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Section 3.1 Divisors on algebraic curves

Some definitions and examples of divisors on algebraic curves.

Throughout this section, we let \(X\) be an algebraic curve over an algebraically closed field \(k\text{.}\)

Subsection 3.1.1 Definitions and examples

Definition 3.1.1.

A divisor on \(X\) is a finitely-supported, formal sum

\begin{equation*} \sum_{p \in X} a_p \cdot p \end{equation*}

with integer coefficients. The set of all divisors is the Abelian group \(\Div(X)\)

Definition 3.1.2.

The degree of a divisor \(D = \sum a_p \cdot p\) is

\begin{equation*} \deg(D) = \sum a_p \in \Z. \end{equation*}

The group of all divisors of degree zero is \(\Div^0(X) = \ker(\deg)\text{.}\)

Definition 3.1.3.

The divisor associated to a rational function \(f \in k(X)\) is

\begin{equation*} \divisor(f) = \sum_{p \in X} \ord_p(f) \cdot p \end{equation*}

where \(\ord_p(f)\) is the order of a zero or pole of \(f\text{,}\) for instance, by looking at its order as a Laurent series. A divisor of this form is called a principal divisor. The group of all principal divisors is the group \(\Prin(X)\text{.}\) The Jacobian of \(X\) is the quotient \(\Jac(X) = \Div(X) / \Prin(X)\text{.}\)

Two divisors \(D_0, D_1\) are said to be linearly equivalent if their difference is a principal divisor and write \(D_0 \sim D_1\text{.}\)

Remark 3.1.4.

If \(X\) is proper, then every rational function has degree \(0\) and we define \(\Jac^0(X) = \Div^0(X)/\Prin(X)\text{.}\) Here we have

\begin{equation*} \Jac(X) \cong \Jac^0(X) \times \Z. \end{equation*}

Example 3.1.5. Divisors on \(\A^1\).

On \(\A^1\text{,}\) we can specify zeroes and poles of rational functions freely. That is, for \(D = a_1 \cdot p_1 + \dots + a_n \cdot p_n\text{,}\) we have

\begin{equation*} D = \divisor\left( (x - p_1)^{a_1} \cdots (x - p_n)^{a_n} \right) \end{equation*}

Hence \(\Jac(\A^1) = 0\text{.}\)

Example 3.1.6. Divisors on \(\P^1\).

On \(\P^1\text{,}\) as we said above, every rational function has degree \(0\text{.}\) Therefore

\begin{equation*} \Jac^0(\P^1) = 0 \text{ and } \Jac(\P^1) \cong \Z. \end{equation*}

Example 3.1.7. Divisors on \(\Spec \O_K\).

If \(K\) is a finite extension of \(\Q\) and \(\O_K\) is the ring of integral elements of \(K\text{,}\) then the Jacobian of \(\Spec \O_K\) is the same thing as the class group of \(\O_K\text{.}\)

Example 3.1.8. Elliptic curves.

An elliptic curve over \(k\) is a projective smooth genus-one curve \(E\) with a special marked point '\(p_0\text{.}\)'

Every elliptic curve is isomorphic to its Jacobian with

\begin{equation*} p \mapsto p - p_0 \in \Jac^0(E). \end{equation*}

This is called the Abel-Jacobi map.

Subsection 3.1.2 Linear Series

Definition 3.1.9.

The line bundle associated to a divisor \(D\) is the sheaf \(\L(D)\) where for any open set \(U\) of \(X\text{,}\)

\begin{equation*} \L(D)(U) = \{f \in k(X) : \divisor(f)|_U + D \ge 0\}. \end{equation*}

Remark 3.1.10.

The global sections of \(\L(D)\) form a finite dimensional vector space. If \(\ell(D)\) is the dimension of this vector space, then we have

\begin{equation*} \ell(D) - \ell(K - D) = \deg(D) + 1 - g. \end{equation*}

This is known as the Riemann-Roch Theorem. Here \(K\) is a divisor known as the canonical divisor. In particular, we have Riemann's Inequality

\begin{equation*} \ell(D) \ge \deg(D) + 1 - g \end{equation*}

which is sharp if (but not only if) \(\deg(D) > \deg(K) = 2g - 2\text{.}\)

Definition 3.1.11.

Let \(s_0, \dots, s_n\) generate the global sections of some line bundle \(\L\text{.}\) Then for each point \(x \in X\text{,}\) these global sections generate the ring of functions at \(x\text{.}\) In particular, we never simultaneously have \(s_0(x) = s_1(x) = \dots = s_n(x) = 0\text{.}\)

Therefore, we can define a map \(s : X \to \P^n\) given by

\begin{equation*} s(x) = [s_0(x) : s_1(x) : \cdots : s_n(x)]. \end{equation*}

Remark 3.1.12.

Conversely, suppose we have a map \(s : X \to \P^n = \Proj k[x_0,\dots,x_n]\text{.}\) Then we can pull back the functions \(x_0, x_1, \dots, x_n\) on \(\P^n\) to functions \(s_0, \dots, s_n\) on \(X\text{.}\)

The functions \(s_0,\dots,s_n\) generate the sheaf \(s^*(\O(1))\text{.}\)

Example 3.1.13. Veronese embedding.

