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Let $k$ be a field, $X$ be a connected smooth projective $k$-scheme. Let $f:X\rightarrow X$ be a finite $k$-morphism that is surjective on the underlying topological spaces. Suppose $f$ has degree $n$.

Is it possible that there does not exist a coherent locally free sheaf $F$ on $X$ such that $H^0(X, F)=n\,\mathrm{rank}(F)$?

I think for curves, one can find line bundles of any $H^0$.

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You can always find such a sheaf.

Take a very ample line bundle $L$ on $X$, with $H^0(X, \, L)=a \geq n$, and set $$F:=L^{\oplus n-1} \oplus \mathcal{O}_X^{\oplus a-n}.$$

Then $$H^0(X,\, F)=n(a-1)=n \; \mathrm{rank}(F).$$ Furthermore, the role of $f \colon X \to X$ is irrelevant here.

Remark. It seems to me that an interesting question is the existence of an irreducible locally free sheaf on $X$ with the same property.

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  • $\begingroup$ I just was not sure there exists a finite cover of any given degree, that is why I put that restriction. $\endgroup$
    – user138661
    Commented Apr 24, 2019 at 8:09
  • $\begingroup$ do you think this can be done for smooth proper schemes? The Picard group has to be non-trivial but I am not sure if you have large $H^0$ line bundles. $\endgroup$
    – user138661
    Commented Apr 24, 2019 at 11:59

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