Quasi-Projective Varieties

In §Affine Varieties and §Projective Varieties, we examined geometric objects defined as subsets of affine space and projective space, respectively. However, the most natural objects in algebraic geometry belong to a larger category encompassing both. In this section, we define quasi-projective varieties and show that they include both affine and projective varieties. We also define morphisms between quasi-projective varieties and show that they agree with the existing notions.

Definition of Quasi-projective Variety

Open subsets of projective space are natural geometric objects. For instance, the standard open set \(U_0\) obtained by removing the line \(\x_0=0\) from \(\mathbb{P}^2\) is not a projective variety, but it is still an object defined by polynomials, and is even an affine variety.

Definition 1 An open subset \(X \subseteq Y\) of a projective variety \(Y \subseteq \mathbb{P}^n\) is called a quasi-projective variety.

Of course, \(X\) inherits the topology of \(Y\), and this topology is also called the Zariski topology. By definition, it is obvious that the notion of a quasi-projective variety includes all projective varieties. Our first proposition is that any affine variety is quasi-projective.

Proposition 2 Any affine variety is a quasi-projective variety.

Proof

Let an arbitrary affine variety \(X\subseteq \mathbb{A}^n\) be given. We already know that the embedding

\[i:\mathbb{A}^n\rightarrow \mathbb{P}^n;\qquad (x_1,\ldots, x_n)\mapsto [1:x_1:\cdots:x_n]\]

exists. (§Projective Varieties, ⁋Proposition 9) Now consider the image \(i(X)\) of \(X\) in \(\mathbb{P}^n\), and the closure \(\overline{i(X)}\) of \(i(X)\) in \(\mathbb{P}^n\). Then \(\overline{i(X)}\) is a projective variety, and within it, since \(i(X)\) is expressed as

\[i(X)=\overline{i(X)}\cap U_0\]

it is an open set. This completes the proof.

Unpacking the proof above, one easily sees that the Zariski topology defined on a quasi-projective variety coincides with the Zariski topology previously defined on an affine variety, and the same holds for projective varieties.

In general, in both the affine and projective cases, open subsets of a given variety more often fail to be affine or projective varieties. Quasi-projective varieties form a much broader category than these, and the following holds.

Proposition 3 An open subset of a quasi-projective variety \(X\) is a quasi-projective variety. Also, an irreducible closed subset of \(X\) is a quasi-projective variety.

Proof

Suppose \(X\) is an open subset of a projective variety \(Y\subseteq \mathbb{P}^n\). Since an open subset of \(X\) is also an open subset of \(Y\), it is obvious that any open subset of \(X\) is a quasi-projective variety. Thus it suffices to show that any irreducible closed subset of \(X\) is quasi-projective. To this end, write

\[X=Y\cap U,\qquad \text{$U$ open in $\mathbb{P}^n$}\]

and suppose an arbitrary irreducible closed subset of \(X\), say \(Z\), is given. Now consider the closure of \(Z\) in \(\mathbb{P}^n\), \(\overline{Z}\); since \(Z\) is irreducible, \(\overline{Z}\) is also an irreducible closed set. Moreover, since \(Z\subseteq X\subseteq Y\) and \(Y\) is a closed subset of \(\mathbb{P}^n\), we have \(\overline{Z}\subseteq Y\), and therefore \(\overline{Z}\) is itself a projective variety. On the other hand, since \(Z\) is a closed subset of \(X\), for some closed subset of \(\mathbb{P}^n\), \(C\), we have \(Z=X\cap C\), and then from \(Z\subseteq X\cap \overline{Z}\subseteq X\cap C=Z\) we obtain

\[Z=X\cap \overline{Z}=(Y\cap U)\cap \overline{Z}=\overline{Z}\cap U\]

From this, \(Z\) is an open subset of the projective variety \(\overline{Z}\).

Regular Functions and Regular Maps

Henceforth, unless stated otherwise, by a variety we shall always mean a quasi-projective variety. Our geometric intuition is that any point of a variety \(X\) always has an affine open neighborhood. Since this was already proved in each of the affine and projective cases, we need only extend it to the quasi-projective case.

Proposition 4 For any variety \(X\) and any \(x\in X\), there exists an open covering of \(X\) consisting of affine varieties.

Proof

First, since \(X\) is quasi-projective, there exists a projective variety \(Y\subseteq \mathbb{P}^n\) such that \(X\) is an open subset of \(Y\). Now \(X\) can be covered by the sets \(X\cap U_i\) using standard affine charts, where each \(X\cap U_i\) is an open subset of the affine variety \(Y\cap U_i\). (§Projective Varieties, ⁋Proposition 10) Now, by §Affine Varieties, ⁋Proposition 6, any open subset of an affine variety can be covered by principal open sets, and these are affine by §Affine Varieties, ⁋Proposition 7, which completes the proof.

Now by the above proposition, we can make the following definition.

