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# The cohomology of the relative dualizing sheaf of anarithmeticsurfacearelativecurve

Let $X\to S$ be a semi-stable curve over $S=\mathrm{Spec} \ \mathbf{Z}$. Let $\omega$ be the relative dualizing sheaf of $X\to S$. Let $g$ be the genus of the generic fibre. Assume that $g\geq 2$.

I know that $\omega$ is ample. Even better, a theorem of Deligne and Mumford says that $\omega^{\otimes 3}$ is very ample.

So there is some integer $n$ depending on $X\to S$ such that $H^1(X,\omega^{\otimes m}) =0$ for all $m\geq n$.

Can we find such an $n$? Is it independent of $g$? Does $n=3$ work?

If not, can we find such an $n$ and bound it in terms of $g$?

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Let $X\to S$ be a semi-stable curve over $S=\mathrm{Spec} \ \mathbf{Z}$. Let $\omega$ be the relative dualizing sheaf of $X\to S$. Let $g$ be the genus of the generic fibre. Assume that $g\geq 2$.

I know that $\omega$ is ample. Even better, a theorem of Deligne and Mumford says that $\omega^{\otimes 3}$ is very ample.

So there is some integer $n$ depending on $X\to S$ such that $H^1(X,\omega^{\otimes m}) =0$ for all $m\geq n$.

Can we find such an $n$? Is it independent of $g$$? g? Does n=3 work? If not, can we find such an n and bound it in terms of g? 2 deleted 54 characters in body Let X\to S be a semi-stable minimal regular arithmetic surface curve over S=\mathrm{Spec} \ \mathbf{Z}. Let \omega be the relative dualizing sheaf of X\to S. Let g be the genus of the generic fibre. Assume that g\geq 2. I know that \omega is ample. Even better, a theorem of Deligne and Mumford says that \omega^{\otimes 3} is very ample. So there is some integer n depending on X\to S such that H^1(X,\omega^{\otimes m}) =0 for all m\geq n. Can we find such an n? Can Is it be independent of the "genus" of X\to S? (The genus being the genus of the generic fibre.) g$$? Does$n=3$work? If not, can we find such an$n$and bound it in terms of the genus of$X\to S$?g$?

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