Linear Systems

Previously, we defined a (Weil) divisor on a variety \(X\) in §Divisors, ⁋Definition 1. By the definition of the Zariski topology, this can essentially be thought of as the zero set of some function defined on \(X\), with the order of this zero added, and to define this well even in cases such as \(\mathbb{P}^n\), we generalized the function to a section of a suitable line bundle.

On the other hand, since divisors also allow negative coefficients, this zero set may not be a zero set, but rather a zero of negative order, that is, a pole. In such cases, we can find an effective divisor linearly equivalent to the given divisor and then investigate its properties. (§Divisors, ⁋Definition 7)

Above, for convenience of exposition, we have discussed only Weil divisors, but a similar argument can be made for Cartier divisors, and the resulting definition is as follows.

Definition 1 On a variety \(X\), a Weil divisor \(D=\sum n_i D_i\) is effective if for all \(i\), \(n_i\geq 0\). A Cartier divisor \(\{(U_i, f_i)\}\) is effective if for all \(i\), \(f_i\) is regular on \(U_i\).

Our goal, then, is to examine what effective divisors exist in the divisor class of a divisor \(D\). To this end, for the divisor \(D\), consider the line bundle \(\mathcal{L}=\mathcal{O}_X(D)\) it defines. (§Line Bundles and Vector Bundles, ⁋Definition 17) For \(\mathcal{L}\), each nonzero global section \(s\in \Gamma(X, \mathcal{L})\) has no poles and thus defines an effective divisor \(\divisor(s)\), and since we can verify that this differs from the original \(D\) only by a trivialization, it is linearly equivalent to \(D\). In other words, to find an effective divisor linearly equivalent to \(D\), it suffices to look at the nonzero global sections of \(\mathcal{O}_X(D)\). However, a point to note is that \(\divisor(s)\) does not depend on a nonzero scalar multiple of \(s\), and for this reason, the object we should be interested in is not \(\Gamma(X, \mathcal{L})\) itself but its projectivization.

Definition 2 On a variety \(X\), for a line bundle \(\mathcal{L}\), the complete linear system of \(\mathcal{L}\) is the projectivization of the global section space of \(\mathcal{L}\), \(\Gamma(X, \mathcal{L})\):

\[\lvert \mathcal{L} \rvert = \mathbb{P}(\Gamma(X, \mathcal{L}))\]

A linear system for \(\mathcal{L}\) is a nonempty projective subspace of \(\lvert \mathcal{L} \rvert\). That is, for a subspace \(V \subseteq \Gamma(X, \mathcal{L})\), it is of the form \(\mathbb{P}(V) \subseteq \lvert \mathcal{L} \rvert\).

Linear systems on projective space

Earlier, by the calculation following §Line Bundles and Vector Bundles, ⁋Example 16, we saw that \(\Gamma(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(d))\) is isomorphic to the space of degree \(d\) homogeneous polynomials \(\mathbb{K}[\x_0, \ldots, \x_n]_d\). Since each element of this space defines in \(\mathbb{P}^n\) a degree \(d\) hypersurface, we can understand the complete linear system of \(\mathcal{O}_{\mathbb{P}^n}(d)\)

\[\lvert \mathcal{O}_{\mathbb{P}^n}(d)\rvert=\mathbb{P}(\Gamma(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(d)))\cong \mathbb{P}(\mathbb{K}[\x_0,\ldots, \x_n]_d)\cong \mathbb{P}^{\binom{n+d}{d} - 1}\]

geometrically as a family of degree \(d\) hypersurfaces in \(\mathbb{P}^n\).

Example 3 For convenience, fix \(n=2\). Then the family of degree \(1\) hypersurfaces, that is, lines in \(\mathbb{P}^2\), is isomorphic to \(\mathbb{P}^2\) itself. Looking more closely, in

\[\lvert \mathcal{O}_{\mathbb{P}^2}(1)\rvert\cong \mathbb{P}(\mathbb{K}[\x_0,\x_1,\x_2]_1)\cong \mathbb{P}^{\binom{3}{1}-1}=\mathbb{P}^2\]

a point of the right-hand side \(\mathbb{P}^2\), \([a_0:a_1:a_2]\), defines in \(\mathbb{P}^2\) a line \(Z(a_0\x_0+a_1\x_1+a_2\x_2)\).