Definition 5 For a quasi-projective variety \(X\), a function \(f: X \rightarrow \mathbb{K}\) is called regular if there exists an open affine cover of \(X\), \(\{U_i\}\), such that for each \(i\), the restriction

\[f\vert_{U_i}:U_i\rightarrow\mathbb{K}\]

is an element of the coordinate ring of the affine variety \(U_i\), \(\mathbb{K}[U_i]\). The sheaf of all regular functions on \(X\) is denoted \(\mathcal{O}_X\), or more simply \(\mathcal{O}\). ([Topology] §Sheaves, ⁋Definition 1)

Then the following are examples of regular functions.

Example 6 Let us examine some examples of regular functions.

1. On an affine variety \(X\), we have \(\mathcal{O}(X) = \mathbb{K}[X]\). In Definition 5, we can choose \(X\) itself as an open affine cover, so \(\mathbb{K}[X]\subseteq \mathcal{O}(X)\), and the reverse inclusion is the fact that a function written as a rational expression in a neighborhood of each point is again an element of the coordinate ring, that is, the equivalence of §Affine Varieties, ⁋Definition 11 and §Affine Varieties, ⁋Definition 14.

2. On \(\mathbb{P}^n\), we have \(\mathcal{O}(\mathbb{P}^n) = \mathbb{K}\). To verify this, consider the standard open cover \(U_i = \{x_i \ne 0\}\). In particular, a regular function on \(U_0\) is an element of \(\mathbb{K}[\x_1/\x_0, \ldots, \x_n/\x_0]\), and a regular function on \(U_1\) is an element of \(\mathbb{K}[\x_0/\x_1, \x_2/\x_1, \ldots, \x_n/\x_1]\). Thus, if a function \(f\) is regular on all of \(\mathbb{P}^n\), then on \(U_0\) this function is a polynomial in the \(\mathrm{s}_i=\x_i/\x_0\), and on \(U_1\) it is a polynomial in \(\mathrm{t}_i=\x_i/\x_1\). However, on \(U_0\cap U_1\), we know that these coordinate functions satisfy the equations

   \[\mathrm{t}_0=\frac{1}{\mathrm{s}_1},\qquad \mathrm{t}_j=\frac{\mathrm{s}_j}{\mathrm{s}_1}\quad (j=2,\ldots, n)\]

   Therefore, if

   \[f\vert_{U_0}=p(\mathrm{s}_1, \ldots, \mathrm{s}_n),\qquad f\vert_{U_1}=q(\mathrm{t}_0,\mathrm{t}_2,\ldots, \mathrm{t}_n)\]

   then, by the fact that they must define the same function on \(U_0\cap U_1\) and by the above equations, we know that

   \[p(\mathrm{s}_1,\ldots, \mathrm{s}_n)=q\left(\frac{1}{\mathrm{s}_1},\frac{\mathrm{s}_2}{\mathrm{s}_1}, \ldots,\frac{\mathrm{s}_n}{\mathrm{s}_1}\right)\]

   must hold. Now, for the expression on the right-hand side to be a polynomial, \(q\) must be a constant function so that the \(\mathrm{s}_1\) in the denominators disappear, and from this we know that \(p\) and \(q\) must be constant functions. Similarly, applying the same argument to all charts \(U_i,U_j\) yields the desired result.

Now we define morphisms between varieties, that is, regular maps. There are several ways to define this, but for now, given a variety we always assume a projective space containing it, and since morphisms between projective spaces have already been defined, we use this to define them as follows. (§Projective Varieties, ⁋Definition 15)

Definition 7 A function \(\varphi:X \rightarrow Y\) between two varieties \(X \subseteq \mathbb{P}^n\) and \(Y \subseteq \mathbb{P}^m\) is called a morphism (or regular map) if for every \(x\in X\) there exist an open neighborhood \(U \subseteq X\) of \(x\) and homogeneous polynomials \(F_0, \ldots, F_m\) of the same degree such that

\[\varphi(q) = [F_0(q) : \cdots : F_m(q)] \in \mathbb{P}^m\]

holds for all \(q \in U\).

So far, when discussing varieties we have assumed an embedding into affine space or an embedding into projective space, and the above definition is also an extension of this assumption. Through this we can perform concrete calculations, but this definition can hardly be called intrinsic. The following proposition shows that this definition is naturally interpreted from the perspective of regular functions.

Proposition 8 A continuous map \(\varphi: X \rightarrow Y\) is a morphism if and only if for every affine open set of \(Y\), \(V\), and every regular function \(f \in \mathcal{O}_Y(V)\), \(f \circ \varphi: \varphi^{-1}(V) \rightarrow \mathbb{K}\) is a regular function.

The heart of the proof is that a morphism is locally expressed by homogeneous polynomials, and since a regular function on \(Y\) can (considering the dehomogenization process) be written, for homogeneous polynomials of the same degree \(d\), \(F,G\), in the form \(F/G\), their composition must also be a regular function. Then, using this, we can show the following.

Proposition 9 Regular maps between affine varieties are exactly the regular maps when these are regarded as quasi-projective varieties.

This is essentially a consequence of §Affine Varieties, ⁋Definition 15.

Properties of Regular Maps

Proposition 10 A regular map \(\varphi: X \rightarrow Y\) is continuous.

