# Properties of specific globally generated bundles on a surface

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.

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 $$E=L\oplus O$$.
• 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. Commented May 22, 2023 at 0:40
• @JasonStarr Sorry, I think $H^0(L^8\otimes E)=0$ gives no reasonable answer. Commented May 22, 2023 at 1:19