1.2 Projective varieties and homogeneous coordinates

Projective varieties and homogeneous coordinates extend our geometric toolbox beyond affine space. They let us study points at infinity and handle parallel lines intersecting, giving us a more complete picture of algebraic curves and surfaces.

This topic builds on affine varieties, introducing projective space and homogeneous polynomials. We'll see how to convert between affine and projective varieties, and explore the projective closure of affine varieties for a unified view of geometry.

Projective space and its properties

Definition and notation

• Projective space is an extension of affine space that allows for the inclusion of points at infinity, providing a more complete geometric setting for studying algebraic varieties
• The n-dimensional projective space over a field $K$, denoted as $P^n(K)$ or $P^n$, is the set of equivalence classes of $(n+1)$-tuples $(x_0, ..., x_n)$ of elements of $K$, not all zero, under the equivalence relation $(x_0, ..., x_n) \sim (\lambda x_0, ..., \lambda x_n)$ for all nonzero $\lambda$ in $K$
  • The elements of projective space are called points, and the equivalence classes are called homogeneous coordinates
  • Example: In $P^2(\mathbb{R})$, the points $(1, 2, 3)$, $(2, 4, 6)$, and $(-1, -2, -3)$ are all equivalent and represent the same point in projective space

Properties and relationship to affine space

• Projective space has the property that parallel lines intersect at a point at infinity, unlike in affine space where parallel lines never intersect
  • This allows for a more unified treatment of geometric objects and their intersections
• Projective space is covered by affine spaces $U_i = \{[x_0, ..., x_n] : x_i \neq 0\}$, each isomorphic to the affine space $A^n$, allowing for the study of local properties of projective varieties
  • Example: In $P^2$, the affine spaces $U_0 = \{[1, x_1, x_2]\}$, $U_1 = \{[x_0, 1, x_2]\}$, and $U_2 = \{[x_0, x_1, 1]\}$ cover the projective plane
• The hyperplane at infinity in $P^n$ is defined by the equation $x_0 = 0$ and consists of the points added to the affine space $A^n$ to create the projective space $P^n$
  • These points at infinity represent the directions of lines and planes in affine space

Projective varieties with polynomials

Homogeneous polynomials and their zero sets

• A polynomial $f(x_0, ..., x_n)$ is homogeneous of degree $d$ if $f(\lambda x_0, ..., \lambda x_n) = \lambda^d f(x_0, ..., x_n)$ for all nonzero $\lambda$ in the base field $K$
  • Example: The polynomial $f(x, y, z) = x^2 + y^2 - z^2$ is homogeneous of degree 2
• A projective variety is the zero set of a collection of homogeneous polynomials in projective space
• The zero set of a collection of homogeneous polynomials $\{f_a\}$ is the set $V(\{f_a\}) = \{[x_0, ..., x_n] \in P^n : f_a(x_0, ..., x_n) = 0 \text{ for all } a\}$
  • Example: The zero set of the homogeneous polynomial $f(x, y, z) = x^2 + y^2 - z^2$ in $P^2$ is the projective conic $V(f) = \{[x, y, z] \in P^2 : x^2 + y^2 - z^2 = 0\}$

Homogeneous ideals and their varieties

• The ideal generated by a collection of homogeneous polynomials $\{f_a\}$ is the set $I(\{f_a\}) = \{\sum g_a f_a : g_a \text{ are homogeneous polynomials}\}$, which is a homogeneous ideal
• The projective variety $V(I)$ associated to a homogeneous ideal $I$ is the zero set of all polynomials in $I$
  • Example: If $I = \langle x^2, xy \rangle$ in $K[x, y, z]$, then $V(I) = \{[0, 0, 1], [0, 1, 0]\}$ in $P^2$
• The ideal-variety correspondence states that for any projective varieties $V$ and $W$, we have $V \subseteq W$ if and only if $I(W) \subseteq I(V)$

Affine vs Projective varieties

Converting affine varieties to projective varieties

• An affine variety $V(f_1, ..., f_m)$ in $A^n$ can be homogenized to a projective variety $V(f_1^h, ..., f_m^h)$ in $P^n$ by introducing a new variable $x_0$ and homogenizing each polynomial $f_i$ to $f_i^h$
• To homogenize a polynomial $f(x_1, ..., x_n)$ of degree $d$, multiply each monomial by an appropriate power of $x_0$ to make the total degree of each term equal to $d$, resulting in the homogeneous polynomial $f^h(x_0, ..., x_n)$
  • Example: The affine variety $V(xy - 1)$ in $A^2$ can be homogenized to the projective variety $V(xy - z^2)$ in $P^2$
• The affine variety $V(f_1, ..., f_m)$ is isomorphic to the intersection of the projective variety $V(f_1^h, ..., f_m^h)$ with the affine space $U_0 = \{[1, x_1, ..., x_n]\}$

Converting projective varieties to affine varieties

• To dehomogenize a projective variety $V(f_1, ..., f_m)$ in $P^n$ to an affine variety in $A^n$, intersect $V$ with the affine space $U_0$ and dehomogenize each polynomial $f_i$ by setting $x_0 = 1$
  • Example: The projective variety $V(xy - z^2)$ in $P^2$ can be dehomogenized to the affine variety $V(xy - 1)$ in $A^2$ by intersecting with $U_0 = \{[1, x, y]\}$
• Each affine space $U_i$ in the cover of $P^n$ provides a different affine view of the projective variety $V$, allowing for the study of its local properties

Projective closure of affine varieties

Definition and construction

• The projective closure of an affine variety $V$ is the smallest projective variety containing $V$, denoted as $\overline{V}$
• To find the projective closure of an affine variety $V(f_1, ..., f_m)$ in $A^n$, homogenize each polynomial $f_i$ to $f_i^h$ and take the zero set in $P^n$, i.e., $\overline{V} = V(f_1^h, ..., f_m^h)$ in $P^n$
  • Example: The projective closure of the affine variety $V(y - x^2)$ in $A^2$ is the projective variety $V(yz - x^2)$ in $P^2$

Points at infinity and compactification

• The intersection of the projective closure $\overline{V}$ with the hyperplane at infinity $\{x_0 = 0\}$ in $P^n$ gives the points at infinity of the affine variety $V$
  • These points represent the limiting behavior of the affine variety in different directions
  • Example: The point at infinity of the affine variety $V(y - x^2)$ is $[0, 0, 1]$, which corresponds to the vertical direction
• The projective closure allows for the study of the behavior of an affine variety "at infinity" and provides a compactification of the affine variety
  • Compactification is important in algebraic geometry as it allows for the application of powerful topological and analytical techniques to the study of varieties