3.4 Projective closure and homogenization

Projective closure and homogenization are key concepts in bridging affine and projective geometry. They allow us to extend affine varieties into projective space, adding points at infinity to study their global structure and behavior.

By homogenizing equations, we can transform affine varieties into projective ones. This process reveals hidden properties, like asymptotes and singularities, and enables us to apply powerful tools from projective geometry to analyze affine varieties more comprehensively.

Projective closure of an affine variety

Definition and properties

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• The projective closure of an affine variety $V$ is the smallest projective variety $\bar{V}$ containing $V$ as a dense open subset
• Obtained by embedding the affine variety into projective space and taking the closure in the Zariski topology
• Adds points at infinity to the original affine variety, completing it in a sense (compactification)
• The projective closure is unique up to isomorphism and is independent of the choice of embedding
• Allows for the study of the global structure and behavior of the affine variety

Relationship between affine and projective varieties

• Every affine variety can be embedded into a projective space, and its projective closure can be obtained
• The original affine variety is a dense open subset of its projective closure
• The complement of the affine variety in its projective closure consists of the points at infinity
• The projective closure provides a natural way to compactify an affine variety and study its behavior at infinity

Homogenization of defining equations

Process of homogenization

• To compute the projective closure, first homogenize the defining equations of the affine variety by introducing a new variable $x_0$
• Homogenization is done by multiplying each monomial in the equations by an appropriate power of $x_0$ to make all terms have the same total degree
• The resulting homogeneous equations define a projective variety, which is the projective closure of the original affine variety
• The original affine variety can be recovered by setting $x_0 = 1$ in the homogeneous equations (dehomogenization)

Examples of homogenization

• For the affine variety defined by $x^2 + y^2 = 1$, the homogenized equation is $x^2 + y^2 = z^2$
• The affine variety defined by $xy = 1$ has the homogenized equation $xy = z^2$
• The affine variety defined by $y = x^2$ has the homogenized equation $yz = x^2$

Geometric meaning of points at infinity

Interpretation of points at infinity

• The points at infinity added in the projective closure correspond to the directions in which the affine variety "goes off to infinity"
• These points at infinity can be thought of as the limit points of the affine variety when viewed from increasingly distant vantage points
• The behavior of the variety near the points at infinity can provide information about its global structure and asymptotic properties
• The points at infinity can be studied by looking at the intersection of the projective closure with the hyperplane at infinity defined by $x_0 = 0$

Examples of points at infinity

• For the affine variety defined by $y = x^2$, there is a single point at infinity $[0:1:0]$, corresponding to the vertical direction
• The affine variety defined by $xy = 1$ has two points at infinity: $[1:0:0]$ and $[0:1:0]$, corresponding to the horizontal and vertical directions
• The affine variety defined by $x^2 + y^2 = 1$ has no real points at infinity, but has two complex points at infinity $[i:1:0]$ and $[-i:1:0]$

Homogenization for studying varieties at infinity

Applications of homogenization

• Homogenization allows for the study of the behavior of affine varieties near their points at infinity
• By analyzing the homogeneous defining equations, one can determine the multiplicity and structure of the points at infinity
• The intersection multiplicity of the projective closure with the hyperplane at infinity can provide information about the singularities and tangent directions of the variety at infinity
• Homogenization techniques can be used to study the degree and genus of projective curves, as well as the intersection theory of projective varieties

Importance of studying varieties at infinity

• The study of varieties at infinity using homogenization is particularly useful in understanding the global topology and geometry of the varieties
• The behavior at infinity can reveal important properties such as asymptotes, branches, and singularities
• Studying the points at infinity can provide insight into the overall structure and shape of the variety
• Homogenization allows for the application of powerful tools from projective geometry to study affine varieties in a more complete and unified setting