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.

Definition of Projective Space

We now define projective varieties, another important class of algebraic varieties. We begin with the following.

Definition 1 We define the projective \(n\)-space \(\mathbb{P}^n_{\mathbb{K}}\) over a field \(\mathbb{K}\) as follows. As a set,

\[\mathbb{P}^n = (\mathbb{K}^{n+1} \setminus \{0\}) / \sim\]

where the equivalence relation \(\sim\) is given by

\[(x_0, \ldots, x_n) \sim (y_0, \ldots, y_n) \iff \text{$x_i = \lambda y_i$ for some $\lambda \in \mathbb{K}^\times$, for all $i$}\]

When there is no risk of confusion, we write simply \(\mathbb{P}^n\).

An equivalence class \([(x_0, \ldots, x_n)]\) is usually denoted by \([x_0 : \cdots : x_n]\), and these are called homogeneous coordinates. The \(x_0, \ldots, x_n\) are called coordinates, and at least one of them must be nonzero. The key point of homogeneous coordinates is that they determine only a ratio. That is, for every \(\lambda \in \mathbb{K}^\times\), we have \([x_0 : \cdots : x_n] = [\lambda x_0 : \cdots : \lambda x_n]\).

Homogeneous Polynomials and Projective Space

As in the affine case, we must now equip \(\mathbb{P}^n\) with a topology. Again, we will define closed sets as zero sets of polynomials, but care is needed: since \(\mathbb{P}^n\) is defined as a quotient set, a polynomial does not in general define a function on \(\mathbb{P}^n\). That is, for arbitrary \(F \in \mathbb{K}[x_0, \ldots, x_n]\), although \([x_0 : \cdots : x_n] = [\lambda x_0 : \cdots : \lambda x_n]\), in general

\[F(x_0, \ldots, x_n) \neq F(\lambda x_0, \ldots, \lambda x_n)\]

and if \(\mathbb{K}\) is infinite, the only polynomial whose evaluation at every point of \(\mathbb{P}^n\) is well-defined independent of the representative is a constant polynomial. However, if we are only interested in the zero set defined by a polynomial, this problem is resolved. For a homogeneous polynomial \(F\) of degree \(d\),

\[F(\lambda x_0, \ldots, \lambda x_n) = \lambda^d F(x_0, \ldots, x_n)\]

so

\[F(\lambda x_0, \ldots, \lambda x_n) = 0 \iff F(x_0, \ldots, x_n) = 0\]

Therefore, the zero set of a homogeneous polynomial is well defined on projective space.

Definition 2 A polynomial \(F \in \mathbb{K}[\x_0, \ldots, \x_n]\) is said to be homogeneous of degree \(d\) if for all \(\lambda \in \mathbb{K}\),

\[F(\lambda \x_0, \ldots, \lambda \x_n) = \lambda^d F(\x_0, \ldots, \x_n)\]

Although we have made the definition appear complicated, if \(\mathbb{K}\) is infinite, this is essentially the condition that when the polynomial is expressed as a sum of monomials, every monomial has degree \(d\). We may then define the following.

Definition 3 Given homogeneous polynomials \(F_1, \ldots, F_k \in \mathbb{K}[\x_0, \ldots, \x_n]\), we define the projective algebraic set \(Z(F_1, \ldots, F_k)\) by

\[Z(F_1, \ldots, F_k) = \{[x_0 : \cdots : x_n] \in \mathbb{P}^n \mid F_1(x) = \cdots = F_k(x) = 0\}\]

Among projective algebraic sets, those that cannot be expressed as a union of finitely many strictly smaller projective algebraic sets are called projective varieties.

As explained above, since each \(F_i\) is homogeneous, one verifies that this is well defined.

Meanwhile, we recall that when dealing with affine varieties, it sufficed to consider only ideals rather than arbitrary subsets of \(\mathbb{K}[\x_1, \ldots, \x_n]\). In the projective case, the same philosophy applies, but with the additional hypothesis of homogeneity, leading to the notion of a homogeneous ideal.

