Irreducible Subsets of Projective Space

Throughout this article, k is an algebraically closed field.

We wish to consider irreducible closed subsets of $\mathbb P^n_k$ . For that we need the following preliminary result.

Lemma 1.
Let $A$ be a graded ring; a proper homogeneous ideal $\mathfrak p \subsetneq A$ is prime if and only if:
$a, b \in A \text{ homogeneous}, ab \in \mathfrak p \implies a \in \mathfrak p \text{ or } b\in\mathfrak p$ .

Proof

Suppose $\mathfrak p$ is not prime so there exists $a, b\in A-\mathfrak p$ such that $ab\in \mathfrak p$ . Since $a\not\in \mathfrak p$ , among all homogeneous components of a pick $a_d$ of maximum degree such that $a_d \not\in\mathfrak p$ ; similarly pick $b_e$ fo b so $b_e \not\in\mathfrak p$ .

The degree-(d+e) component of ab is congruent to $a_d b_e$ mod $\mathfrak p$ . Since $\mathfrak p$ is homogeneous we have $a_d b_e \in \mathfrak p$ , and by the given condition this means $a_d \in \mathfrak p$ or $b_e \in \mathfrak p$ , a contradiction. ♦

Exercise A

Prove that a proper homogeneous ideal $\mathfrak q$ of a graded ring A is primary if and only if

$a, b \in A \text{ homogeneous }, ab\in\mathfrak q \implies a \in \mathfrak q \text{ or } b\in r(\mathfrak q)$ .

Proposition 1.
Suppose the closed subset $V\subseteq \mathbb P^n_k$ corresponds to the homogeneous radical ideal $\mathfrak a \subseteq B$ , $\mathfrak a \ne B_+$ , where $B = k[T_0, \ldots, T_n]$ .
Then V is irreducible if and only if $\mathfrak a$ is prime.

Proof

(⇒) Suppose V is irreducible; let $\mathfrak a = I_0(V)$ . If $f, g \in B - I_0(V)$ are homogeneous, then $C := V_0(\mathfrak a + fB)$ and $D := V_0(\mathfrak a + gB)$ are closed subsets of $\mathbb P^n_k$ properly contained in V. Since V is irreducible the following shows $fg\not\in\mathfrak a$ :

$V \supsetneq C \cup D = V_0(\mathfrak a + fB) \cup V_0(\mathfrak a + gB) = V_0((\mathfrak a + fB)(\mathfrak a + gB)) \supseteq V_0(\mathfrak a + fgB)$ .

(⇐) Let $V = V_0(\mathfrak p)$ where $\mathfrak p$ is prime. Let $C, D\subseteq V$ be closed subsets with union V. Now write $C = V_0(\mathfrak a)$ and $D = V_0(\mathfrak b)$ for homogeneous radical ideals $\mathfrak a$ and $\mathfrak b$ . Then $V = C\cup D = V_0(\mathfrak a \cap \mathfrak b)$ . Since $\mathfrak a\cap \mathfrak b$ is a homogeneous radical ideal, $\mathfrak p = \mathfrak a \cap \mathfrak b$ . By exercise B here, $\mathfrak a = \mathfrak p$ or $\mathfrak b = \mathfrak p$ . ♦

Corollary 1.
Let $V\subseteq \mathbb P^n_k$ be a non-empty closed subset. Then $V$ is irreducible if and only if $\mathrm{cone}(V)$ is irreducible.

Proof

By proposition 1, V is irreducible if and only if $I_0(V)$ is prime. But $I_0(V) = I(\mathrm{cone}(V))$ (from an exercise here) so the result follows. ♦

Quasi-projective Varieties

Recall that a projective variety is a closed subset of some $\mathbb P^n_k$ .

Definition.
A quasi-projective variety is an open subset of a projective variety.

This merely defines it as a set; we need a geometric structure on it.

First, let $F_0, \ldots, F_m \in k[T_0, \ldots, T_n]$ be homogeneous polynomials of the same degree. If $\mathbf v \in \mathbb P^n$ is such that not all $F_i(\mathbf v) = 0$ , then we can define a function on an open subset U of $\mathbb P^n$ containing $\mathbf v$ as follows:

$U \longrightarrow \mathbb P^m,\quad \mathbf v' \mapsto ( F_0(\mathbf v') : F_1(\mathbf v') : \ldots : F_m(\mathbf v') )$ .

