Let $X$ be non-hyperelliptic of genus $g \geq 3$. Show that there exist $D=p_1+\cdots+p_{g+1}$ in $X$ such that the linear system $|D|$ is base point free and $\dim |D| = 1$.
My attempt:
Because $X$ is non-hyperelliptic, by Geometric Riemann Roch we have $\dim\text{span}(p_1,\cdots,p_{g+1})=\deg(D)-\dim|D|-1=g-1$. This shows that The indued map by canonical line bundle $\phi_K:X\to\mathbb{P}^{n-1} $ is embedding and surjective. Then it's remain to find a base point free $D$, or $h^0 (D\times\mathcal{O}(-p))=h^0(D)-1$ for any $p$. Then how do I proceed?
This is true for any general points $p_i$. Show that for general points $p_i$, $1\leq i\leq g$, $h^0(K-p_1-p_2-\cdots-p_i)=g-i$.