Definition 4 An ideal \(\mathfrak{a} \subseteq \mathbb{K}[\x_0, \ldots, \x_n]\) is said to be homogeneous if \(\mathfrak{a}\) is generated by homogeneous polynomials.

For a homogeneous ideal \(\mathfrak{a}\), if we define its zero set \(Z(\mathfrak{a})\) as the set of points where all homogeneous polynomials in \(\mathfrak{a}\) vanish, then as in the affine case, we can define the Zariski topology. For this, we need the following proposition.

Proposition 5 For homogeneous ideals \(\mathfrak{a}, \mathfrak{b}, \mathfrak{a}_i \subseteq \mathbb{K}[\x_0, \ldots, \x_n]\), the following hold:

1. \(Z(0) = \mathbb{P}^n\), \(Z(1) = \emptyset\),
2. \(\bigcap_i Z(\mathfrak{a}_i) = Z\left(\sum_i \mathfrak{a}_i\right)\),
3. \(Z(\mathfrak{a}) \cup Z(\mathfrak{b}) = Z(\mathfrak{a} \cap \mathfrak{b}) = Z(\mathfrak{a}\mathfrak{b})\).

Proof

The only difference from §Affine Varieties, ⁋Proposition 4 is that the polynomials considered here are all homogeneous, but the proof itself is identical, so we omit it.

As in the affine case, this shows that there exists a topology on projective space \(\mathbb{P}^n\) whose closed sets are the projective algebraic sets, and we may endow each projective variety with the induced subspace topology. We call this topology the Zariski topology. (We first examined the Zariski topology in the affine case in §Affine Varieties.)

Projective Nullstellensatz

Definition 6 The homogeneous ideal of a subset \(X \subseteq \mathbb{P}^n\) is defined to be the ideal generated by

\[\{F \in \mathbb{K}[\x_0, \ldots, \x_n] \mid F \text{ is homogeneous and } F(x) = 0 \text{ for all } x \in X\}\]

and is denoted by \(I(X)\).

Theorem 7 (Projective Nullstellensatz) Let \(\mathbb{K}\) be an algebraically closed field and let \(\mathfrak{a} \subseteq \mathbb{K}[\x_0, \ldots, \x_n]\) be a homogeneous ideal. Then

1. \(Z(\mathfrak{a}) = \emptyset \iff \sqrt{\mathfrak{a}} \supseteq (\x_0, \ldots, \x_n)\),
2. \(I(Z(\mathfrak{a})) = \sqrt{\mathfrak{a}}\) (if \(Z(\mathfrak{a}) \ne \emptyset\)).

The difference from the affine case is that \(Z(\mathfrak{a}) = \emptyset\) does not mean \(\mathfrak{a} = (1)\), but rather that \(\sqrt{\mathfrak{a}}\) contains the irrelevant ideal \((\x_0, \ldots, \x_n)\). This is because \((\x_0, \ldots, \x_n)\) corresponds to the origin of \(\mathbb{K}^{n+1}\), which is excluded from the definition of projective space.

Standard Affine Cover

Projective space \(\mathbb{P}^n\) can be covered by \(n+1\) copies of affine space. This is one of the most important ways to understand projective space.

Definition 8 For \(i = 0, 1, \ldots, n\), we define the \(i\)-th standard open set \(U_i\) by

\[U_i = \{[x_0 : \cdots : x_n] \in \mathbb{P}^n \mid x_i \ne 0\}\]

Endow each \(U_i\) with the subspace topology inherited from \(\mathbb{P}^n\). Then the following holds.

Proposition 9 Each \(U_i\) is homeomorphic to affine space \(\mathbb{A}^n\) (in the subspace topology).

Proof

For notational convenience, we treat the case \(i=0\). Define the map \(\varphi_0: U_0 \rightarrow \mathbb{A}^n\) by

\[\varphi_0([x_0 : x_1 : \cdots : x_n]) = \left(\frac{x_1}{x_0}, \ldots, \frac{x_n}{x_0}\right)\]

The inverse map \(\psi_0: \mathbb{A}^n \rightarrow U_0\) is given by

\[\psi_0(a_1, \ldots, a_n) = [1 : a_1 : \cdots : a_n]\]

That these are mutual inverses is obvious from the definitions. We now show that both \(\varphi_0\) and \(\psi_0\) are continuous.

