The Riemann–Roch Theorem for Curves

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.

We now examine in greater detail the complete linear system $H^0(X, \mathcal{L})$ of a line bundle $\mathcal{L}$, which we encountered in §Linear Systems, ⁋Definition 2. We could have introduced this immediately after defining linear systems, but the proof requires Serre duality, so we postponed it.

The Riemann–Roch Theorem

Definition 1 For a divisor $D$ on a smooth projective curve $C$, we define the Riemann–Roch dimension as

\[\ell(D) = \dim H^0(C, \mathcal{O}_C(D)).\]

In general, we regard $\mathcal{O}_X(D)$ as the sheaf of rational functions that may have poles of order at most $1$ along $D$; from this perspective, $H^0(C, \mathcal{O}_C(D))$ can be viewed as a 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, $H^0(C, \mathcal{O}_C(D))$ is the collection of divisors linearly equivalent to a given divisor $D$, and by projectivizing it we obtain the complete linear system $\lvert \mathcal{O}_X(D)\rvert$ of $\mathcal{O}_C(D)$. For convenience, we write this as $\lvert D\rvert$ in this post. Then the Riemann–Roch dimension above is one more than the projective dimension of $\lvert D\rvert$.

Now fix a point $p\in C$. By definition, the elements of $\lvert D\rvert$ passing through $p$ correspond to those sections $s$ in $H^0(C,\mathcal{O}_C(D))$ satisfying $s(p)=0$. That is, such an $s$ is an element of $H^0(C, \mathcal{O}_C(D))$ with $\divisor(s)-p\geq 0$, and through this we verify that the collection of exactly these elements is precisely the space of global sections of

\[\mathcal{O}_C(D-p)\cong \mathcal{O}_C(D)\otimes \mathcal{O}_C(-p).\]

Therefore, if equality holds in $H^0(C,\mathcal{O}_C(D-p))\subseteq H^0(C, \mathcal{O}_C(D))$, then 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 used this to define a regular map $\varphi_D:C\rightarrow \mathbb{P}^{\ell(D)-1}$; from this perspective, the difference between $\ell(D)$ and $\ell(D-p)$ can be viewed as a correction term imposed by the point $p$ on the divisor $D$.

The Riemann–Roch theorem, which we study in this post and the next, extends this idea in a certain sense: the canonical class $K_C$ plays a global role replacing that of the point $p$. Specifically, the formula we aim 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 (Serre duality pairing, projective case)). Recall here that the canonical divisor $K_C$ is the divisor corresponding to the canonical line bundle. Then the following lemma allows us to 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 $C$,

\[H^i(C, \mathcal{F}) = 0 \quad (i \ge 2)\]

holds.

Proof

Fix an embedding $C\hookrightarrow \mathbb{P}^N$. Then by a dimension count, there exist hyperplanes $H_1,H_2$ such that $C\cap H_1\cap H_2=\emptyset$. Thus, setting $U_i=C\setminus H_i$, we know that these form an affine open cover of $C$.

Now consider the Čech cohomology with respect to $\{U_1,U_2\}$. As was briefly introduced right after §Sheaf Cohomology, ⁋Proposition 12, any affine open cover of a projective variety satisfies the hypotheses of §Sheaf Cohomology, ⁋Theorem 11 (Leray), and therefore the sheaf cohomology we seek reduces exactly to the computation for this affine open cover. The Čech complex is then 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.

Therefore, by this result, for any divisor $D$,

\[\rchi(\mathcal{O}_C(D)) = h^0(C, \mathcal{O}_C(D)) - h^1(C, \mathcal{O}_C(D))\tag{$2$}\]

holds. Here $h^i$ is shorthand for the dimension of $H^i$.