The map is well-defined: indeed if $F_i(\mathbf v) \ne 0$ we can find an open neighbourhood U of $\mathbf v$ such that $0 \not\in F_i(U)$ . Also, if we replace projective coordinates $(t_0 : \ldots : t_n)$ with $(\lambda t_0 : \ldots : \lambda t_n)$ , then each $F_i(\lambda t_0, \ldots, \lambda t_n) = \lambda^d F_i(t_0, \ldots, t_n)$ where $d = \deg F_i$ so

$\begin{aligned} (F_0(\lambda t_0, \ldots, \lambda t_n) : \ldots : F_m(\lambda t_0, \ldots, \lambda t_n)) &= (\lambda^d F_0(t_0, \ldots, t_n) : \ldots: \lambda^d F_m(t_0, \ldots, t_n)) \\ &= (F_0(t_0, \ldots, t_n) : \ldots : F_m(t_0, \ldots, t_n)).\end{aligned}$

We write $(F_0 : \ldots : F_m) : U\to \mathbb P^m$ for the resulting function.

Definition.
Let $V \subseteq \mathbb P^n$ and $W \subseteq \mathbb P^m$ be quasi-projective varieties and $\phi : V\to W$ be a function.
We say $\phi$ is regular at $\mathbf v \in V$ if there is an open neighbourhood U of $\mathbf v$ in V such that
$\phi|_U = (F_0 : \ldots : F_m)$ for some homogeneous $F_i \in k[T_0, \ldots, T_n]$ of the same degree.
We say $\phi$ is regular if it is regular at every $\mathbf v\in V$ , in which case we also say $\phi : V\to W$ is a morphism of quasi-projective varieties.

From the above definitions, we obtain the category of all quasi-projective varieties and morphisms between them.

Example 1

First consider the case where $V\subseteq \mathbb A^n$ and $W\subseteq \mathbb A^m$ are closed subsets.

E.g., let $V\subseteq \mathbb A^3$ . A regular map $\phi : V\to \mathbb A^1$ in the earlier sense can be expressed as a polynomial $f(X, Y, Z)$ , e.g. take $f = X^3 - Y^2 + 3Z$ . Via embeddings $\mathbb A^3 \hookrightarrow \mathbb P^3$ and $\mathbb A^1 \hookrightarrow \mathbb P^1$ taking $(x, y, z) \mapsto (1:x:y:z)$ and $t \mapsto (1:t)$ respectively, f can be written in terms of homogeneous coordinates as

$(T_0 : T_1 : T_2 : T_3) \mapsto (T_0^3 : T_1^3 - T_2^2 T_0 + 3 T_3 T_0^2)$

since it is the homogenization of the map $(\frac{T_1}{T_0}, \frac{T_2}{T_0}, \frac{T_3}{T_0}) \mapsto (\frac{T_1}{T_0})^3 - (\frac{T_2}{T_0})^2 + 3(\frac{T_3}{T_0})$ . This generalizes to an arbitrary regular map of closed subsets $\phi : (V\subseteq \mathbb A^n) \to (W \subseteq \mathbb A^m)$ .

Conversely we have:

Lemma 2.
Let $\phi :V\to W$ be regular under the new definition. Then there exist polynomials $f_1, \ldots, f_m \in k[X_1, \ldots, X_n]$ which represent $\phi$ .

Proof

We will prove this for the case where V is irreducible.

For each of $1\le i\le m$ , let $\pi_i : \mathbb A^m \to \mathbb A^1$ be projection onto the i-th coordinate. Then $\pi_i \circ \phi : V \to \mathbb A^1$ is regular under the new definition, and by proposition 2 here (and its preceding discussion) $\pi_i \circ \phi$ can be represented as a polynomial $f_i(X_1, \ldots, X_n)$ . Hence we see that

$\phi(\mathbf v) = (f_1 (\mathbf v), \ldots, f_m(\mathbf v))$ for polynomials $f_1, \ldots, f_m \in k[X_1, \ldots, X_n]$ . ♦

Example 2

Take the map $\phi : \mathbb P^1 \to \mathbb P^3$ given by

$\phi : (T_0 : T_1) \mapsto (T_0^3 : T_0^2 T_1 : T_0 T_1^2 : T_1^3)$

Note that the same set of polynomials $(F_0, F_1, F_2, F_3)$ works globally over the whole of $\mathbb P^1$ .

Example 3

Suppose $\mathrm{char} k \ne 2$ . Let $V\subset \mathbb P^2$ be the closed subset defined by $T_0^2 = T_1^2 + T_2^2$ . We define a map $\phi : V \to \mathbb P^1$ as follows

Outside the point (1 : 1 : 0), take $(T_0 : T_1 : T_2) \mapsto (T_0 - T_1 : T_2)$ .
Outside the point (1 : -1 : 0), take $(T_0 : T_1 : T_2) \mapsto (T_2 : T_0 + T_1)$ .

p1_cover

The map agrees outside those two points since $(T_0 - T_1 : T_2) = (T_2 : T_0 + T_1)$ due to the equality $T_0^2 = T_1^2 + T_2^2$ .