First, to show the continuity of \(\varphi_0\), consider a closed set \(Z(f)\) in \(\mathbb{A}^n\). Then

\[\varphi_0^{-1}(Z(f)) = \left\{[x_0 : \cdots : x_n] \in U_0 \mid f\left(\frac{x_1}{x_0}, \ldots, \frac{x_n}{x_0}\right) = 0\right\}\]

Now if \(f\) is a polynomial of degree \(d\), then

\[F(\x_0,\ldots, \x_n)=\x_0^d f(\x_1/\x_0, \ldots, \x_n/\x_0)\]

is a homogeneous polynomial, and \(\varphi_0^{-1}(Z(f)) = Z(F) \cap U_0\). This is a closed set in the subspace topology on \(U_0\).

Now we show the continuity of the inverse \(\psi_0\). Consider a closed set \(Z(F) \cap U_0\) in \(U_0\), where \(F\) is a homogeneous polynomial of degree \(d\). Then

\[\psi_0^{-1}(Z(F) \cap U_0) = \{(x_1, \ldots, x_n) \in \mathbb{A}^n \mid F(1, x_1, \ldots, x_n) = 0\}\]

Since \(F(1, \x_1, \ldots, \x_n)\) is a polynomial in \(\mathbb{K}[\x_1, \ldots, \x_n]\), the set \(\psi_0^{-1}(Z(F) \cap U_0)\) is closed in \(\mathbb{A}^n\).

Therefore, since \(\varphi_0\) and \(\psi_0\) are mutual inverses and both are continuous, \(\varphi_0\) is a homeomorphism.

Intuitively, we may think of \(U_i\) as the set of points where the coordinate \(x_i\) is not at infinity. Also, \(\mathbb{P}^n = U_0 \cup \cdots \cup U_n\), and by the above proposition each \(U_i \cong \mathbb{A}^n\). Since the key ingredient in the proof above was the following proposition, we separate it out.

Proposition 10 For a projective variety \(X \subseteq \mathbb{P}^n\) and a standard open set \(U_i\), if \(X \cap U_i\) is nonempty, then it is an affine variety in \(U_i \cong \mathbb{A}^n\).

Proof

For notational convenience, we treat the case \(i=0\). Let \(X = Z(F_1, \ldots, F_k)\) with each \(F_j\) homogeneous of degree \(d_j\). Set

\[f_j(\x_1, \ldots, \x_n) = F_j(1, \x_1, \ldots, \x_n)\]

Then by the homogeneity of \(F_j\), for any point with \(x_0 \ne 0\),

\[F_j(x_0, x_1, \ldots, x_n) = x_0^{d_j} f_j\left(\frac{x_1}{x_0}, \ldots, \frac{x_n}{x_0}\right)\]

Hence \([x_0 : \cdots : x_n] \in U_0\) belonging to the zero set of all \(F_j\) is equivalent to the point \((x_1/x_0, \ldots, x_n/x_0)\) in \(\mathbb{A}^n\) (the image of this point under the homeomorphism \(\varphi_0\) of Proposition 9) belonging to the zero set of all \(f_j\), and from this we obtain \(\varphi_0(X \cap U_0) = Z(f_1, \ldots, f_k) \subseteq \mathbb{A}^n\).

It remains to verify irreducibility. Since \(U_0\) is an open subset of \(\mathbb{P}^n\), \(X \cap U_0\) is a nonempty open subset of \(X\) by assumption. A nonempty open subset of an irreducible space is irreducible, so \(X \cap U_0\) is irreducible, and since a homeomorphism preserves irreducibility, \(Z(f_1, \ldots, f_k)\) is also irreducible. Thus \(X \cap U_0\) is an affine variety.