On the other hand, from a topological perspective we know well that the Euler characteristic of a compact Riemann surface $S$ of genus $g$ is given by

\[\rchi(S)=2(1-g).\]

Here the Euler characteristic can be thought of via triangulation as $V-E+F$ with vertex count $V$, edge count $E$, and face count $F$¹, or it can be regarded as defined through differential-geometric tools such as the Gauss–Bonnet theorem.

In algebraic geometry we generally take the underlying field $\mathbb{K}$ to be 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 algebraic-geometric Euler characteristic is given by substituting $D=0$ into formula ($2$) above:

\[\rchi(\mathcal{O}_{C_S})=h^0(C_S, \mathcal{O}_{C_S})-h^1(C_S, \mathcal{O}_{C_S}).\]

Meanwhile, just as one-dimensional holes appear via $H^1$ in topology, the one-dimensional holes in algebraic geometry—that is, the genus, which is a two-dimensional hole from the topological perspective—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 of the topological Euler characteristic is no coincidence, and can be verified via Hodge theory, but this is irrelevant to our present goal so we set it aside. What matters is that the algebraic-geometric Euler characteristic and the value $1-g$ obtained when computing it for a curve are not arbitrary; they are a reinterpretation of the topological result.

The Riemann–Roch theorem, the subject of this post, adds one further step. The computation above was for the trivial sheaf $\mathcal{O}_{C_S}$ with nothing attached, so we now consider the sheaf $\mathcal{O}_{C_S}(D)$ twisted by an arbitrary divisor $D$. The conclusion of the Riemann–Roch theorem is that a correction term of magnitude $\deg D$ appears.

Proposition 3 (Riemann–Roch for curves) For a divisor $D$ on a smooth projective curve $C$,

\[\ell(D) - \ell(K_C - D) = \deg D + 1 - g\]

holds. 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 is 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 we saw above $\rchi(\mathcal{O}_C)=1-g$; combining this with the previous formula yields the desired result. For a general (not necessarily effective) divisor, one writes $D$ as a difference of effective divisors and applies the same additivity argument.

The proof above is clean, but its geometric content is compressed inside the Euler characteristic, so it may not be very intuitive. To remedy this, let us read the equation term by term. First, by definition

\[\ell(D) = \dim H^0(C, \mathcal{O}_C(D)),\]

and geometrically the space on the right-hand side is the collection of meromorphic functions satisfying

\[\divisor(f)+D\geq 0.\]

That is, $D$ forces the poles of $f$ to occur only within the support of $D$, and the order of the pole at each point $p$ is at most $\operatorname{ord}_p D$; thus as $\deg D$ grows, the allowed pole orders increase and so $\ell(D)$ grows.

Moreover, since in our situation $C$ is one-dimensional, an (effective) divisor has the form $D=\sum n_i p_i$, and using this we can obtain more quantitatively the following inequality:

\[\ell(D)\leq \deg(D)+1\tag{$3$}\]

Specifically, if $D = \sum n_i p_i$ is effective, 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 map that, when the principal part of the Laurent expansion of $f$ at $p_i$ is written as

\[\frac{a_{-n_i}}{(x-p_i)^{n_i}}+\frac{a_{-n_i+1}}{(x-p_i)^{n_i-1}}+\cdots +\frac{a_{-1}}{x-p_i},\]

sends

\[f\mapsto (a_{-n_i}, \ldots, a_{-1})\]

considered simultaneously for all $p_i$. 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 inequality to become an equality the linear map must be surjective, but this does not always hold. To see this, consider the long exact sequence obtained from the short exact sequence examined in the proof of Proposition 3:

\[0\longrightarrow H^0(C,\mathcal{O}_C)\overset{i^\ast}{\longrightarrow} H^0(C,\mathcal{O}_C(D)) \overset{p^\ast}{\longrightarrow} H^0(C,\mathcal{O}_D) \overset{\delta}{\longrightarrow} H^1(C,\mathcal{O}_C)\overset{i^\ast}{\longrightarrow} H^1(C,\mathcal{O}_C(D))\to 0.\]

