This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.
We now take a closer look at the complete linear system \(H^0(X, \mathcal{L})\) of a line bundle \(\mathcal{L}\), which we examined in §Linear Systems, ⁋Definition 2. We could have introduced this right after defining linear systems, but we postponed it because the proof requires Serre duality.
The Riemann–Roch Theorem
Definition 1 For a divisor \(D\) on a smooth projective curve \(C\), we define the Riemann–Roch dimension by
\[\ell(D) = \dim H^0(C, \mathcal{O}_C(D)).\]In general, we think of \(\mathcal{O}_X(D)\) as the sheaf of rational functions that may have poles of order \(1\) along \(D\); from this point of view, \(H^0(C, \mathcal{O}_C(D))\) can be regarded as the space of functions defined on \(X\).
Recall that this space \(H^0(C, \mathcal{O}_C(D))\) was first introduced in §Linear Systems, ⁋Definition 2. According to that definition, the space \(H^0(C, \mathcal{O}_C(D))\) is the collection of divisors linearly equivalent to a given divisor \(D\), and by projectivizing it we obtained the complete linear system \(\lvert \mathcal{O}_X(D)\rvert\) of \(\mathcal{O}_C(D)\). For convenience, we will write this as \(\lvert D\rvert\) in this post. Then the Riemann–Roch dimension defined above is one more than the projective dimension of \(\lvert D\rvert\).
Now fix a point \(p\in C\). Then the elements of \(\lvert D\rvert\) passing through \(p\) can be regarded, by definition, as those sections among the elements of \(H^0(C,\mathcal{O}_C(D))\) satisfying \(s(p)=0\). In other words, such an \(s\) is an element of \(H^0(C, \mathcal{O}_C(D))\) satisfying \(\divisor(s)-p\geq 0\), and through this we can verify that the collection of exactly such elements is the global sections of
\[\mathcal{O}_C(D-p)\cong \mathcal{O}_C(D)\otimes \mathcal{O}_C(-p).\]Therefore, if equality holds in \(H^0(C,\mathcal{O}_C(D-p))\subseteq H^0(C, \mathcal{O}_C(D))\), this means that every element of \(\lvert D\rvert\) passes through \(p\), so \(p\) becomes a base point of \(\lvert D\rvert\). On the other hand, if \(\lvert D\rvert\) is basepoint-free, then in §Linear Systems we were able to use it to define a regular map \(\varphi_D:C\rightarrow \mathbb{P}^{\ell(D)-1}\); from this viewpoint, the difference between \(\ell(D)\) and \(\ell(D-p)\) can be seen as a correction term that the point \(p\) imposes on the divisor \(D\).
Then the Riemann–Roch theorem, which we will examine in this post and the next, extends this in a certain sense: the canonical class \(K_C\) plays a global role replacing that of the point \(p\). Specifically, the formula we wish to prove is
\[\ell(D)-\ell(K_C-D)=\deg D+1-g,\]where \(\deg D+1\) is the expected value, \(g\) is the loss coming from the topological data of the curve, and \(\ell(K_C-D)\) is a correction term arising from the positional relationship between the canonical class and \(D\).
To see this, we first apply Serre duality to obtain
\[H^1(C, \mathcal{O}_C(D)) \cong H^0(C, \omega_C \otimes \mathcal{O}_C(-D))^\vee = H^0(C, \mathcal{O}_C(K_C - D))^\vee\tag{$1$}\](§Serre Duality, ⁋Proposition 2). Recall here that the canonical divisor \(K_C\) is the divisor corresponding to the canonical line bundle. Then by the following lemma, we can deduce that only two terms appear in the Euler characteristic of \(\mathcal{O}_C(D)\). In this post we assume that \(\mathbb{K}\) is an infinite field.
Lemma 2 For any coherent sheaf \(\mathcal{F}\) on a smooth projective curve \(C\),
\[H^i(C, \mathcal{F}) = 0 \quad (i \ge 2).\]Proof
Fix an embedding \(C\hookrightarrow \mathbb{P}^N\). Then by a dimension count there exist hyperplanes \(H_1,H_2\) such that \(C\cap H_1\cap H_2=\emptyset\). Hence, setting \(U_i=C\setminus H_i\), we know that these form an affine open cover of \(C\).
