7.1 Projective space and homogeneous polynomials

Projective space and homogeneous polynomials are key concepts in algebraic geometry. They extend Euclidean space to include points at infinity, allowing for a more complete study of geometric properties. This framework is crucial for understanding projective varieties and their transformations.

Homogeneous polynomials have terms of equal total degree, making them invariant under scaling. This property is essential when working with projective coordinates, where points are defined up to scalar multiplication. Together, these concepts form the foundation for studying projective varieties and their relationships to affine spaces.

Projective space and coordinates

Definition and properties of projective space

• Projective space is an extension of Euclidean space that includes points at infinity
  • Allows for the study of geometric properties invariant under projective transformations
• The projective space of dimension n, denoted as $P^n$, is defined as the set of equivalence classes of $(n+1)$-tuples $(x_0, x_1, ..., x_n)$ of elements from a field $K$, not all zero
  • Under the equivalence relation $(x_0, x_1, ..., x_n) \sim (\lambda x_0, \lambda x_1, ..., \lambda x_n)$ for any non-zero $\lambda$ in $K$
• Elements of projective space are called points, and coordinates $(x_0, x_1, ..., x_n)$ are called homogeneous coordinates
• Projective space has the property that any two distinct points determine a unique line, and any two distinct lines intersect in a unique point

Coordinate system in projective space

• Points in projective space are represented by homogeneous coordinates $(x_0, x_1, ..., x_n)$
  • Coordinates are defined up to scalar multiplication by a non-zero element of the field $K$
• The equivalence class of $(x_0, x_1, ..., x_n)$ is denoted by $[x_0 : x_1 : ... : x_n]$
  • For example, in $P^2$, the points $[1 : 2 : 3]$, $[2 : 4 : 6]$, and $[-1 : -2 : -3]$ represent the same point
• The points with $x_0 = 0$ form the hyperplane at infinity, denoted by $H_\infty$
  • These points represent the directions in which lines and curves approach infinity

Homogeneous polynomials

Definition and properties of homogeneous polynomials

• A polynomial is called homogeneous if all of its terms have the same total degree
  • The total degree of a term is the sum of the exponents of its variables
• A homogeneous polynomial of degree $d$ in $n+1$ variables can be written as: $F(x_0, x_1, ..., x_n) = \sum a_{i_0, i_1, ..., i_n} x_0^{i_0} x_1^{i_1} ... x_n^{i_n}$, where $i_0 + i_1 + ... + i_n = d$
• Homogeneous polynomials are invariant under scaling of the variables
  • For any non-zero $\lambda$ in $K$, $F(\lambda x_0, \lambda x_1, ..., \lambda x_n) = \lambda^d F(x_0, x_1, ..., x_n)$

Examples of homogeneous polynomials

• The polynomial $F(x, y, z) = x^2 + y^2 - z^2$ is homogeneous of degree 2
• The polynomial $G(x, y, z) = x^3 - 3xyz$ is homogeneous of degree 3
• The polynomial $H(x, y, z) = x^2 + xy + z$ is not homogeneous, as its terms have different total degrees

Projective varieties

Definition and properties of projective varieties

• A projective variety is a subset of projective space that is the zero set of a collection of homogeneous polynomials
• Given a set of homogeneous polynomials ${F_1, F_2, ..., F_m}$ in $K[x_0, x_1, ..., x_n]$, the projective variety $V(F_1, F_2, ..., F_m)$ is defined as the set of points $P$ in $P^n$ such that $F_i(P) = 0$ for all $i = 1, 2, ..., m$
• The dimension of a projective variety is the maximum number of algebraically independent linear polynomials that vanish on the variety
• The degree of a projective variety is the number of points in the intersection of the variety with a generic linear subspace of complementary dimension

Examples of projective varieties

• The projective variety $V(xz - y^2) \subset P^2$ is a projective conic (a curve of degree 2)
• The projective variety $V(xy, xz, yz) \subset P^3$ is a set of three points: $[1 : 0 : 0]$, $[0 : 1 : 0]$, and $[0 : 0 : 1]$
• The projective variety $V(x^2 + y^2 + z^2) \subset P^2$ is empty, as the sum of squares is always non-negative and can only be zero if all variables are zero, which is not allowed in projective space

Affine vs Projective spaces

Relationship between affine and projective spaces

• Affine space is a vector space without a fixed origin, while projective space is an extension of affine space that includes points at infinity
• There is a natural embedding of affine space $A^n$ into projective space $P^n$, given by $(x_1, ..., x_n) \mapsto [1 : x_1 : ... : x_n]$
  • $[x_0 : x_1 : ... : x_n]$ denotes the equivalence class of $(x_0, x_1, ..., x_n)$ in projective space
• The complement of the image of $A^n$ in $P^n$ is called the hyperplane at infinity, denoted by $H_\infty$, and is defined by the equation $x_0 = 0$

Homogenization and dehomogenization of varieties

• Affine varieties can be homogenized to obtain projective varieties, and projective varieties can be dehomogenized to obtain affine varieties
• The homogenization of an affine variety $V(f_1, ..., f_m) \subset A^n$ is the projective variety $V(F_1, ..., F_m) \subset P^n$
  • $F_i$ is the homogenization of $f_i$ obtained by introducing a new variable $x_0$ and multiplying each term by a power of $x_0$ to make the polynomial homogeneous
• The dehomogenization of a projective variety $V(F_1, ..., F_m) \subset P^n$ with respect to $x_i$ is the affine variety $V(f_1, ..., f_m) \subset A^n$
  • $f_j$ is obtained from $F_j$ by setting $x_i = 1$
• For example, the affine variety $V(y - x^2) \subset A^2$ can be homogenized to obtain the projective variety $V(yz - x^2) \subset P^2$
  • Conversely, the projective variety $V(yz - x^2) \subset P^2$ can be dehomogenized with respect to $z$ to obtain the affine variety $V(y - x^2) \subset A^2$