For a slightly more complicated and geometric example, in the complete linear system defined on \(\mathbb{P}^2\) by the line bundle \(\mathcal{O}_{\mathbb{P}^2}(2)\),

\[\lvert \mathcal{O}_{\mathbb{P}^2}(2)\rvert\cong \mathbb{P}(\mathbb{K}[\x_0,\x_1,\x_2]_2)\cong \mathbb{P}^{\binom{4}{2}-1}=\mathbb{P}^5\]

let us consider a 1-dimensional subspace (that is, a \(1\)-dimensional linear system). A typical example is the pencil of two conics. On \(\mathbb{P}^2\), consider two conics \(C_1=Z(F_1)\), \(C_2=Z(F_2)\). Then, as long as \(C_1\) and \(C_2\) are not the same conic, their linear combination

\[Z(\lambda F_1+\mu F_2)\]

is another curve in \(\mathbb{P}^2\) of degree \(2\), and by definition all these conics pass through \(C_1\cap C_2\). Here, the point \([\lambda:\mu]\in \mathbb{P}^1\) parametrizes these conics.

For a more concrete example, consider the two conics in \(\mathbb{P}^2\)

\[C_1: Z((\x_0-2\x_2)^2+\x_1^2-9\x_2^2),\qquad C_2: Z((\x_0+2\x_2)^2+\x_1^2-9\x_2^2)\]

These expressions look complicated in themselves, but when restricted to \(U_2\), they are the equations of two circles

\[(\x-2)^2+\y^2=9,\qquad (\x+2)^2+\y^2=9\]

On \(U_2\), these meet at \((x,y)=(0,\pm\sqrt{5})\), computed from the two equations above, and outside \(U_2\) — that is, on the line at infinity¹ of \(\mathbb{P}^2\) — they meet at the two points \([1:i:0]\), \([1: -i:0]\). The pencil \(Z(\lambda F_1+\mu F_2)\) above is then the collection of conics passing through this intersection \(C_1\cap C_2\).

Meanwhile, a general degree 2 homogeneous polynomial is of the form

\[F(\x_0,\x_1,\x_2) = a_{00}\x_0^2 + a_{11}\x_1^2 + a_{22}\x_2^2 + a_{01}\x_0\x_1 + a_{02}\x_0\x_2 + a_{12}\x_1\x_2\]

and this is precisely why \(\Gamma(\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(2))\) is a \(6\)-dimensional space. On the other hand, if we add the condition that it must pass through the four-point set \(C_1\cap C_2\) computed above, each of these four points imposes one constraint, eliminating one parameter each, so we see that 2 parameters are needed to represent it. More concretely, the following four conditions

\[0=F(0,\sqrt{5},1)=5a_{11}+a_{22}+\sqrt{5}a_{12}\] \[0=F(0,-\sqrt{5},1)=5a_{11}+a_{22}-\sqrt{5}a_{12}\] \[0=F(1,i,0)=a_{00}-a_{11}+ia_{01}\] \[0=F(1,-i,0)=a_{00}-a_{11}-ia_{01}\]

force \(a_{12}=0\), \(a_{01}=0\), \(5a_{11}=-a_{22}\), \(a_{00}=a_{11}\), so the actual variables are the two \(a_{00}\), \(a_{02}\). That is, this collection of conics will form, in \(\Gamma(\mathbb{P}^2,\mathcal{O}_{\mathbb{P}^2}(2))\), a 2-dimensional subspace \(V\), and projectivizing this yields, as represented by \([\lambda:\mu]\), the space \(\mathbb{P}^1\).

