Let $X$ be a reduced projective scheme of pure dimension 1 over the field $k$. Let $\pi: X \to \mathbb{P}_k^1$ be a finite, flat and surjective morphism onto the projective line. 

> What is the minimal $d \in \mathbb{Z}_{\geq 1}$ such that $H^1(X,\pi^*\mathcal{O}_{\mathbb{P}_k^1}(d)) = 0$?

I could already show that $d \geq -\chi(\mathcal{O}_X)/\deg(\pi)$ is necessary. But I was wondering whether $d \in O(-\chi(\mathcal{O}_X)/\deg(\pi))$ holds.

I am grateful for any kind of help or maybe even an example of such a situation where $d \notin O(-\chi(\mathcal{O}_X)/\deg(\pi))$ holds.