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 $H^0(C, \mathcal{O}_C)=\bigoplus_i \mathbb{K}^{n_i}$. Moreover, the linear map ($4$) examined above actually coincides with $p^\ast$ in this long exact sequence, and from this the cokernel of $p^\ast$ can be computed via 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

\[\dim\coker p^\ast =\dim \ker (i^\ast: H^1(C,\mathcal{O}_C)\twoheadrightarrow H^1(C,\mathcal{O}_C(D)))=\dim H^1(C,\mathcal{O}_C)-\dim H^1(C,\mathcal{O}_C(D)).\]

Applying formula (1) here, we obtain

\[\dim\coker p^\ast=\dim H^1(C,\mathcal{O}_C)-\dim H^0(C,\mathcal{O}_C(K_C-D))^\vee=g-\ell(K_C-D).\]

From inequality ($3$), the difference between $\deg(D)+1$ and $\ell(D)$ is precisely the dimension of this cokernel, so these computations recover the result of Proposition 3. In other words, $\ell(K_C-D)$ measures how far $\ell(D)$ falls from its upper bound $\deg D+1$; this was originally a counting problem for $1$-forms vanishing along $D$, but we have rewritten it 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 $\ell(K_C-D)=0$, and the Riemann–Roch theorem gives

\[\ell(D)=\deg D+1-g.\]

That is, as the genus grows, the space determined by a divisor of the same degree becomes smaller. In the general case, however, the influence of $\ell(K_C-D)$ is superimposed on this, and one must be careful that the term $\ell(K_C-D)$ can vary depending not only on the degree of $D$ but also on the relationship between $D$ and the canonical class $K_C$.

As another special case, substituting $D=0$ gives

\[\ell(0)-\ell(K_C)=\deg D+1-g,\]

and since $\deg D=0$ and $\ell(0)=1$, we find $\ell(K_C)=g$. Substituting $D=K_C$ then gives

\[\ell(K_C)-\ell(0)=\deg K_C +1-g,\]

from which we recover the computation $\deg(K_C)=2g-2$ from §Canonical Line Bundle, ⁋Example 10. In that example the degree-genus formula was invoked as a well-known fact to obtain $\deg(K_C)$ (and historically this is the more natural route), but shortly we will see in Proposition 7 (Degree-genus formula) that the degree-genus formula is in fact a special case of the Riemann–Roch theorem.

In any case, summarizing the computations so far, we may 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 $D$, which disappears for large degree and reflects geometric information of $K_C$ for 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 Line Bundle, ⁋Example 8). On the other hand, in (§Line Bundles and Vector Bundles, ⁋Example 12) we showed that the global sections of $\mathcal{O}_{\mathbb{P}^1}(d)$ are the homogeneous polynomials of degree $d$, so we know that

\[\ell(dH) = d+1 \quad (d \ge 0), \qquad \ell(dH) = 0 \quad (d < 0)\]

holds. Verifying 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 both sides agree at $d+1$.

Example 5 (Elliptic curve) In the genus $1$ case $g = 1$, the above computations give $\deg K_C=2g-2=0$ 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 since $\deg K_C=0$ and the only effective divisor of degree $0$ is $0$, we have $K_C\sim 0$. 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: from inequality ($3$) we must have $\ell(D)=0$ or $\ell(D)=1$. If $\ell(D)=1$, then there exists a unique effective divisor linearly equivalent to $D$, but its degree is $0$ so it is $0$. Therefore $D\sim 0$, and conversely if $D\sim 0$ then $\mathcal{O}_C(D)\cong \mathcal{O}_C$, so $\ell(D)=1$. That is, the term $\ell(K_C-D)$ equals $1$ precisely when $D$ is linearly equivalent to $0$, and is $0$ otherwise.

Since $K_C \sim 0$, the Riemann–Roch theorem becomes 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. This shows that $g=1$ is the simplest non-trivial case in the progression where the influence of the correction term becomes increasingly complicated as the genus grows.