Of course, Definition 2 applies equally to any variety, whether \(X\) is projective space or a quasi-projective variety. However, the reason we went to such lengths in computing Example 3 above is that for any quasi-projective variety \(X\subseteq \mathbb{P}^n\), if \(D\) comes from some \(\mathcal{O}_{\mathbb{P}^n}(d)\), then the language of homogeneous polynomials can be used as is. That is, in this case the restriction map

\[\Gamma(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(d)) \rightarrow \Gamma(X, \mathcal{O}_{\mathbb{P}^n}(d)\vert_X)\]

sends a homogeneous polynomial \(F \in \mathbb{K}[\x_0, \ldots, \x_n]_d\) to a section on \(X\), and its kernel is the homogeneous component of \(I(X)\) of degree \(d\), \(I(X)_d\). Therefore, if this restriction map is surjective, then with

\[\Gamma(X, \mathcal{O}_{\mathbb{P}^n}(d)\vert_X) \cong \mathbb{K}[\x_0, \ldots, \x_n]_d / I(X)_d\]

essentially the same computations as in \(\mathbb{P}^n\) are possible. In fact, it is known that for a projective variety \(X\), this surjectivity holds when \(d\) is sufficiently large. In particular, since \(F - G \in I(X)\) defines the same intersection, the parameter space becomes \(\mathbb{P}(V/(V \cap I(X)))\).

Base Locus

In the remainder of this post, given a linear system on \(X\), \(L=\mathbb{P}(V)\), and a basis \(F_0,\ldots, F_r\) of \(V\), we use them to define the embedding

\[\varphi_L:X\rightarrow \mathbb{P}^r;\qquad x\mapsto [F_0(x):\cdots:F_r(x)]\]

Of course, this is not always possible. For instance, in Example 3, if we choose the following two basis elements corresponding to \((a_{00},a_{02})=(1,0), (0,1)\):

\[G_1(\x_0,\x_1,\x_2)=\x_0^2+\x_1^2-5\x_2^2, \qquad G_2(\x_0,\x_1,\x_2)=\x_0\x_2\]

then this “embedding” becomes

\[\mathbb{P}^2\rightarrow \mathbb{P}^1;\qquad [\x_0:\x_1:\x_2]\mapsto [\x_0^2+\x_1^2-5\x_2^2:\x_0\x_2]\]

First of all, since this is a map from \(\mathbb{P}^2\) to the smaller space \(\mathbb{P}^1\), we know that something is wrong, and this is because there is a locus where the two functions \(G_1,G_2\) simultaneously become \(0\).

The embedding \(\varphi_L\) above actually depends on the choice of basis of \(V\), but many properties that \(\varphi_L\) possesses do not. For example, as just seen, the points of \(X\) where all the basis elements vanish do not depend on the choice of basis.

To describe this rigorously, we define the support of a Weil divisor \(D = \sum n_i D_i\) as \(\operatorname{Supp}(D) = \bigcup_{n_i \neq 0} D_i\). That is, the support is the union of the prime divisors whose coefficient in the divisor is not \(0\). Using this, the following is well-defined.

Definition 4 For a linear system \(L \subseteq \lvert \mathcal{L} \rvert\), the base locus \(\operatorname{Bs}(L)\) is the closed subset shared by all elements of \(L\). Specifically, when \(L = \mathbb{P}(V)\) with \(V \subseteq \Gamma(X, \mathcal{L})\),

\[\operatorname{Bs}(L) = \bigcap_{s \in V \setminus \{0\}} \operatorname{Supp}(\divisor(s))\]

where \(\divisor(s)\) is the zero divisor of the section \(s\).

In particular, in the calculation of hypersurfaces in \(\mathbb{P}^n\), for \(V \subseteq \mathbb{K}[\x_0, \ldots, \x_n]_d\), this is the same as \(\operatorname{Bs}(L) = \bigcap_{[F] \in L} Z(F)\). Then the definition we wanted to make is the following.

Definition 5 \(L\) is said to be basepoint-free if \(\operatorname{Bs}(L) = \emptyset\). That is, for any point \(p \in X\), not passing through \(p\), there always exists an element of \(L\).