Proof

Let \(C \subseteq Y\) be a closed set. Then there exist, in \(\mathbb{K}[\x_0, \ldots, \x_m]\), homogeneous polynomials \(G_1, \ldots, G_r\) such that \(C = Y \cap Z(G_1, \ldots, G_r)\), and since the image of \(\varphi\) is contained in \(Y\), we have \(\varphi^{-1}(C) = \varphi^{-1}(Z(G_1, \ldots, G_r))\).

Now for any \(x \in X\), choose an open neighborhood \(U \subseteq X\) and homogeneous polynomials \(F_0, \ldots, F_m\) of the same degree given by Definition 7. Since on \(U\) we have \(\varphi = [F_0 : \cdots : F_m]\),

\[\varphi^{-1}(C) \cap U = \{q \in U \mid G_1(F_0(q), \ldots, F_m(q)) = \cdots = G_r(F_0(q), \ldots, F_m(q)) = 0\}\]

and each \(G_k(F_0, \ldots, F_m)\) is again a homogeneous polynomial in \(\x_0, \ldots, \x_n\), so the right-hand side is the intersection of a closed subset of \(\mathbb{P}^n\) and \(U\). That is, \(\varphi^{-1}(C) \cap U\) is closed in \(U\), and since such \(U\) cover \(X\), \(\varphi^{-1}(C)\) is closed in \(X\).

Proposition 11 The composition of regular maps is a regular map. The identity map is a regular map.

Proof

Let \(\varphi: X \rightarrow Y\) and \(\psi: Y \rightarrow Z\) be regular maps. If \(W \subseteq Z\) is an open set and \(f \in \mathcal{O}(W)\), then since \(\psi\) is regular, \(f \circ \psi \in \mathcal{O}(\psi^{-1}(W))\). Now since \(\varphi\) is regular, \((f \circ \psi) \circ \varphi \in \mathcal{O}(\varphi^{-1}(\psi^{-1}(W)))\). That is,

\[f \circ (\psi \circ \varphi) \in \mathcal{O}((\psi \circ \varphi)^{-1}(W))\]

so \(\psi \circ \varphi\) is a regular map.

For the identity map \(\id_X: X \rightarrow X\), since \(f \circ \id_X = f\), it is trivially a regular map.

Thus quasi-projective varieties and regular maps form a category.

Proposition 12 The restriction of a regular map to a closed subset is a regular map. The restriction of a regular map to an open subset is also a regular map.

Proof

Let \(\varphi: X \rightarrow Y\) be a regular map and let \(Z \subseteq Y\) be a closed subset. Consider \(\psi = \varphi\vert_{\varphi^{-1}(Z)}: \varphi^{-1}(Z) \rightarrow Z\). Since \(\varphi\) is continuous by Proposition 10, \(\psi\) is also continuous. Now if \(f\) is a regular function on an open subset \(V\) of \(Z\), then \(f\) extends to a regular function on an open subset of \(Y\) in a neighborhood of each point of \(V\). Specifically, choosing an affine open set \(W\) of \(Y\) containing a point of \(V\), \(W\cap Z\) is a closed subset of the affine variety \(W\), so regular functions on it are restrictions of elements of \(\mathbb{K}[W]\) (Example 6), and since \(f\) is written in a neighborhood of that point as a quotient of such functions, it suffices to lift the numerator and denominator to \(\mathbb{K}[W]\). Then \(f \circ \psi = (f \circ \varphi)\vert_{\varphi^{-1}(Z)}\), and since whether a function is regular is determined locally, from the fact that \(f \circ \varphi\) is regular on these neighborhoods we obtain that \(f\circ\psi\) is also regular.

The case of an open subset is simpler. If \(U \subseteq Y\) is an open subset, then whenever \(f\) is regular on \(V \subseteq U\), \(f \circ \varphi\) is regular on \(\varphi^{-1}(V)\).

Definition 13 A morphism \(\varphi: X \rightarrow Y\) is an isomorphism if there exists an inverse map \(\psi: Y \rightarrow X\) such that \(\psi\) is also a morphism.

The notion of isomorphism means geometrically that two varieties are the same. That is, isomorphic varieties cannot be distinguished from the perspective of regular functions.

Example 14 We have already examined morphisms between projective varieties and morphisms between affine varieties separately, so functions linking projective and affine varieties are new. For instance, the canonical surjection

\[\mathbb{A}^{n+1}\setminus\{(0,\ldots, 0)\}\rightarrow \mathbb{P}^n;\qquad (x_0,\ldots, x_n)\mapsto [x_0:\cdots:x_n]\]

is a morphism between quasi-projective varieties. Also, the canonical inclusion

\[\iota_i:\mathbb{A}^n\hookrightarrow \mathbb{P}^n\]

is also a morphism between quasi-projective varieties.

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References

[Har] J. Harris, Algebraic Geometry: A First Course, Springer, 1992.
[Sha] I. R. Shafarevich, Basic Algebraic Geometry I: Varieties in Projective Space, Springer, 2013.