Clifford's theorem on special divisors

Remarks

The inequality comes from adding linear systems. Since $D$ and $K - D$ are both effective up to linear equivalence, the addition map $\abs{D} \times \abs{K - D} \to \abs{K}$ is finite-to-one, so $\dim \abs{D} + \dim \abs{K - D} \le \dim \abs{K} = g - 1$. Adding the Riemann–Roch theorem $\ell(D) - \ell(K - D) = \deg D - g + 1$ gives $2(\ell(D) - 1) \le \deg D$. The hypothesis that $D$ is special is needed: for $\deg D > 2g - 2$, Riemann–Roch gives $\ell(D) - 1 = \deg D - g$, which exceeds $\deg D / 2$ once $\deg D > 2g$. Example: on a hyperelliptic curve with degree-$2$ map divisor $g^1_2$, the divisor $D = r \cdot g^1_2$ with $r \le g - 1$ has $\ell(D) = r + 1$ and $\deg D = 2r$, so equality holds.

Going deeper

For a student

Riemann–Roch computes $\ell(D)$ exactly for nonspecial divisors, and Clifford's theorem is the first general bound in the special range $0 \le \deg D \le 2g - 2$. It is used to show that a non-hyperelliptic curve has a canonical embedding, and that a divisor of degree $2$ moving in a pencil forces the curve to be hyperelliptic. Brill–Noether theory refines the bound for a general curve, where the true dimensions are much smaller.