The key property of a basepoint-free linear system is as follows. If \(L=\mathbb{P}(V)\) is basepoint-free, then a basis of \(V\), \(F_0,\ldots,F_r\), satisfies \(\bigcap Z(F_i)\cap X=\emptyset\), so using this we can define the following regular map:

\[\varphi_L:X\rightarrow\mathbb{P}^r,\quad p\mapsto[F_0(p):\cdots:F_r(p)]\]

Our original interest in linear systems was to find, for a given divisor \(D\), an effective divisor linearly equivalent to \(D\), and the following proposition gives a direct answer to this.

Proposition 6 In the above situation, a hyperplane in \(\mathbb{P}^r\), \(H\), defines an effective divisor belonging to \(L\).

To verify this, it suffices to check that for a hyperplane in \(\mathbb{P}^r\), \(H: a_0\x_0+\cdots+a_r\x_r=0\), the preimage \(\varphi_L^{-1}(H)\) coincides with the zero set of the global section

\[\sigma=a_0F_0+\cdots+a_rF_r\in V\]

that is, with \(\divisor(\sigma)\). Let us look at a more concrete example.

Example 7 Let us examine the two examples on \(\mathbb{P}^2\) seen in Example 3. First, consider the complete linear system

\[\lvert \mathcal{O}_{\mathbb{P}^2}(1)\rvert=\mathbb{P}(\mathbb{K}[\x_0,\x_1,\x_2]_1)\]

If we choose for the vector space \(\mathbb{K}[\x_0,\x_1,\x_2]_1\) the basis \(\x_0,\x_1,\x_2\), since \(\x_0,\x_1,\x_2\) are simultaneously \(0\) at no point of \(\mathbb{P}^2\), this is basepoint-free. The \(\varphi_L\) defined by this choice of basis is simply the identity.

In the case of the base locus of the two conics, as we saw above, the base locus is not empty. In fact, the base locus is the four intersection points of \(C_1\cap C_2\) already examined in Example 3, and geometrically, since each element of the pencil shares exactly the four intersection points of \(C_1\cap C_2\), we see that this matches the definition of the base locus.

The above example intuitively shows the origin of the name basepoint, but since \(\varphi_L\) is the identity, Proposition 6 does not actually have much meaning. Let us look at a more non-trivial example.

Example 8 On \(\mathbb{P}^1\), when \(d \ge 1\), the map defined by the complete linear system \(\lvert \mathcal{O}_{\mathbb{P}^1}(d) \rvert\) is

\[\nu_d: \mathbb{P}^1 \rightarrow \mathbb{P}^d, \quad [s : t] \mapsto [s^d : s^{d-1}t : \cdots : t^d]\]

This shows that the Veronese embedding examined in §Projective Varieties, ⁋Example 16 can be recovered in the language of complete linear systems.

For example, in \(\mathbb{P}^d\), considering the hyperplane \(H_0: \x_0 = 0\),

\[\nu_d^{-1}(H_0) = \{[s:t] \in \mathbb{P}^1 \mid s^d = 0\}\]

so scheme-theoretically, giving the point \([0:1]\) multiplicity \(d\), this becomes the effective divisor \(d\cdot[0:1]\). As another example, for the hyperplane \(H_1: \x_0 - \x_d = 0\),

\[\nu_d^{-1}(H_1) = \{[s:t] \in \mathbb{P}^1 \mid s^d - t^d = 0\}\]

and if \(\operatorname{char}\mathbb{K}\nmid d\), then \(s^d - t^d\) factors into a product of \(d\) distinct linear factors (for example, if \(\mathbb{K}=\mathbb{C}\), then \(s^d-t^d=\prod_{k=0}^{d-1}(s-\zeta^k t)\)), so \(\nu_d^{-1}(H_1)\) is an effective divisor on \(\mathbb{P}^1\) consisting of \(d\) distinct points. In any case, these preimages are effective divisors belonging to \(\lvert \mathcal{O}_{\mathbb{P}^1}(d)\rvert\) of degree \(d\).