Example 11 To interpret the above proposition geometrically, let \(\mathbb{K}=\mathbb{R}\) and consider the conic \(X = Z(\x_0^2 + \x_1^2 - \x_2^2)\) in \(\mathbb{P}^2\).

This conic is the cone \(\x_0^2 + \x_1^2 = \x_2^2\) in \(\mathbb{A}^3\) expressed in homogeneous coordinates. We can see how \(X\) looks in the standard open sets from Proposition 10. That is, to see what \(X\) looks like in \(U_i\), we simply substitute \(1\) for \(\x_i\) and regard the remaining \(n\) variables as coordinates on \(\mathbb{A}^n\). In particular, we obtain the following:

1. In \(U_0\) and \(U_1\), the conic \(X\) is the hyperbolas \(1+y^2-z^2=0\) and \(x^2+1-z^2=0\).
2. In \(U_2\), the conic \(X\) is the circle \(x^2+y^2=1\).

This happens because the equation \(\x_0^2 + \x_1^2 = \x_2^2\) in \(\mathbb{A}^3\) defines a cone, and its traces cut by the planes \(\x_0=1\), \(\x_1=1\), \(\x_2=1\) become hyperbolas and a circle.

On the other hand, we can also interpret this directly in \(\mathbb{P}^2\). To do so, we construct \(\mathbb{P}^2\) as follows. For points with \(\x_2 \neq 0\), we radially project onto the upper hemisphere satisfying \(\x_2 > 0\), and for points with \(\x_2 = 0\), we identify antipodal points. Through this, we may think of \(\mathbb{P}^2\) as the “line at infinity” \(\mathbb{P}^1\) together with the plane \(\mathbb{A}^2\) corresponding to the surface of the upper hemisphere. Then the given cone first becomes a circle contained in the upper hemisphere via the radial projection, and from this we see that \(X\) appears as a circle in \(\mathbb{P}^2\).

Of course, we could have constructed \(\mathbb{P}^2\) by radially projecting points satisfying \(\x_0 \neq 0\) onto the upper hemisphere with \(\x_0 > 0\), and taking points with \(\x_0 = 0\) as \(\mathbb{P}^1\). In this process, two semicircles would be drawn on the upper hemisphere, but the boundary points of these two semicircles would be identified as the same in the process of identifying points with \(\x_0 = 0\), so in this picture too \(X\) becomes a circle.

From this perspective, viewing \(X\) in \(U_i\) corresponds to removing the line at infinity \(\x_i = 0\) from \(\mathbb{P}^2\). If we view \(X\) in \(U_2\), then as we saw above, \(X\) does not meet the line at infinity \(\x_2 = 0\), so removing this line leaves a complete circle. However, if for example we remove the line at infinity \(\x_1 = 0\), then \(X\) meets this line at two points, and so removing these two points from the circle \(X\) and unfolding it yields a hyperbola.

Affine Cone

The preceding example shows how to view a curve on projective space in each affine open chart, but one may still find it somewhat unintuitive. Another way to understand a projective variety as a geometric object in affine space is to consider its affine cone.

Definition 12 The affine cone \(C(X) \subseteq \mathbb{A}^{n+1}\) of a projective variety \(X \subseteq \mathbb{P}^n\) is defined as follows:

\[C(X) = \{(x_0, \ldots, x_n) \in \mathbb{A}^{n+1} \setminus \{0\} \mid [x_0 : \cdots : x_n] \in X\} \cup \{0\}\]

That is, \(C(X)\) is the union of the points in \(\mathbb{A}^{n+1}\) that appear when all points of \(X\) are expressed in homogeneous coordinates, together with the origin.

Example 13 The affine cone \(C(X)\) of the conic \(X = Z(\x_0^2 + \x_1^2 - \x_2^2) \subseteq \mathbb{P}^2\) from Example 11 is the cone \(\x_0^2 + \x_1^2 = \x_2^2\) in \(\mathbb{A}^3\).

The following then holds, and the proofs are not difficult either.