Isomorphisms

Definition.
Consider the category of all quasi-projective k-varieties, with morphisms defined as above. Two such varieties are said to be isomorphic if they are isomorphic in the category.
A quasi-projective variety is said to be
projective if it is isomorphic to a closed subset of some $\mathbb P^n_k$ (this generalizes the existing definition of projective varieties);
affine if it is isomorphic to an affine k-variety (closed subspace of some $\mathbb A^n$ );
quasi-affine if it is isomorphic to an open subset of an affine variety.

Example 4

In example 3 above, we get an isomorphism $\phi : V\to \mathbb P^1$ since we have the reverse map

$\mathbb \psi : \mathbb P^1 \to V, \quad (U_0 : U_1) \mapsto (U_0^2 + U_1^2 : U_1^2 - U_0^2 : 2U_0 U_1)$ .

As an exercise, prove that $\phi\circ \psi = 1_{\mathbb P^1}$ and $\psi\circ \phi = 1_V$ .

Definition.
The coordinate ring of a quasi-projective variety V is the set
$k[V] := \{ f : V\to \mathbb A^1 : f \text{ regular } \}$ ,
taken to be a k-algebra via point-wise addition and multiplication:
$f, g : V\to\mathbb A^1 \text{ regular } \implies \begin{cases} (f+g) :V \to \mathbb A^1, \ &\mathbf v \mapsto f(\mathbf v) + g(\mathbf v), \\ (fg) : V\to \mathbb A^1, \ &\mathbf v \mapsto f(\mathbf v)g(\mathbf v). \end{cases}$

Note

As before, a regular map $\phi:V\to W$ of quasi-projective varieties induces a ring homomorphism $\phi^* : k[W] \to k[V]$ . By lemma 2, when V is affine $k[V]$ agrees with our earlier version (we proved this in the case where V is irreducible).

Example 5

For each $g\in GL_{n+1}(k)$ , we have an automorphism

$\phi_g : \mathbb P^n_k \longrightarrow \mathbb P^n_k, \quad (t_0 : \ldots : t_n) \mapsto (\sum_{j=0}^n g_{0j} t_j : \ldots : \sum_{j=0}^n g_{nj} t_j).$

Note that $\phi_{gh} = \phi_g \circ \phi_h$ for $g, h \in GL_{n+1}(k)$ . Also $\phi_g = 1$ if and only if g is a scalar multiple of the identity matrix, so we get an injective homomorphism $PGL_{n+1}(k) = GL_{n+1}(k)/k^* \hookrightarrow \mathrm{Aut} \mathbb P^n_k$ . In fact this is an isomorphism of groups.

E.g. when n = 1, we get the Möbius transformations:

$\left[\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in GL_2 k \right] : (t_0 : t_1) \mapsto (at_0 + bt_1 : ct_0 + dt_1).$

Example 6

We have an isomorphism between the quasi-affine variety $\mathbb A^1 - \{0\}$ and $V = \{(x,y) \in \mathbb A^2 : xy = 1\}$ via the maps

$\mathbb A^1 - \{0\} \to V, \ x \mapsto (x, \frac 1 x), \quad V \to \mathbb A^1 - \{0\}, \ (x, y) \mapsto x.$

Hence $\mathbb A^1-\{0\}$ is an affine variety even though it is not closed in $\mathbb A^1$ . From the isomorphism we also have:

$k[\mathbb A^1 - \{0\}] \cong k[V] = k[X, Y]/(XY - 1) \implies k[\mathbb A^1 - \{0\}] = k[X, \frac 1 X].$

Example 7

Let $V = \mathbb A^2 -\{(0, 0)\}$ . We will show that V is not affine. Indeed consider the injection $\phi : V \hookrightarrow \mathbb A^2$ which induces

$\phi^* : k[X, Y] \cong k[\mathbb A^2] \longrightarrow k[V]$ .

The map is injective since V is dense in $\mathbb A^2$ . Let us show that it is surjective. Suppose $f \in k[V]$ so that $f: V \to \mathbb A^1$ is regular. Write

$V = U \cup U',$ where $U = (\mathbb A^1 - \{0\}) \times \mathbb A^1, \ U' = \mathbb A^1 \times (\mathbb A^1 - \{0\})$ .

By example 6, we have $f|_U \in k[U] \cong k[X, Y, \frac 1 X]$ and $f|_{U'} \in k[U']\cong k[X, Y, \frac 1 Y]$ . Since $U, U', U\cap U'$ are all dense in V we have injections $k[V] \to k[U] \to k[U\cap U']$ and $k[V] \to k[U'] \to k[U\cap U']$ so that $f \in k[X, Y, \frac 1 X] \cap k[X, Y, \frac 1 Y]$ . It is easy to show that this means $f\in k[X, Y]$ .

Hence $\phi$ induces an isomorphism of the coordinate rings $k[\mathbb A^2] \to k[V]$ . If V is affine, by proposition 1 here $\phi$ would be an isomorphism of varieties, which is a contradiction since $\phi$ is not surjective.

Commutative Algebra 62

Irreducible Subsets of Projective Space