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Let $X$ be a smooth projective surface (other than the projective plane) and $E$ be a rank $r \geq 2$ globally generated vector bundle over $X$ such that $\text{det}(E) \cong L$. If $H^0(X, E) \neq 0$ and $H^0(X, E\otimes L^*)=0$, then can we determine some specific properties of $E$?

For example can we deduce that $E$ is decomposable?

Any suggestion (including some specific cases e.g. $r=2$ over some specific surface) are welcome.

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Let me deal with the case $r=2$.

Since $H^0(E\otimes L^*)\neq 0$ one gets an inclusion $L\to E$. Taking saturation, one has an exact sequence $0\to L(D)\to E\to I(-D)\to 0$ where $D\geq 0$ is a divisor and $I$ is an ideal sheaf defining finite set of points, possibly empty. Using the fact that $E$ is globally generated, we have $I(-D)$ is globally generated and then one easily checks that $D=0$ and $I=O$. Again, since $E$ is globally generated, at least one section of $E$ must map to a non-zero section of $O$, which gives a splitting, showing $E=L\oplus O$.

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    $\begingroup$ That is a great answer to a question that the OP should have asked. However, the OP actually asked about the case when $E\otimes L^*$ has trivial vector space of global sections. $\endgroup$ Commented May 22, 2023 at 0:40
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    $\begingroup$ @JasonStarr Sorry, I think $H^0(L^8\otimes E)=0$ gives no reasonable answer. $\endgroup$
    – Mohan
    Commented May 22, 2023 at 1:19
  • $\begingroup$ @Mohan, Thank you very much for the answer in the non- zero section case. Can we put some conditions on the surface or the bundle to obtain some other property of the bundle? ( in the zero section case) e.g. second Chern class etc? $\endgroup$
    – Sherlock
    Commented May 22, 2023 at 5:52
  • $\begingroup$ Mohan, I completely agree. $\endgroup$ Commented May 22, 2023 at 10:39

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