Now consider the Čech cohomology for \(\{U_1,U_2\}\). As briefly introduced just after §Sheaf Cohomology, ⁋Proposition 12, any affine open cover of a projective variety satisfies the hypothesis of §Sheaf Cohomology, ⁋Theorem 11, and therefore the sheaf cohomology we seek reduces precisely to the computation for this affine open cover. Now the Čech complex is simply
\[\check{C}(\mathcal{U}, \mathcal{F}):\qquad \mathcal{F}(U_1)\oplus \mathcal{F}(U_2)\rightarrow \mathcal{F}(U_1\cap U_2)\rightarrow 0,\]a complex of length 1, so \(\check{H}^i = 0\ (i \ge 2)\) follows immediately.
Hence, by this result, for any divisor \(D\) we have
\[\rchi(\mathcal{O}_C(D)) = h^0(C, \mathcal{O}_C(D)) - h^1(C, \mathcal{O}_C(D)).\tag{$2$}\]Here \(h^i\) is shorthand notation for the dimension of \(H^i\).
Meanwhile, from the topological viewpoint we are well aware that the Euler characteristic of a compact Riemann surface \(S\) of genus \(g\) is given by
\[\rchi(S)=2(1-g).\]In this case the Euler characteristic can be thought of using a triangulation as \(V-E+F\) with \(V\) the number of vertices, \(E\) the number of edges, and \(F\) the number of faces[^1], or it can be regarded as defined by differential-geometric tools such as the Gauss–Bonnet theorem.
In algebraic geometry we generally think of the underlying field \(\mathbb{K}\) as the complex numbers, so the genus \(g\) compact Riemann surface above is nothing but a one-dimensional curve \(C_S\) from the algebraic-geometric viewpoint. Then the Euler characteristic in the algebraic-geometric sense is given by substituting \(D=0\) into formula (\(2\)) above:
\[\rchi(\mathcal{O}_{C_S})=h^0(C_S, \mathcal{O}_{C_S})-h^1(C_S, \mathcal{O}_{C_S}).\]On the other hand, just as one-dimensional holes appear via \(H^1\) in topology, the one-dimensional holes in algebraic geometry—that is, the two-dimensional holes from the topological viewpoint—called the genus, are defined by \(g=h^1(C_S, \mathcal{O}_{C_S})\), and since global sections are only constant functions, the Euler characteristic of \(C_S\) is expressed as
\[\rchi(\mathcal{O}_{C_S})=h^0(C_S, \mathcal{O}_{C_S})-h^1(C_S, \mathcal{O}_{C_S})=1-g.\]That this value is half the topological Euler characteristic is no accident, and it can be verified through Hodge theory, but this is irrelevant to our goal so we will skip it for now. What is important is that the Euler characteristic in algebraic geometry, and the value \(1-g\) obtained when computing it on a curve, is not something out of the blue but a reinterpretation of the result from topology.
The Riemann–Roch theorem, the subject of this post, adds one further step to this. The above computation was for the trivial sheaf \(\mathcal{O}_{C_S}\) with nothing attached, so we consider the sheaf \(\mathcal{O}_{C_S}(D)\) twisted by an arbitrary divisor \(D\). Then the result of the Riemann–Roch theorem is that a correction term of size \(\deg D\) appears.
Proposition 3 (Riemann–Roch for curves) For a divisor \(D\) on a smooth projective curve \(C\),
\[\ell(D) - \ell(K_C - D) = \deg D + 1 - g.\]Here \(g\) is the genus of \(C\) and \(K_C\) is the canonical divisor.
Proof
By the computations and definitions above,
\[\rchi(\mathcal{O}_C(D)) = h^0(C, \mathcal{O}_C(D)) - h^1(C, \mathcal{O}_C(D)) = \ell(D) - \ell(K_C - D).\]On the other hand, for an effective divisor \(D\) there exists a short exact sequence
\[0 \to \mathcal{O}_C \to \mathcal{O}_C(D) \to \mathcal{O}_D \to 0,\]and then by additivity of the Euler characteristic \(\rchi(\mathcal{O}_C(D)) = \rchi(\mathcal{O}_C) + \rchi(\mathcal{O}_D)\).
