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 . 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 V(x2+y2−1)⊆A2 can be homogenized to a projective variety V(x2+y2−z2)⊆P2
To homogenize a polynomial f(x1,…,xn), introduce a new variable x0 and multiply each monomial by an appropriate power of x0 to make the degree of all monomials equal to the degree of f
Example: The polynomial f(x,y)=x2+xy+y is homogenized to f∗(x0,x1,x2)=x12+x1x2+x0x2
The homogenization of an affine variety V(f1,…,fs)⊆An is the projective variety V(f1∗,…,fs∗)⊆Pn, where fi∗ is the homogenization of fi
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(x2+y2−z2)⊆P2 can be dehomogenized to an affine variety V(x2+y2−1)⊆A2 by setting z=1
To dehomogenize a F(x0,x1,…,xn), set one of the variables (usually x0) equal to 1 and simplify the resulting polynomial
Example: The homogeneous polynomial F(x0,x1,x2)=x12+x1x2+x0x2 is dehomogenized to f(x,y)=x2+xy+y by setting x0=1
The dehomogenization of a projective variety V(F1,…,Fs)⊆Pn with respect to x0 is the affine variety V(F1(1,x1,…,xn),…,Fs(1,x1,…,xn))⊆An
Affine vs Projective Representations
Converting Affine Varieties to Projective Varieties
To convert an affine variety V(f1,…,fs)⊆An to its projective closure, homogenize each polynomial fi to obtain fi∗ and consider the projective variety V(f1∗,…,fs∗)⊆Pn
Example: The affine variety V(x2+y2−1)⊆A2 has the projective closure V(x2+y2−z2)⊆P2
The projective closure of an affine variety V⊆An is the smallest projective variety in Pn containing V
Converting Projective Varieties to Affine Varieties
To convert a projective variety V(F1,…,Fs)⊆Pn to its affine part with respect to x0, dehomogenize each polynomial Fi by setting x0=1 and consider the affine variety V(F1(1,x1,…,xn),…,Fs(1,x1,…,xn))⊆An
Example: The projective variety V(x2+y2−z2)⊆P2 has the affine part V(x2+y2−1)⊆A2 with respect to z
The affine part of a projective variety V⊆Pn with respect to x0 is the intersection of V with the affine space An, obtained by setting x0=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)⊆A2 has two branches that approach the lines x=0 and y=0 at infinity, which can be seen in its projective closure V(xy−z2)⊆P2
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 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−x2)⊆A2 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
Key Terms to Review (12)
Affine coordinates: Affine coordinates are a system of coordinates used in affine geometry that define points in a space relative to a set of basis vectors. They allow for the representation of geometric objects and transformations without needing to consider distances or angles, focusing instead on the relationships between points. This is particularly useful in the context of algebraic geometry, especially during homogenization and dehomogenization, where the interplay between projective and affine spaces becomes crucial.
Affine variety: An affine variety is a subset of affine space that is defined as the common zero set of a collection of polynomials. It represents the solution set to polynomial equations, allowing for the study of geometric properties using algebraic techniques, and serves as a fundamental building block in algebraic geometry.
Algorithm for homogenization: An algorithm for homogenization is a systematic method used to convert a polynomial equation into a homogeneous polynomial by introducing an additional variable, typically denoted as 't'. This transformation allows the study of properties of the polynomial in projective space, making it easier to analyze solutions and their geometric interpretations, especially when dealing with intersection theory and algebraic varieties.
Computational methods: Computational methods are techniques and algorithms used to solve mathematical problems through numerical approximations and simulations rather than purely analytical solutions. These methods are essential for processing and analyzing complex mathematical structures, such as polynomials and geometric objects, in a way that is often more feasible than traditional symbolic computations. By utilizing computational methods, mathematicians can explore, visualize, and manipulate algebraic varieties, making them invaluable in the study of algebraic geometry.
Dual varieties: Dual varieties are geometric constructs that relate to the original varieties by representing the set of hyperplanes tangent to the original variety at all its points. This concept plays a crucial role in the context of homogenization and dehomogenization, connecting projective spaces to their duals and allowing for the analysis of properties such as tangent spaces and singularities. Understanding dual varieties enhances the study of intersection theory and helps in various applications of algebraic geometry.
Homogeneous Polynomial: A homogeneous polynomial is a polynomial whose terms all have the same total degree. This property allows it to have a consistent form when represented in projective space, enabling various applications in geometry, algebra, and computational methods.
Homogenization process: The homogenization process refers to the technique used in algebraic geometry to convert a polynomial into a homogeneous polynomial by introducing an additional variable, typically denoted as 't'. This transformation helps in studying the properties of the polynomial at infinity and allows for the application of projective geometry techniques.
Intersection Theory: Intersection theory is a branch of algebraic geometry that studies the intersection of algebraic varieties, focusing on the properties and dimensions of their intersections. It provides a framework to count and analyze how geometric objects intersect, which is essential for solving polynomial equations and understanding the structure of varieties. This theory connects algebraic concepts with geometric intuition, making it a powerful tool in various mathematical contexts.
Parametrization of curves: Parametrization of curves refers to the representation of a curve using a parameter, usually denoted as 't', which describes the coordinates of points on the curve in terms of this single variable. This method allows for a more flexible and comprehensive description of curves, making it easier to analyze and manipulate them mathematically. By converting a curve into parametric equations, it's possible to express not only geometric properties but also their behavior under transformations like homogenization and dehomogenization.
Projective Coordinates: Projective coordinates are a system of coordinates used in projective geometry that enable the representation of points in a projective space. Unlike traditional Cartesian coordinates, projective coordinates introduce a notion of 'points at infinity,' allowing for a unified treatment of parallel lines and facilitating the study of geometric properties invariant under projection. This concept is crucial when dealing with homogenization and dehomogenization processes, which transform coordinates to handle the complexities of projective spaces.
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.
Resultant computation: Resultant computation is a mathematical technique used to eliminate variables from a system of polynomial equations, helping to find solutions that satisfy all equations simultaneously. This process is significant in algebraic geometry for determining conditions under which polynomials share common roots, and it connects deeply with concepts like homogenization, where equations are transformed into a uniform degree, and the historical development of algebraic methods.