Riemann--Roch and Serre duality on a curve

On a smooth projective curve, $\chi(\OO(D))=\deg D+1-g$, and Serre duality turns the correction term into sections of $K-D$.Suggest a slogan

(Riemann--Roch)

For $D$ a divisor on a smooth projective curve $X$ of genus $g$ over $k = \bar{k}$, \[ \ell(D) - \ell(K - D) = \deg D + 1 - g , \] where $\ell(D) = h^0(X, \OO(D))$ and $K$ is a canonical divisor [@Har10a, Theorem IV.1.3].

(Serre duality)

$H^1(X, \OO(D)) \cong \dualof{H^0(X, \OO(K-D))}$, so $\ell(K-D) = h^1(D)$ and Riemann–Roch reads \[ \chi(\OO(D)) = \deg D + 1 - g . \]

(Hyperplane sections)

If $X \subseteq \PP^n$ is a curve of degree $d$ and $D = X \cap H$ is a hyperplane section, then $\mcl(D) \cong \OO_X(1)$, $\deg D = d$, and Riemann–Roch gives \[ \chi(\mcl(D)) = d + 1 - p_a . \]

Over a field $k$ that is not algebraically closed, adding a closed point $P$ to $D$ raises $\chi(\OO(D))$ by $\deg P = [\kappa(P) : k]$, and the formula reads $\chi(\OO(D)) = \deg D + \chi(\OO_X)$ with $\deg D = \sum n_P \deg P$ (D-CRVDEGREES).

The second form is the one to state first when asked, because it says what the theorem is: the Euler characteristic is linear in the divisor, and the genus is the constant of integration. Duality is what converts the unknown $h^1$ into a countable $h^0$.

Two immediate consequences, both asked:

$D = 0$ gives $\ell(K) = g$: the space of global regular differentials on a curve of genus $g$ has dimension exactly $g$. This is the answer to the question about holomorphic differentials on a Riemann surface, and it is also how $g$ can be defined so that Riemann–Roch becomes a statement rather than a tautology.
$D = K$ gives $\deg K = 2g-2$, which is the input to Riemann–Hurwitz and to every genus computation.

For $\deg D > 2g - 2$ the correction term vanishes and $\ell(D) = \deg D + 1 - g$ exactly; this is the range in which everything is computable and where “very ample” questions are settled.

State Riemann--Roch.

State Serre duality for a curve.

What is the dimension of the space of holomorphic differentials on a curve of genus $g$?

Dependencies

T-IJW1KThe cohomology of $\OO_{\PP^n}(d)$uses
PR-Y5S7VThe degree of a divisor on a projective curveuses

Backlinks

T-COHSDSerre duality in dimension $n$related-to
PR-VGA2LCurves of genus $0$uses
D-CRVGONLinear series $g^r_d$, and the gonality of a curveuses
T-LKT0URiemann--Hurwitzuses
D-CRVSPECSpecial divisors and the index of specialityuses
T-CRVJACFUNThe Jacobian represents $\Pic^0(X/-)$, and is a smooth proper group scheme of dimension $g$uses
T-U5QSYThe Jacobian, Abel's theorem, and the Abel--Jacobi mapuses
PR-CRVDEGBDThe degree thresholds $2g$ and $2g+1$uses
D-G1AEHThe genus in its several sensesuses
PR-CRVGRPThe group law on an elliptic curve, and multiplication by $n$uses
T-D8TUXWhen the canonical divisor is very ampleuses

What links to this

Vanishing and duality