Here \(\mathcal{O}_D\) is a skyscraper sheaf of degree \(\deg D\), so \(\rchi(\mathcal{O}_D) = \deg D\), and as examined above \(\rchi(\mathcal{O}_C)=1-g\); combining this with the previous formula yields the desired result. For a general (non-effective) divisor, one expresses \(D\) as a difference of effective divisors and applies the same additivity argument.
The above proof is clean, but its geometric content is compressed inside the Euler characteristic, so it may not be intuitively clear. To remedy this, let us read the equality term by term. First, by definition
\[\ell(D) = \dim H^0(C, \mathcal{O}_C(D)),\]and the space \(H^0(C, \mathcal{O}_C(D))\) on the right-hand side is geometrically the collection of meromorphic functions satisfying
\[\divisor(f)+D\geq 0.\]That is, \(D\) forces the poles of \(f\) to occur only inside the support of \(D\), and at each point \(p\) the order of the pole is at most \(\operatorname{ord}_p D\); therefore as \(\deg D\) grows the allowed orders of poles also grow, so \(\ell(D)\) increases.
Moreover, in our situation \(C\) is one-dimensional, so an (effective) divisor is of the form \(D=\sum n_i p_i\), and using this we can obtain the following quantitative inequality:
\[\ell(D)\leq \deg(D)+1.\tag{$3$}\]Specifically, suppose \(D = \sum n_i p_i\) is effective. Then we can consider the linear map sending \(f\in H^0(C, \mathcal{O}_C(D))\) to its principal part at each point \(p_i\):
\[H^0(C, \mathcal{O}_C(D)) \longrightarrow \bigoplus_i \mathbb{K}^{n_i}.\tag{$4$}\]Intuitively, this is the function that, when the principal part of the Laurent polynomial of \(f\) at the point \(p_i\) is expressed as
\[\frac{a_{-n_i}}{(x-p_i)^{n_i}}+\frac{a_{-n_i-1}}{(x-p_i)^{n_i-1}}+\cdots +\frac{a_{-1}}{x-p_i},\]considers
\[f\mapsto (a_{-n_i}, \ldots, a_{-1})\]for all \(p_i\) at once. Then the dimension of the target of the above linear map is \(\sum n_i = \deg D\), and the kernel of this map is the global sections without poles, i.e. \(H^0(C, \mathcal{O}_C) = \mathbb{K}\), from which we obtain \(\ell(D) \leq 1 + \deg D\). If \(D\) is not effective but \(\ell(D) > 0\), then \(D\) is linearly equivalent to some effective divisor, so the same inequality holds.
In general, for this formula to become an equality the linear map must be surjective, but this does not always hold. To verify this, consider the short exact sequence
\[0\longrightarrow \mathcal{O}_C\overset{i}{\longrightarrow} \mathcal{O}_C(D)\overset{p}{\longrightarrow} \mathcal{O}_D\longrightarrow 0\]examined in the proof of Proposition 2 and the resulting long exact sequence
\[0\longrightarrow H^0(C,\mathcal{O}_C)\overset{i^\ast}{\longrightarrow} H^0(C,\mathcal{O}_C(D)) \overset{p^\ast}{\longrightarrow} H^0(C,\mathcal{O}_D) \overset{\delta}{\longrightarrow} H^1(C,\mathcal{O}_C)\overset{i^\ast}{\longrightarrow} H^1(C,\mathcal{O}_C(D))\to 0.\]Here \(C\) is a curve and \(D=\sum n_i p_i\), so \(\mathcal{O}_D\) is a skyscraper sheaf of degree \(D\) with support \(\lvert D\rvert\), whence we know that \(H^0(C, \mathcal{O}_C)=\bigoplus_i \mathbb{K}^{n_i}\).