Example 6 ($g=2$) Now consider the one-step more complicated situation of $g=2$. Here $\deg K_C = 2g - 2 = 2$ 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 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 in $\mathbb{P}^1$, is 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 single point $p$ and examine how $\ell(D)$ varies with $d$. For small $d$, where $\ell(K_C-D)$ is still active, special phenomena appear, but as $d$ grows, $\ell(D)$ stabilizes linearly.

For $d=1$, if $\ell(p)\ge 2$ then a degree 1 map $C\to\mathbb{P}^1$ would exist, implying $C\cong\mathbb{P}^1$, which contradicts $g=2$; 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 $\ell(D)=d-1$.

The canonical map $\varphi_{K_C}: C \rightarrow \mathbb{P}^1$ for $g=2$ examined in the above example was a 2:1 branched covering. More generally, among curves of genus $g \ge 2$, those admitting a degree 2 covering to $\mathbb{P}^1$ are called hyperelliptic curves, and those admitting no such covering 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 $g\geq 2$. We verified above that $\deg K_C = 2g - 2$ and $h^0(K_C) = g$, so the codomain of $\varphi_{K_C}$ is $\mathbb{P}^{g-1}$. However, this is generally not a closed embedding, as we already saw in the $g=2$ case where it becomes a $2:1$ covering map. Computing this concretely, one finds that $\varphi_{K_C}$ factors as the composition of the Veronese embedding $\mathbb{P}^1 \hookrightarrow \mathbb{P}^{g-1}$ and the hyperelliptic covering $C\rightarrow \mathbb{P}^1$.

Degree-genus formula

In §Canonical Line Bundle, ⁋Example 10 we asserted the following proposition as a well-known fact to show that $\deg K_C=2g-2$, but now we can give a rigorous proof. Note, however, that this proceeds in exactly the opposite direction to that example: there the adjunction formula and degree-genus formula were used to prove $\deg K_C=2g-2$, whereas now we derive the degree-genus formula from $\deg K_C=2g-2$ together with the adjunction formula. The degree of $K_C$ was already obtained from Riemann–Roch before Example 4 (without using the degree-genus formula).

Proposition 7 (Degree-genus formula) For a smooth plane curve $C \subset \mathbb{P}^2$ of degree $d$,

\[g(C) = \frac{(d-1)(d-2)}{2}\]

holds.

Proof

By the adjunction formula (§Canonical Line Bundle, ⁋Proposition 9), $K_C = (K_{\mathbb{P}^2} + C)\vert_C = (d-3)H\vert_C$. Therefore $\deg K_C = d(d-3)$, and substituting this into $\deg K_C = 2g - 2$ yields

\[d(d-3) = 2g - 2 \implies g = \frac{d(d-3) + 2}{2} = \frac{(d-1)(d-2)}{2}.\]

This formula directly computes the geometric properties of plane curves. For example, a smooth plane cubic has genus 1, so it coincides with the elliptic curve treated in Example 5 (Elliptic curve). On the other hand, for $d = 1, 2$ we have $g = 0$, reflecting that lines and conics are both birationally equivalent to $\mathbb{P}^1$ (§Rational Maps, ⁋Proposition 10).

Example 8 Computing the genus by degree $d$, for degree 3 (cubic) we have $g = \frac{2 \cdot 1}{2} = 1$, an elliptic curve; for degree 4 (quartic) we have $g = \frac{3 \cdot 2}{2} = 3$, and for degree 5 (quintic) we have $g = \frac{4 \cdot 3}{2} = 6$. Since the genus grows rapidly with degree, higher-degree smooth plane curves have increasingly complex topological structure.

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.

This coincides with defining the Euler characteristic as the alternating sum of cohomology. ↩

Share on

Twitter Facebook LinkedIn

K

The Riemann–Roch Theorem

Degree-genus formula

Share on

댓글남기기