Intereting Posts

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Combinatorial Proof with Summation Identity
Find $\lim _{x\to \infty }\left(x\left(\ln\left(x+1\right)-\ln x\right)\right) $
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Could someone give me a definition of globally generated vector bundle?

A rapid search gives me the definition of globally generated sheaves, but I am in the middle of a long work and don’t really have time to learn all basics of sheaves theory and the connection to vector bundles. I just need a definition that uses the terms that come with the definition of vector bundle, so that I can check if the conditions are satisfied in my cases.

I am sorry if this seems a lazy questions. Of course I appreciate an answer that, in addition, lets me understand the connection.

- Quasicoherent sheaves as smallest abelian category containing locally free sheaves
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- Question about Tate resolution and cohomology groups of nonzero coherent sheaves.
- Hartshorne exercise II.5.12(b)

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- Question about Tate resolution and cohomology groups of nonzero coherent sheaves.
- Is the determinant bundle the pullback of the $\mathcal O(1)$ on $\mathbb P^n$ under the Plücker embedding?
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- Hartshorne exercise II.5.12(b)
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- (Anti-) Holomorphic significance?
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- Quasicoherent sheaves as smallest abelian category containing locally free sheaves
- Why is every holomorphic line bundle over $\mathbb{C}$ trivial?

A holomorphic vector bundle $E$ is *globally generated* if there exist holomorphic sections $s_1, \dots, s_r$ such that for all $x \in X$, $s_1(x), \dots, s_r(x)$ span $E_x$.

In terms of the sheaf-theoretic notion you came across, a holomorphic vector bundle $E$ is globally generated if $\mathcal{O}(E)$, its sheaf of holomorphic sections, is globally generated as a sheaf.

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- Clever use of Pell's equation
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- I Need Help Understanding Quantifier Elimination
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- Exercise $1.8$ of chapter one in Hartshorne.