Relative projektive Kurve/Verhalten des Hilbert-Kunz slopes/Fakt

Let

$C\longrightarrow \operatorname {Spec} {\left(\mathbb {Z} _{g}\right)}$

be a smooth projective relative curve and let ${}{\mathcal {T}}$ be a vector bundle over ${}C$. For the generic fiber ${}C_{0}$ let

${}\mu _{HK}({\mathcal {T}}_{0})=\sum _{\ell }r_{\ell }\mu _{\ell }^{2}\,,$

where the ${}\mu _{\ell }$ are the slopes (and ${}r_{\ell }$ the ranks) in the Harder-Narasimhan filtration of ${}{\mathcal {T}}_{0}$, and for every prime number let

${}{\bar {\mu }}_{HK}({\mathcal {T}}_{p})={\frac {\sum _{k}r_{k}\mu _{k}^{2}}{p^{e}}}\,,$

where the ${}\mu _{k}$ are the slopes in the strong Harder-Narasimhan filtration of ${}F^{e*}({\mathcal {T}}_{p})$ on the fiber ${}C_{p}$. Then

${}\lim _{p\rightarrow \infty }{\bar {\mu }}_{HK}({\mathcal {T}}_{p})=\mu _{HK}({\mathcal {T}}_{0})\,$

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