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Let $X$ be a smooth projective curve of genus $g$ over $\mathbb{C}$, $K_{X}$ be a cononical sheaf on $X$ and $\mathcal{E}$ be a locally free sheaf on $X$ s.t. $H^{0}(X,\mathcal{E}^{*})=\operatorname{Hom}(\mathcal{E},\mathcal{O}_{X})=0$.

I want to show a following lemma,

$h^{0}(X,\mathcal{E}\otimes K_{X})\geq h^{0}(X,\mathcal{E})+g-1$

Consider $\mathcal{P}=\mathbb{P}(\mathcal{E})$ the projective bundle associated to $\mathcal{E}$. $\pi\colon \mathbb{P}(\mathcal{E}) \rightarrow X$ is a canonical projection and $\mathcal{O}(1)$ is the hyperplane bundle. The lemma is equivalent to

$h^{0}(\mathcal{P},\mathcal{O}(1)\otimes \pi^{*}K_{X})\geq h^{0}(\mathcal{P},\mathcal{O}(1))+h^{0}(\mathcal{P},\pi^{*}K_{X})-1$

So, I'm trying to prove

a map $\mathbb{P}(H^{0}(\mathcal{P},\mathcal{O}(1))\times\mathbb{P}(H^{0}(\mathcal{P},\pi^{*}K_{X}))\rightarrow \mathbb{P}(H^{0}(\mathcal{P},\mathcal{O}(1)\otimes \pi^{*}K_{X}))$ obtained by adding effective divisor is finite to one.

But, I have trouble here. I suspect this is not true generally, and do I need some extra condition ?
Thanks in advance.

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    $\begingroup$ Of course it is true: there are only finitely many ways to split an effective divisor as a sum of two effective divisors. By the way, this is called the H. Hopf lemma (not to be confused with the E. Hopf lemma), and you can find it in many places, e.g. Arbarello et al. $\endgroup$
    – abx
    Commented Dec 7, 2020 at 5:20
  • $\begingroup$ Thanks. Your comment is helpful for me. I'll search Arbarello's book. $\endgroup$
    – Aoki
    Commented Dec 8, 2020 at 12:01

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