Dehomogenization

• Dehomogenization converts a projective algebraic variety into an affine algebraic variety by setting one of the variables equal to 1
  • Example: A projective variety $V(x^2 + y^2 - z^2) \subseteq \mathbb{P}^2$ can be dehomogenized to an affine variety $V(x^2 + y^2 - 1) \subseteq \mathbb{A}^2$ by setting $z = 1$
• To dehomogenize a homogeneous polynomial $F(x_0, x_1, \ldots, x_n)$, set one of the variables (usually $x_0$) equal to 1 and simplify the resulting polynomial
  • Example: The homogeneous polynomial $F(x_0, x_1, x_2) = x_1^2 + x_1x_2 + x_0x_2$ is dehomogenized to $f(x, y) = x^2 + xy + y$ by setting $x_0 = 1$
• The dehomogenization of a projective variety $V(F_1, \ldots, F_s) \subseteq \mathbb{P}^n$ with respect to $x_0$ is the affine variety $V(F_1(1, x_1, \ldots, x_n), \ldots, F_s(1, x_1, \ldots, x_n)) \subseteq \mathbb{A}^n$

Affine vs Projective Representations

Converting Affine Varieties to Projective Varieties

• To convert an affine variety $V(f_1, \ldots, f_s) \subseteq \mathbb{A}^n$ to its projective closure, homogenize each polynomial $f_i$ to obtain $f_i^*$ and consider the projective variety $V(f_1^*, \ldots, f_s^*) \subseteq \mathbb{P}^n$
  • Example: The affine variety $V(x^2 + y^2 - 1) \subseteq \mathbb{A}^2$ has the projective closure $V(x^2 + y^2 - z^2) \subseteq \mathbb{P}^2$
• The projective closure of an affine variety $V \subseteq \mathbb{A}^n$ is the smallest projective variety in $\mathbb{P}^n$ containing $V$

Converting Projective Varieties to Affine Varieties

• To convert a projective variety $V(F_1, \ldots, F_s) \subseteq \mathbb{P}^n$ to its affine part with respect to $x_0$, dehomogenize each polynomial $F_i$ by setting $x_0 = 1$ and consider the affine variety $V(F_1(1, x_1, \ldots, x_n), \ldots, F_s(1, x_1, \ldots, x_n)) \subseteq \mathbb{A}^n$
  • Example: The projective variety $V(x^2 + y^2 - z^2) \subseteq \mathbb{P}^2$ has the affine part $V(x^2 + y^2 - 1) \subseteq \mathbb{A}^2$ with respect to $z$
• The affine part of a projective variety $V \subseteq \mathbb{P}^n$ with respect to $x_0$ is the intersection of $V$ with the affine space $\mathbb{A}^n$, obtained by setting $x_0 = 1$
• The affine part of a projective variety and the projective closure of an affine variety are related by the operations of homogenization and dehomogenization

Applications of Homogenization

Simplifying the Study of Affine Varieties

• Homogenization can simplify the study of affine varieties by working in the projective setting, where the geometry is more uniform and some computations are easier
  • Example: Bézout's theorem, which states that the number of intersection points of two plane curves (counting multiplicities) is equal to the product of their degrees, is more easily stated and proved in the projective setting
• Homogenization can determine the behavior of an affine variety at infinity by studying the added points in the projective closure
  • Example: The affine variety $V(xy - 1) \subseteq \mathbb{A}^2$ has two branches that approach the lines $x = 0$ and $y = 0$ at infinity, which can be seen in its projective closure $V(xy - z^2) \subseteq \mathbb{P}^2$

Applying Projective Results to Affine Varieties

• Dehomogenization allows the application of results obtained in the projective setting to affine varieties
  • Example: If a projective variety is irreducible, then its affine part is also irreducible
• Dehomogenization can analyze the local properties of a projective variety by considering its affine parts
  • Example: The singularities of a projective variety can be studied by examining the singularities of its affine parts
• Homogenization and dehomogenization can establish a correspondence between affine and projective varieties, enabling the transfer of properties and results between the two settings

Benefits of Projective Space

Uniform and Symmetric Setting

• Projective space provides a more uniform and symmetric setting for studying algebraic varieties, as it treats points at infinity on an equal footing with finite points
  • Example: In the projective plane, parallel lines always intersect at a point at infinity, whereas in the affine plane, they do not intersect
• Many geometric properties and results are simpler and more elegant in the projective setting, such as Bézout's theorem and the intersection theory of varieties

Compactness and Simplification

• Projective space is compact, which can simplify certain arguments and proofs involving algebraic varieties
  • Example: The compactness of projective space can be used to prove that every non-constant polynomial map between projective varieties is surjective
• Some computations, such as the computation of degrees and the application of resultants, are more straightforward in the projective setting
  • Example: The degree of a projective variety can be computed using the Hilbert polynomial, which is easier to work with than the degree of an affine variety

Analyzing Affine Varieties at Infinity

• Working in projective space allows for the study of the behavior of affine varieties at infinity, providing a more complete understanding of their geometry
  • Example: The projective closure of the affine variety $V(y - x^2) \subseteq \mathbb{A}^2$ contains an additional point at infinity, $(0:1:0)$, which corresponds to the vertical asymptote of the parabola
• Projective techniques, such as the use of homogeneous coordinates and the projective closure, can be used to analyze and solve problems involving affine varieties
  • Example: The intersection of two affine varieties can be computed by homogenizing their defining equations, computing the intersection of their projective closures, and then dehomogenizing the result