Proposition 14 The affine cone \(C(X)\) of a projective variety \(X \subseteq \mathbb{P}^n\) satisfies the following properties:

1. (Homogeneity) \(C(X)\) consists of lines passing through the origin. That is, if \((x_0, \ldots, x_n) \in C(X)\) and \(\lambda \in \mathbb{K}\), then \((\lambda x_0, \ldots, \lambda x_n) \in C(X)\).

2. (Algebraic structure) If \(X = Z(F_1, \ldots, F_k)\), then \(C(X) = Z(F_1, \ldots, F_k) \subseteq \mathbb{A}^{n+1}\). Here the \(F_i\) are regarded as polynomials on \(\mathbb{A}^{n+1}\).

3. (Correspondence) The correspondence \(X \leftrightarrow C(X)\) gives a one-to-one correspondence between projective varieties and nonzero irreducible affine algebraic sets consisting of lines passing through the origin.

Through this proposition, we can indirectly understand properties of \(X\) by studying properties of the affine cone \(C(X)\).

Morphisms of Projective Varieties

Finally, we define morphisms of projective varieties. Earlier, when we defined projective algebraic sets, we saw that polynomials do not in general define functions on projective space; a similar phenomenon occurs when defining morphisms, and the solution is again homogeneous polynomials.

Definition 15 A function \(\varphi: X \rightarrow Y\) is called a morphism between projective varieties \(X \subseteq \mathbb{P}^n\) and \(Y \subseteq \mathbb{P}^m\) if for each point \(x \in X\), there exists an open subset \(U\) of \(X\) containing \(x\) and suitable homogeneous polynomials \(F_0, \ldots, F_m \in \mathbb{K}[\x_0, \ldots, \x_n]\) of the same degree such that for all \(y \in U\),

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

If \(F_0, \ldots, F_m\) are all homogeneous polynomials of the same degree \(d\), then since \(F_i(\lambda x) = \lambda^d F_i(x)\),

\[[F_0(\lambda x) : \cdots : F_m(\lambda x)] = [\lambda^d F_0(x) : \cdots : \lambda^d F_m(x)] = [F_0(x) : \cdots : F_m(x)]\]

so one verifies that well-definedness is guaranteed. On the other hand, as in §Affine Varieties, ⁋Definition 17, if the inverse map \(\psi: Y \rightarrow X\) of a morphism \(\varphi: X \rightarrow Y\) exists and is also a morphism, then we call \(\varphi\) an isomorphism, and two projective varieties for which such a \(\varphi\) exists are said to be isomorphic to each other. The following examples are representative morphisms.

Example 16 First, the Veronese embedding (of degree 2) from \(\mathbb{P}^1\) to \(\mathbb{P}^2\) defined by

\[[x:y] \mapsto [x^2: xy: y^2]\]

is a morphism between projective spaces. As another example, the Segre embedding from \(\mathbb{P}^1 \times \mathbb{P}^1\) to \(\mathbb{P}^3\) is given by the formula

\[([x:y], [u:v]) \mapsto [xu: xv: yu: yv]\]

However, since we have not yet defined the product of two projective varieties, we postpone verifying that this actually defines a morphism to later.

Example 17 Twisted cubic in \(\mathbb{P}^3\)

\[C = \{[1 : t : t^2 : t^3] \mid t \in \mathbb{K}\} \cup \{[0 : 0 : 0 : 1]\}\]

is the common zero locus of the three quadratic polynomials

\[\x_0 \x_2 - \x_1^2, \quad \x_0 \x_3 - \x_1 \x_2, \quad \x_1 \x_3 - \x_2^2\]

and is isomorphic to \(\mathbb{P}^1\). In fact, extending the concept of the Veronese embedding examined in Example 16 to \(d=3\),

\[[x:y] \mapsto [x^3: x^2y: xy^2: y^3]\]

becomes an isomorphism from \(\mathbb{P}^1\) to \(C\).

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References

[Har] J. Harris, Algebraic Geometry: A First Course, Springer, 1992.
[Sha] I. R. Shafarevich, Basic Algebraic Geometry I: Zarieties in Projective Space, Springer, 2013.
[Ful] W. Fulton, Algebraic Curves, 2008.