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I'm stuck with a very basic question in locally free sheaves (vector bundles).

Suppose we have a 5-dimensional vector space $V$ over $k$ (of characteristic zero) and let $\mathcal F=\Omega_{\mathbb{P}(V)}^1(2)$ the twisted cotangent sheaf. Then we know the space of its global sections: $$\Gamma(\mathbb{P}(V),\mathcal{F})=\mathrm{Hom}_k(\bigwedge^2 W,k)$$

I read in a paper that choosing such a global section $h$ there is a "nowhere vanishing section" $\sigma_h : \mathcal{O}_{\mathbb{P}(V)}\to \mathcal{F}$. Here's the questions:

  • Assuming it can be done, how the morphism $\sigma_h$ can be defined?
  • Is this a "general" property in the sense of algebraic geometry? Since every element in $\bigwedge^2 W$ can be represented as a skew-symmetric matrix of odd dimension, $h$ will be necessarily singular and the kernel is 1-dimensional (the Pfaffian). So this property holds outside a line. Is that correct?
  • Why call that morphism a section?

Thank you for the help. I think many of my troubles are due to my algebraically-minded attitude towards these objects, losing much geometric intuition.

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    In general, $H^0(X,F)=Hom(\mathcal{O}_X,F)$ (and more generally for higher cohomology groups, cf. Hartshorne Proposition III.6.3). – 2017-02-01
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    Thank you, that Proposition is everywhere! – 2017-02-01
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    It is NOT true that there is a nowhere vanishing section. A typical section vanishes at one point, but some sections vanish on $\mathbb{P}^2 \subset \mathbb{P}(V)$. – 2017-02-01

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