Let \(X = \P^r\) and let \(\L = \O(d)\) with \(d > 0\text{.}\) Then the set of monomials of degree \(d\) is a basis for the space of global sections of \(\O(d)\text{.}\) The induced map

\begin{equation*} s : \P^r \to \P^n, \quad n = \binom{r + d}{d} - 1 \end{equation*}

is called the Veronese map.

For \(r = 2\) and \(d = 2\) this is the map

\begin{equation*} [x:y:z] \to [x^2 : y^2 : z^2 : xy : xz : yz] \end{equation*}

from \(\P^2 \to \P^5\text{.}\)

For \(r = 1\) and \(d = 3\text{,}\) the image of

\begin{equation*} [x:y] \to [x^3 : x^2y : xy^2 : y^3] \in \P^3 \end{equation*}

is the twisted cubic.

Remark 3.1.14.

Under this construction, linear forms on \(\P^n\) correspond to \(d\)-forms on \(\P^r\text{.}\) For instance, the twisted cubic has the property that every plane intersects it at exactly 3 points.

This fact can be used to show that every graph can be embedded in \(\R^3\) using straight lines for edges by putting the vertices on the twisted cubic. If two lines were to cross that would form a plane which intersects the twisted cubic in 4 points.

More generally, given an embedding \(s : X \to \P^n\text{,}\) every hyperplane \(H \in \P^{n,*}\) intersects \(X\) at some divisor \(D\) (a set of points with multiplicities). We also see that such a divisor has non-negative coefficients. This motivates the following definition.

Definition 3.1.15.

A divisor \(E\) with all non-negative coefficients is called effective and we write \(E \ge 0\) to indicate this.

Given a divisor \(D\text{,}\) the set of all effective divisors \(E \ge 0\) which are linearly equivalent to \(D\) is called a complete linear system and is denoted \(|D|\text{.}\)

Remark 3.1.16.

Recall that we defined (Definition 3.1.9)

\begin{equation*} \L(D)(U) = \{f \in \O_X(U) : \divisor(f) + D \ge 0\}. \end{equation*}

It follows that divisors of \(|D|\) correspond to global sections of \(\L(D)\text{.}\)

We have \(\divisor(s) = \divisor(s')\) if \(\divisor(s/s') = 0\text{.}\) When \(X\) is proper, we have \(\divisor(s/s') = 0 \iff s/s' \in k^*\text{.}\) So divisors of global sections define an element of \(|D|\) only up to a non-zero scalar. In other words, what we have is

\begin{equation*} |D| \cong (\Gamma(X, \L(D)) - \{0\})/k^*. \end{equation*}

Remark 3.1.17. Another definition of line bundle.

In Definition 3.1.9 we defined \(\L(D)\) and called it a "line bundle." This is a special case of a more general definition.

A sheaf \(\L\) is called a line bundle if there exists an open cover \(X = \bigcup U_i\) for which \(\L(U_i) \cong \O_X(U_i)\) for all \(i\text{.}\)

When \(X\) is smooth, then every divisor on \(X\) is locally principal (because \(\O_X\) is locally a unique factorization ring). So there exists an open cover \(X = \bigcup U_i\) and functions \(f_i \in \O_X(U_i)\) such that

\begin{equation*} D|_{U_i} = \divisor(f_i). \end{equation*}

Recall that a section \(s \in \L(D)(U_i)\) has to satisfy \(\divisor(s) + \divisor(f_i) \ge 0.\) Therefore, a global section locally takes the form of \(s = f/f_i\) with \(f \in k[X]\text{.}\)

The functions \(f_i\) determine the isomorphism \(\L(D)(U_i) \to \O_X(U_i).\) Conversely, every line bundle \(\L\) is isomorphic to \(\L(D)\) where the functions \(f_i \in \O_X(U_i)\) are the reciprocals of the image of \(1 \in \L(U_i)\) under the isomorphism \(\L(U_i) \to \O_X(U_i)\text{.}\)

Remark 3.1.18.

The things which we have called "divisors" are just one type of divisor called Weil divisors. A second type of divisor exists in the case where \(X\) is not smooth and those are Cartier divisors. A Cartier divisor is a Weil divisor which is locally principal. As discussed in the last remark, these notions are equivalent on smooth curves.

For not-necessarily-smooth curves (and other varieties), the set of Cartier divisors forms a subgroup \(\CDiv(X)\) of the divisor group and here we define \(\Pic(X) = \CDiv(X)/\Prin(X)\) and \(\Pic^0(X) = \CDiv^0(X)/\Prin(X)\text{.}\) This is called the Picard group. It coincides with the Jacobian for smooth varieties.

A variety does not necessarily need to be smooth in order for every Weil divisor to be Cartier.

Consider the exact sequence

\begin{equation*} 0 \to \L(D - P) \to \L(D) \to k(P) \to 0. \end{equation*}

Taking global sections gives

\begin{equation*} 0 \to \Gamma(X, \L(D - P)) \to \Gamma(X, \L(D)) \to k \end{equation*}

so we see that \(\ell(D - P)\) is either \(\ell(D)\) or \(\ell(D) - 1\text{.}\)

Now suppose \(P\) is not a base point of \(\L(D)\) which exists since a global section \(s \neq 0\) can only have finitely many zeroes. Then with this same \(s\text{,}\) we have \(s \notin \Gamma(X,\L(D - P))\) since \(s\) does not have a zero at \(P\text{.}\) So \(\Gamma(X,\L(D - P))\) is a proper subspace of \(\Gamma(X,\L(D))\) which finishes the proof.

References 3.1.3 References