Ample line bundle

Although we are assuming that every variety is quasi-projective, varieties can generally be defined more abstractly. This approach has its pros and cons: the good point is that our discussion becomes more flexible, and what we give up is that embedding a variety is no longer trivial.

For example, in our language, to say that \(\mathbb{P}^1\times \mathbb{P}^1\) is a (quasi-projective) variety, we must embed it into some projective space. (§Projective Varieties, ⁋Example 16) Instead, if we do not assume the existence of an ambient projective space in the definition of a variety, then \(\mathbb{P}^1\times \mathbb{P}^1\) automatically becomes a variety without our having to show this, but it is unclear whether a general variety embeds into projective space.

However, even on an abstract variety, we can define line bundles, linear systems, and so on. Then in particular, using Proposition 6, we can define a suitable map to projective space. The importance of the following definition should be understood in this context.

Definition 9 A line bundle \(\mathcal{L}\) (or the corresponding linear system \(\lvert \mathcal{L} \rvert\)) is said to be very ample if the complete linear system \(\lvert \mathcal{L} \rvert = \mathbb{P}(\Gamma(X, \mathcal{L}))\) defines a regular map \(\varphi_{\mathcal{L}}: X \rightarrow \mathbb{P}(\Gamma(X, \mathcal{L})^\ast)\) that is a closed embedding.

Here, the reason the target is not the projectivization of \(\Gamma(X, \mathcal{L})\) but the projectivization of its dual is that, as we saw in Proposition 6, the elements of the linear system correspond to the hyperplanes of the target. If for \(\Gamma(X, \mathcal{L})\) we choose a basis \(s_0,\ldots,s_r\) and take the dual basis as coordinates, then \(\mathbb{P}(\Gamma(X, \mathcal{L})^\ast)\cong \mathbb{P}^r\), and in these coordinates \(\varphi_{\mathcal{L}}\) is given by \(x\mapsto [s_0(x):\cdots:s_r(x)]\). Since \(\varphi_L\) differs by an automorphism of \(\mathbb{P}^r\) depending on the choice of basis, whether this is a closed embedding does not depend on the choice of basis.

The key point in the definition of very ample is that the map is not merely a morphism but a closed embedding. That is, as explained above, even in the world of abstract varieties, we can use this to define a projective variety, and moreover, using a very ample line bundle \(\mathcal{L}\), we can even express \(X\) in explicit coordinates in this ambient projective space.

We know that \(\mathcal{O}_{\mathbb{P}^n}(1)\) is very ample, but \(\mathcal{O}_{\mathbb{P}^n}(-1)\) is not. As we saw in §Line Bundles and Vector Bundles, ⁋Example 16, this is because the twisting direction of \(\mathcal{O}_{\mathbb{P}^n}(-1)\) as the fiber moves along the base does not allow sections to cross the zero section, so no global sections exist. On the other hand, the twisting that \(\mathcal{O}_{\mathbb{P}^n}(1)\) possesses allows this, enabling global sections to exist.

Although this is a very simple example, if there is a space more complicated than \(\mathbb{P}^n\), and the complexity of this space cannot be resolved solely by the twisting of a particular line bundle (even if it is in the right direction), we could add more and more twisting until it is resolved. From this intuition, we define the following.

Definition 10 \(\mathcal{L}\) is said to be ample if there exists some \(m > 0\) such that \(\mathcal{L}^{\otimes m}\) is very ample.

To see the usefulness of this definition, one should think of a space having a line bundle that is ample but not very ample, but it is still somewhat premature to introduce such a space. However, before long we will deal with such a space, and ampleness will then prove its full worth.

