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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.

Edit: If $X$ is integral, then $d$ is in the desired order: We then have $$ H^1(X,\pi^*\mathcal{O}_{\mathbb{P}_k^1}(d)) \cong H^0(X,\omega_X \otimes_{\mathcal{O}_X}\pi^*\mathcal{O}_{\mathbb{P}_k^1}(-d)) $$ which is zero if $\deg \pi^*\mathcal{O}_{\mathbb{P}_k^1}(d) > \deg \omega_X = -2\chi(\mathcal{O}_X)$. So what happens otherwise?

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  • $\begingroup$ Have you examined the hyper elliptic case ? $\endgroup$
    – meh
    Commented Jul 29, 2019 at 14:41
  • $\begingroup$ @aginensky Ain't hyperelliptic curves integral curves? See the 'Edit'-part. $\endgroup$
    – windsheaf
    Commented Jul 30, 2019 at 7:58
  • $\begingroup$ @ windsheaf. Your edit was unclear to me- and maybe others. I interpreted it to mean what about the case of d of small degree. Good luck with your question, but non-integral X leads into a rather technical area that I personally wouldn't think about without some motivation as to why I should care. $\endgroup$
    – meh
    Commented Jul 30, 2019 at 15:05
  • $\begingroup$ @aginensky The edit provides the answer to my question in the integral case. In that case we have $d > -\chi(\mathcal{O}_X)/\deg(\pi)$ is sufficient for the $H^1$-term to vanish. The motivation behind this question is the following: When pushing down $\mathcal{O}_X$ to $\mathbb{P}_k^1$ via $\pi$ it decomposes into a direct some of twisted sheaves $\mathcal{O}_{\mathbb{P}_k^1}(d_i)$. And finding $d$ as in my question is equivalent to give a bound for those integers $d_i$. $\endgroup$
    – windsheaf
    Commented Jul 31, 2019 at 7:36

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