I am doing exercises of Hartshorne’s algebraic geometry. In chapter IV, Ex. 1.5:
For a effective divisor $ D$ on a curve $ X$ of genus $ g$ , show that $ dim|D| \leq deg D$ .
I found a solution says:If $ D$ is effective , then $ \ell(K-D) \leq \ell(K)$ (here K is the canonical divisor).
Anyone know why this is true? Is $ |K-D| \subset |K|$ also true? Thanks.
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