Moreover, knowing that the linear map (\(4\)) examined above actually coincides with \(p^\ast\) in this long exact sequence, we can compute the cokernel of \(p^\ast\) by the following chain of isomorphisms:
\[\coker p^\ast=\frac{H^0(C, \mathcal{O}_D)}{\im p^\ast}=\frac{H^0(C, \mathcal{O}_D)}{\ker\delta}\cong \im\delta\cong\ker i^\ast,\]and in particular its dimension is
\[\dim\coker p^\ast =\dim \ker (i^\ast: H^1(C, \mathcal{O}_C)\twoheadrightarrow H^1(C, \mathcal{O}_C(D)))=\dim H^1(C, \mathcal{O}_C)-\dim H^1(C, \mathcal{O}_C(D)).\]If we apply formula (1) here, we find
\[\dim\coker p^\ast=\dim H^1(C, \mathcal{O}_C)-\dim H^0(C, \mathcal{O}_C(K_C-D))^\vee=g-\ell(K_C-D).\]In inequality (\(3\)) above, the difference between \(\deg(D)+1\) and \(\ell(D)\) is exactly the dimension of this cokernel, so these computations recover the result of Proposition 3. In other words, \(\ell(K_C-D)\) is the quantity measuring how far \(\ell(D)\) falls from its upper bound \(\deg D+1\); this was originally a counting problem of \(1\)-forms vanishing along \(D\), but it has been rewritten as \(\ell(K_C-D)\) using Serre duality.
As an example, consider the case where the degree of \(D\) is very large so that \(\deg(K_C-D)<0\). Then in this case \(\ell(K_C-D)=0\), and therefore the Riemann–Roch theorem gives the formula
\[\ell(D)=\deg D+1-g.\]Thus, as the genus grows, the space created by a divisor of the same degree becomes smaller. However, in the general case the effect of \(\ell(K_C-D)\) is added to this, and what one must note is that the term \(\ell(K_C-D)\) can vary depending not only on the degree of \(D\) but also on what relation \(D\) has to the canonical class \(K_C\).
As another special example, substituting \(D=0\) gives
\[\ell(0)-\ell(K_C)=\deg D+1-g,\]and since \(\deg D=0\) and \(\ell(0)=1\), we see that \(\ell(K_C)=g\). Now putting \(D=K_C\) yields
\[\ell(K_C)-\ell(0)=\deg K_C +1-g,\]and from this we can recover the computation \(\deg(K_C)=2g-2\) from §Canonical Bundle, ⁋Example 10. In that example the degree-genus formula was mentioned as a well-known fact and \(\deg(K_C)\) was derived from it (and this is historically more appropriate), but shortly we will see in Proposition 7 that the degree-genus formula is a special case of the Riemann–Roch theorem.
Anyway, summarizing the computations so far, we can think of \(\ell(D)\) as the dimension of the complete linear system of \(D\), and \(\ell(K_C - D)\) as a correction term that \(K_C\) imposes on \(D\); this correction term disappears in large degree and reflects the geometric information of \(K_C\) in small degree.
Example 4 \(\mathbb{P}^1\): The genus of \(\mathbb{P}^1\) is \(g = 0\), and the canonical divisor is \(K_{\mathbb{P}^1} = -2H\) (§Canonical Bundle, ⁋Example 8). On the other hand, we showed in (§Line Bundles and Vector Bundles, ⁋Example 12) that the global sections of \(\mathcal{O}_{\mathbb{P}^1}(d)\) are homogeneous polynomials of degree \(d\), so we know that
\[\ell(dH) = d+1 \quad (d \ge 0), \qquad \ell(dH) = 0 \quad (d < 0).\]Now checking the Riemann–Roch formula, for \(D = dH\) we have \(\deg D = d\) and \(K_C - D = (-2-d)H\), so
\[\ell(dH) - \ell(-2H-dH) = d + 1 - 0 = d + 1,\]and we can verify that both sides agree as \(d+1\).
Example 5 (Elliptic curve) In the genus \(1\) case \(g = 1\), we know from the above computations that \(\deg K_C=2g-2=0\) and \(\ell(K_C)=g=1\). Since \(\ell(K_C)=1>0\), as we saw earlier there exists an effective divisor linearly equivalent to \(K_C\); but \(\deg K_C=0\) and the only effective divisor of degree \(0\) is \(0\), so \(K_C\sim 0\). Using this, the Riemann–Roch formula becomes
\[\ell(D) - \ell(-D) = \deg D.\]In particular, if \(\deg D > 0\) then \(\ell(-D) = 0\), so \(\ell(D) = \deg D\).
The case \(\deg D=0\) is the small-degree case mentioned above. First, inequality (\(3\)) implies that \(\ell(D)=0\) or \(\ell(D)=1\). If \(\ell(D)=1\), then there exists a unique effective divisor linearly equivalent to \(D\), and since its degree is \(0\) it must be \(0\). Hence \(D\sim 0\), and conversely if \(D\sim 0\) then \(\mathcal{O}_C(D)\cong \mathcal{O}_C\), so \(\ell(D)=1\). Thus the term \(\ell(K_C-D)\) is \(1\) only when \(D\) is linearly equivalent to \(0\), and is \(0\) otherwise.
