Affine and Projective Space
Definition (Affine Space).
Let $k$ be a field. The affine $n$-space over $k$, denoted $\mathbb{A}^n(k)$, is defined as:
A polynomial $f \in k[x_1, \ldots, x_n]$ defines a vanishing set (or affine algebraic set):
\[V_a(f) = \{ (x_1, \ldots, x_n) \in \mathbb{A}^n(k) \mid f(x_1, \ldots, x_n) = 0 \}\]Definition (Projective Space).
The projective space $\mathbb{P}^n(k)$ is the set of lines through the origin in $k^{n+1}$. That is,
where two nonzero vectors $(x_0, \ldots, x_n) \sim (\lambda x_0, \ldots, \lambda x_n)$ for $\lambda \in k^\times$ are considered equivalent. A point in projective space is written in homogeneous coordinates:
\[(x_0 : x_1 : \cdots : x_n)\]Affine Charts and Hyperplanes at Infinity
Projective space $\mathbb{P}^n(k)$ can be covered by affine charts, each corresponding to setting one of the coordinates $x_i \ne 0$. For example, the chart $\{x_0 \ne 0\}$ identifies projective points with affine points via:
\[(x_0 : x_1 : \ldots : x_n) \mapsto \left( \frac{x_1}{x_0}, \ldots, \frac{x_n}{x_0} \right)\]This yields an isomorphism:
\[\mathbb{A}^n(k) \cong \{ x_0 \ne 0 \} \subset \mathbb{P}^n(k)\]The hyperplane at infinity $H$ is defined by $H = \{x_0 = 0\}$. It “completes” affine space by adding points “at infinity,” making the geometry more symmetric.
We observe that $H \simeq \mathbb{P}^{n-1}(k)$, and the full projective space decomposes as:
\[\mathbb{P}^n(k) = \mathbb{A}^n(k) \cup H\]Examples.
- Points in $\mathbb{P}^1(k)$ have coordinates $(x : y)$, not both zero, modulo scaling.
- Affine chart $\{y \neq 0\}$: we may use the representative $(x : 1)$, corresponding to $x \in \mathbb{A}^1(k)$
- Point at infinity: $\infty = (1 : 0)$
Thus, we have $\mathbb{P}^1(k) = \mathbb{A}^1(k) \cup \{ \infty \}$
- Points in $\mathbb{P}^2(k)$ are given by $(x : y : z)$, not all zero, modulo scaling.
- Affine chart $\{z \neq 0\}$: write $(x/z, y/z, 1)$, giving coordinates in $\mathbb{A}^2(k)$
- Line at infinity: $\{(x:y:0)\in\mathbb{P}^2(k)\}\simeq\mathbb{P}^1(k)$
So we view: $\mathbb{P}^2(k) = \mathbb{A}^2(k) \cup \mathbb{P}^1(k)$
Homogeneous Polynomials and Homogenization
Definition (Homogeneous Polynomial).
A homogeneous polynomial $f(x_0, \ldots, x_n)$ of degree $d$ is a polynomial in which every monomial term has total degree $d$. It satisfies the scaling relation:
Such a polynomial defines a projective vanishing set:
\[V_p(f) = \{ (x_0 : \cdots : x_n) \in \mathbb{P}^n(k) \mid f(x_0, \ldots, x_n) = 0 \}\]This is well-defined: if $f$ is homogeneous and $f(\vec{x}) = 0$, then $f(\lambda \vec{x}) = \lambda^d f(\vec{x}) = 0$, so the vanishing condition depends only on the class $(x_0 : \cdots : x_n)$.
Definition (Homogenization).
Given a (non-homogeneous) polynomial $f(x_1, \ldots, x_n)$ of total degree $d$, its homogenization is defined as:
This yields a homogeneous polynomial of degree $d$ in the variables $x_0, \ldots, x_n$.
Setting $x_0 = 1$ recovers the original polynomial $f(x_1, \ldots, x_n)$ — this process is called dehomogenization. In terms of vanishing sets, we have:
\[V_p(f^h) \cap \mathbb{A}^n(k) = V_a(f)\]Lines in $\mathbb{P}^2(k)$
A line in $\mathbb{P}^2(k)$ is defined by a degree-1 homogeneous polynomial:
\[ax + by + cz = 0\]- If at least one of $a$ or $b$ is nonzero, the line intersects the affine plane $\mathbb{A}^2(k)$ (in the chart $z \ne 0$) at the affine line $ax + by + c = 0$, and extends to a point at infinity.
- If $a = b = 0$, then the equation reduces to $cz = 0$, which defines the line at infinity, and does not intersect the affine chart ${z \ne 0}$.
Any two distinct lines in $\mathbb{P}^2(k)$ intersect at exactly one point. Non-parallel lines intersect in the affine chart, while parallel lines intersect at a point on the line at infinity.