Riemann–Roch theorem

Remarks

In sheaf language, $\ell(D) = h^0(X, \mathcal{O}_X(D))$ and $\ell(K - D) = h^1(X, \mathcal{O}_X(D))$ by Serre duality, so the formula reads $\chi(\mathcal{O}_X(D)) = \deg D + 1 - g$. One proof checks the formula for $D = 0$, where it says $h^1(\mathcal{O}_X) = g$. Then it shows that both sides change by $1$ when a point $P$ is added to $D$. This uses the exact sequence $0 \to \mathcal{O}_X(D) \to \mathcal{O}_X(D + P) \to k_P \to 0$. Taking $D = 0$ and $D = K$ gives $\ell(K) = g$ and $\deg K = 2g - 2$. If $\deg D > 2g - 2$, then $\deg(K - D) < 0$, so $\ell(K - D) = 0$ and $\ell(D) = \deg D - g + 1$ exactly. Example: on the projective line, $g = 0$ and $K = -2P$ for any point $P$. For $D = nP$ with $n \ge 0$ the formula gives $\ell(D) = n + 1$. This is the dimension of the space of polynomials of degree at most $n$ in an affine coordinate.

Going deeper

For a student

Riemann–Roch is the main computational tool of the theory of curves. It shows that a divisor of degree at least $2g + 1$ gives an embedding into projective space. It also shows that every curve of genus $0$ is the projective line, and that every curve of genus $1$ with a point is a plane cubic. It feeds directly into Clifford's theorem, the Riemann–Hurwitz formula and Brill–Noether theory. Its higher-dimensional generalizations are the Hirzebruch–Riemann–Roch and Grothendieck–Riemann–Roch theorems.