Since \(K_C \sim 0\), the Riemann–Roch theorem is especially simple on an elliptic curve. If \(\deg D > 0\) the correction term \(\ell(K_C-D)=\ell(-D)\) vanishes, so \(\ell(D)=\deg D\) is completely determined, showing that \(g=1\) is the simplest non-trivial case in the process where the influence of the correction term becomes more complicated as the genus grows.
Example 6 (\(g=2\)) Now let us look at the one-step more complicated situation of \(g=2\). In this case \(\deg K_C = 2g - 2 = 2\) and \(\ell(K_C)=2\), and substituting \(D=p\) into Proposition 3 gives
\[\ell(p)-\ell(K_C-p)=2-g,\]so \(\ell(K_C-p)=\ell(K_C)-1<\ell(K_C)\), and therefore the canonical map
\[\varphi_{K_C}:C\rightarrow \mathbb{P}^1\]is well defined. Then a hyperplane in \(\mathbb{P}^1\), that is, the preimage of a point of \(\mathbb{P}^1\), becomes an effective divisor linearly equivalent to \(K_C\), and from the fact that this is a degree \(2\) map we can write \(K_C\) as a sum of two points \(p_1+p_2\).
Now let us apply Riemann–Roch to multiples \(D=d\cdot p\) of a point \(p\) and examine how \(\ell(D)\) varies with \(d\). For small \(d\), that is, where \(\ell(K_C-D)\) is still alive, special phenomena appear, but as \(d\) grows \(\ell(D)\) stabilizes linearly.
- For \(d=1\), if \(\ell(p)\ge 2\) then there exists a degree 1 map \(C\to\mathbb{P}^1\), which would imply \(C\cong\mathbb{P}^1\), contradicting \(g=2\); hence \(\ell(p)=1\). By Riemann–Roch, \(\ell(K_C-p)=1\).
- For \(d=2\), if \(2p\sim K_C\) then \(\ell(2p)=2\). In this case \(p\) is called a Weierstrass point; this condition corresponds exactly to the situation where the preimage of some point under the canonical map \(\varphi_{K_C}\) is ramified at \(p\). For a generic point \(2p\not\sim K_C\), so \(\ell(2p)=1\).
- For \(d\ge 3\), we have \(\deg(K_C-D)=2-d<0\), so \(\ell(K_C-D)=0\), and therefore \(\ell(D)=d-1\).
In the above example the canonical map \(\varphi_{K_C}: C \rightarrow \mathbb{P}^1\) for \(g=2\) was a \(2:1\) branched covering. More generally, among curves of genus \(g \ge 2\), those for which there exists a degree 2 covering to \(\mathbb{P}^1\) are called hyperelliptic curves, and the others are called non-hyperelliptic curves. Note that by convention the cases of genus \(0,1\) are excluded from hyperelliptic curves.
Now let us examine the behavior of the morphism \(\varphi_{K_C} : C \rightarrow \mathbb{P}^{g-1}\) defined by the complete linear system \(\lvert K_C\rvert\) of the canonical bundle \(K_C\) on a curve \(C\) with \(g\geq 2\). Since we verified above that \(\deg K_C = 2g - 2\) and \(h^0(K_C) = g\), the target of \(\varphi_{K_C}\) is \(\mathbb{P}^{g-1}\). However, this is generally not a closed embedding, as we already saw in the case \(g=2\) where it becomes a \(2:1\) covering map. Computing this concretely, one finds that \(\varphi_{K_C}\) is the composition of the Veronese map \(\mathbb{P}^1 \hookrightarrow \mathbb{P}^{g-1}\) and the hyperelliptic covering \(C\rightarrow \mathbb{P}^1\).
Degree-genus formula
In §Canonical Bundle, ⁋Example 10 we passed over the next proposition by claiming it as a well-known fact in order to show \(\deg K_C=2g-2\), but now we can give a rigorous proof of it. However, this is exactly the opposite of that example: there the adjunction formula and the degree-genus formula were used to prove \(\deg K_C=2g-2\), whereas now we derive the degree-genus formula from the fact that
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