General elements

An element of a linear system is a nonzero section considered up to scalar multiplication (Definition 2), and what each element actually gives us is the zero locus of that section. Then it is natural to ask what kind of space this zero locus looks like. Looking again at the pencil of Example 3, this is generated by \(G_1=\x_0^2+\x_1^2-5\x_2^2\) and \(G_2=\x_0\x_2\), so its elements are the conics \(Z(\lambda G_1+\mu G_2)\). The determinant of the symmetric matrix corresponding to this quadratic form is \(-\lambda(20\lambda^2+\mu^2)/4\), and a conic being singular is equivalent to this value being \(0\), so the only bad elements are those where \([\lambda:\mu]\) is one of the three points \([0:1]\), \([1:\pm 2\sqrt{5}i]\). For example, the element with \(\lambda=0\) splits into two lines \(Z(\x_0)\) and \(Z(\x_2)\) and has a singular point at their intersection. All other elements are smooth conics, and what we want to know is the property that almost all elements have in this way.

If \(\mathcal{L}\) is very ample, this question translates into a question about hyperplanes. This is because \(X\), via \(\varphi_{\mathcal{L}}\), becomes a closed subvariety of \(\mathbb{P}^N = \mathbb{P}(\Gamma(X,\mathcal{L})^\ast)\), and by Proposition 6, the elements of \(\lvert \mathcal{L}\rvert\) are precisely the traces of hyperplanes \(H\subseteq \mathbb{P}^N\) cutting \(X\), namely \(X\cap H\). That is, \(\lvert \mathcal{L}\rvert\) is identified with the dual projective space \((\mathbb{P}^N)^\ast\), and so our question becomes one of how a general hyperplane cuts \(X\). The answer to this is the following theorem.

Proposition 11 (Bertini’s theorem) For a projective variety \(X\subseteq \mathbb{P}^N\), there exists a dense open subset \(U\) of the dual projective space \((\mathbb{P}^N)^\ast\) such that for every \(H\in U\), the following hold.

1. The singular points of \(X\cap H\) are all singular points of \(X\). In particular, if \(X\) is smooth, then \(X\cap H\) is also smooth.
2. If \(\dim X\geq 2\), then \(X\cap H\) is irreducible.

Intuitively, the first result means the following. For \(H\) to create, on \(X\) at a smooth point \(x\), a singular point of \(X\cap H\) is the same as \(H\) being tangent at \(x\) to \(X\), and if we let \(n=\dim X\), then at a fixed \(x\), the hyperplanes \(H\) tangent there form inside \((\mathbb{P}^N)^\ast\) a subspace of dimension \((N-n-1)\), so even collecting these over all \(x\in X\), the dimension is at most \(N-1\). That is, the hyperplanes tangent to \(X\) somewhere cannot fill the \(N\)-dimensional \((\mathbb{P}^N)^\ast\), and the remainder becomes the dense open subset we seek. To make this rigorous, one must actually compute the dimension of the incidence variety formed by collecting such \(H\), and the irreducibility of the second result is a deeper fact, so we take both results without proof.

Henceforth, we will call an element belonging to some dense open subset of a linear system a “general element” of that linear system. Then, translating the above proposition back into the original language, we obtain the following.

Corollary 12 For a smooth projective variety \(X\) and a very ample line bundle \(\mathcal{L}\), a general element of the complete linear system \(\lvert \mathcal{L}\rvert\) is smooth, and if \(\dim X\geq 2\), it is irreducible.

Proof

By Definition 9, \(\varphi_{\mathcal{L}}\) is a closed embedding, so we can view \(X\) as a closed subvariety of \(\mathbb{P}^N\), and by Proposition 6, under this identification, an element of \(\lvert \mathcal{L}\rvert\) is, for a hyperplane \(H\), of the form \(X\cap H\). Applying Proposition 11 (Bertini’s theorem) to this gives the result.

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References

[Hart] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer, 1977.
[Sha] I. R. Shafarevich, Basic Algebraic Geometry I: Varieties in Projective Space, Springer, 2013.

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1. The line at infinity of \(\mathbb{P}^2\) and its geometric intuition have already been analyzed in §Projective Varieties, ⁋Example 11. ↩