Quasi-Projective Varieties

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

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 verify 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 in fact it is 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, \(i(X)\) is an open subset because

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

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, for both the affine and projective cases, open subsets of a given variety more often fail to be affine or projective varieties themselves. Quasi-projective varieties form a much broader category, 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 let an arbitrary irreducible closed subset \(Z\) of \(X\) be given. Now consider the closure \(\overline{Z}\) of \(Z\) in \(\mathbb{P}^n\); since \(Z\) is irreducible, \(\overline{Z}\) is also an irreducible closed set. Moreover, since \(Z\subseteq X\subseteq Y\) and \(Y\) is closed in \(\mathbb{P}^n\), we have \(\overline{Z}\subseteq Y\), and therefore \(\overline{Z}\) is itself a projective variety. On the other hand, since \(Z\) is closed in \(X\), we have \(Z=X\cap C\) for some closed subset \(C\) of \(\mathbb{P}^n\), and then from \(Z\subseteq X\cap \overline{Z}\subseteq X\cap C=Z\) we obtain

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

Thus \(Z\) 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 \(X\) has an affine open neighborhood. This was already proved in the affine and projective cases, so 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 \(X\) by 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, and each \(X\cap U_i\) is an open subset of the affine variety \(Y\cap U_i\). (§Projective Varieties, ⁋Proposition 10) 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 A function \(f: X \rightarrow \mathbb{K}\) on a quasi-projective variety \(X\) is called regular if there exists an open affine cover \(\{U_i\}\) of \(X\) such that for each \(i\), the restriction

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

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

Here are some examples of regular functions.

Example 6 Let us examine some examples of regular functions.

1. For an affine variety \(X\), we have \(\mathcal{O}(X) = \mathbb{K}[X]\). In Definition 5, we may choose \(X\) itself as the open affine cover, so \(\mathbb{K}[X]\subseteq \mathcal{O}(X)\), and the reverse inclusion follows from the fact that a function given locally by a rational expression 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 \(U_0\) it is a polynomial in \(\mathrm{s}_i=\x_i/\x_0\), 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 relations

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

   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 the requirement that these define the same function on \(U_0\cap U_1\), together with the above relations, implies 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 right-hand side to be a polynomial, \(q\) must be a constant function so that the denominators involving \(\mathrm{s}_1\) disappear, and from this we see that \(p\) and \(q\) must be constant functions. 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 do this, but we shall always assume that a variety is embedded in some projective space, and since morphisms between projective spaces have already been defined, we use this to define morphisms 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 projective space, and the above definition also lies in the same vein. Through this we can perform concrete calculations, but this definition can hardly be called intrinsic. The following proposition shows that this definition admits a natural interpretation 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 \(V\) of \(Y\) and every regular function \(f \in \mathcal{O}_Y(V)\), the composition \(f \circ \varphi: \varphi^{-1}(V) \rightarrow \mathbb{K}\) is a regular function.

The heart of the proof is that a morphism is expressed locally by homogeneous polynomials, and regular functions on \(Y\) can be written in the form \(F/G\) for homogeneous polynomials \(F, G\) of the same degree \(d\) (considering the dehomogenization process), so their composition must also be a regular function. 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 homogeneous polynomials \(G_1, \ldots, G_r\) in \(\mathbb{K}[\x_0, \ldots, \x_m]\) such that \(C = Y \cap Z(G_1, \ldots, G_r)\), and since the image of \(\varphi\) lies in \(Y\), we have \(\varphi^{-1}(C) = \varphi^{-1}(Z(G_1, \ldots, G_r))\).

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

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

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 \(\mathbb{P}^n\) with \(U\). That is, \(\varphi^{-1}(C) \cap U\) is closed in \(U\), and since such \(U\) cover \(X\), we conclude that \(\varphi^{-1}(C)\) 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 open and \(f \in \mathcal{O}(W)\), then since \(\psi\) is regular, we have \(f \circ \psi \in \mathcal{O}(\psi^{-1}(W))\). 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 \(\id_X: X \rightarrow X\), we have \(f \circ \id_X = f\), so 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 \(\psi = \varphi\vert_{\varphi^{-1}(Z)}: \varphi^{-1}(Z) \rightarrow Z\). By Proposition 10 \(\varphi\) is continuous, so \(\psi\) 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 \(V\), we have that \(W\cap Z\) 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 \(f\) is given locally as a quotient of such functions, so we simply lift numerator and denominator to \(\mathbb{K}[W]\). Then \(f \circ \psi = (f \circ \varphi)\vert_{\varphi^{-1}(Z)}\), and since regularity is determined locally, the regularity of \(f \circ \varphi\) on these neighborhoods implies that \(f\circ\psi\) is also regular.

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

Definition 13 A morphism \(\varphi: X \rightarrow Y\) is called 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 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.